# Quantum-Classical Hybrid Systems and their Quasifree Transformations

Lars Dammeier<EMAIL_ADDRESS>Reinhard F. Werner
<EMAIL_ADDRESS>Institut für Theoretische Physik, Leibniz
Universität Hannover, Germany

(June, 2023)

###### Abstract

We study continuous variable systems, in which quantum and classical degrees
of freedom are combined and treated on the same footing. Thus all systems,
including the inputs or outputs to a channel, may be quantum-classical
hybrids. This allows a unified treatment of a large variety of quantum
operations involving measurements or dependence on classical parameters. The
basic variables are given by canonical operators with scalar commutators. Some
variables may commute with all others and hence generate a classical
subsystem. We systematically study the class of “quasifree” operations, which
are characterized equivalently either by an intertwining condition for phase-
space translations or by the requirement that, in the Heisenberg picture, Weyl
operators are mapped to multiples of Weyl operators. This includes the well-
known Gaussian operations, evolutions with quadratic Hamiltonians, and “linear
Bosonic channels”, but allows for much more general kinds of noise. For
example, all states are quasifree. We sketch the analysis of quasifree
preparation, measurement, repeated observation, cloning, teleportation, dense
coding, the setup for the classical limit, and some aspects of irreversible
dynamics, together with the precise salient tradeoffs of uncertainty, error,
and disturbance. Although the spaces of observables and states are infinite
dimensional for every non-trivial system that we consider, we treat the
technicalities related to this in a uniform and conclusive way, providing a
calculus that is both easy to use and fully rigorous.

The data defining a quasifree channel are, first, a linear map from the output
phase space to the input phase space, which describes how the Weyl operators
are connected. The second element is a scalar “noise factor”, which is usually
needed to make the channel completely positive. Channels with noise factor 1
are called noiseless. These are homomorphisms in the Heisenberg picture. For
any quasifree channel, the admissible noise functions are in one-to-one
correspondence to states on a certain hybrid system. Since many basic tasks
(e.g., joint measurement, cloning, or teleportation) are encoded in the linear
phase space map, this gives a compact characterization of the possible noises
for channels implementing the task. We establish a general Stinespring-like
decomposition of any quasifree channel into the expansion by an additional
system followed by a noiseless operation. The additional system is itself a
hybrid and in the state characterizing the noise. This allows a clear
distinction between classical and quantum noise of the channel.

Technically, our main contribution is the clarification of the functional
analysis of the spaces of states observables and channels. This required the
resolution of a mismatch in the standard approaches to classical and quantum
systems, respectively, which would have bogged down the theory with many case
distinctions. In the scheme that we propose all hybrid systems and quasifree
operations are treated in a uniform manner. For example, the noise analysis of
dense coding and teleportation become virtually identical. All quasifree
channels can equivalently be considered in the Schrödinger picture or in a
variety of Heisenberg pictures differing by the degree of smoothness demanded
of the observables.

###### Contents

  1. 1 Introduction
    1. 1.1 The basic problem: Good spaces for states and observables
    2. 1.2 Previous works on Canonical Commutation Relations
    3. 1.3 Previous works on hybrid systems
  2. 2 Hybrid states
    1. 2.1 Setup
    2. 2.2 Standard Hilbert space representations and von Neumann algebras
    3. 2.3 Hybrid Uniqueness Theorem
    4. 2.4 Bochner’s Theorem
    5. 2.5 The standard states as a C*-state space
    6. 2.6 Restoring translation symmetry
    7. 2.7 Continuity of state translations
    8. 2.8 $L^{p}$-spaces
    9. 2.9 The squeezed limit
  3. 3 Observables as functions
    1. 3.1 Semicontinuity in C*-algebras
    2. 3.2 Hybrid observables as functions
    3. 3.3 Lévy continuity theorem
    4. 3.4 Tensor product or function space?
    5. 3.5 Eigenvectors of translations
    6. 3.6 Correspondence for spaces of strongly continuous observables
  4. 4 Quasifree channels: Definition and constructions
    1. 4.1 Definition
    2. 4.2 State-channel correspondence
    3. 4.3 Heisenberg pictures for $S$-covariant channels
    4. 4.4 Smoothing channels
    5. 4.5 Squeezing and gauge invariance
    6. 4.6 Composition, concatenation, convolution
    7. 4.7 Noiseless operations
    8. 4.8 Noise factorization and dilations
  5. 5 Basic physical operations
    1. 5.1 States
    2. 5.2 Disturbance
    3. 5.3 Observables
    4. 5.4 Dynamics
    5. 5.5 Classical limit
    6. 5.6 Cloning
    7. 5.7 Instruments
    8. 5.8 Teleportation and dense coding
  6. 6 Summary and Outlook

##  1 Introduction

Canonical variables such as position and momentum can be defined in terms of
their commutation relations. As von Neumann showed [1], for finitely many
degrees of freedom, this uniquely fixes the standard description of a quantum
system in a Hilbert space with explicitly given operators. This approach turns
out to be very useful also for studying operations on such systems. For
example, a symplectic linear transformation of the underlying phase space
gives just another set of operators with the same commutation relations. Hence
by von Neumann’s Theorem, there must be a unitary implementing it. Here we
generalize this approach in two directions: On one hand, we also allow
dissipative operations, which turn some pure states into mixed states, and
which are given by completely positive maps. On the other, we include a
classical part described by operators commuting with all others, allowing for
general quantum-classical hybrid systems. This allows treating problems with a
mixture of classical and quantum information, such as various measurement
scenarios. The defining property of the channels which can be studied in this
way is that they intertwine two actions of the phase space translations or,
equivalently, take Weyl operators into Weyl operators. Such operations have
been called quasifree [2, 3]. However, an analytic treatment in the generality
needed for a practical calculus and including hybrid systems does not seem to
exist. It will be provided in this paper.

While the main aim is to build an easy-to-use general calculus for quasifree
hybrids, we had to go deeper into the functional analysis of such systems than
we had anticipated. The reason is that, although the formal structures for
states and observables for purely quantum systems and for the purely classical
systems (probability) are well established, this cannot be said for the hybrid
combination. Indeed, the standard approaches for the two extreme cases are not
easily merged. These problems are aggravated when discussing operations
(channels) between systems: Should they be described in the Schrödinger or the
Heisenberg picture? On which spaces should these act? Our answer is a setting,
in which both pictures always make sense, for any quasifree channel. So in the
practical calculus of quasifree channels, the technicalities and the
conceptual issues, whose resolution forms the main body of this paper, can be
taken as resolved. When questions of analytic properties of observables and
channels come into play, details can be taken from our paper, but for many
questions, these details can be ignored, and the calculus can be used and
applied straightforwardly and rigorously to a large variety of measuring and
control scenarios. It strictly includes the world of “Gaussian Quantum
Information”. But we no longer need separate definitions of Gaussian states
and Gaussian measurements: Gaussianness is a property of arbitrary channel
types, and it is immediate from our definition that the composition of
Gaussian channels is again Gaussian. However, the quasifree setting is richer,
including, for example, arbitrary channels with no input, i.e., arbitrary
states.

Our paper is organized as follows:
We finish the introductory Sect. 1 with three subsections placing our problem
in context. Logically they can be skipped, i.e., they contain no material
needed to follow the formal development in later chapters. First (Sect. 1.1),
we have some general remarks on how to choose good spaces of states and
observables, how this issue has been viewed traditionally, and why hybrids
pose a special challenge. We then (Sect. 1.2) give a very brief overview of
the rich literature on the canonical commutation relations, mainly for the
historical background and pointers to useful summaries. We do not take
knowledge of this literature as a prerequisite, however. The body of the paper
is mostly self-contained, i.e., we give arguments for the main steps, even
when they could also be covered by a citation. The literature on quantum-
classical hybrids, reviewed in Sect. 1.3, is more disparate, even including
some approaches that fail. We provide a list of pertinent research projects
with brief discussions of some aspects. We also note how these projects relate
to our paper. The three background sections can be summarized by saying that
we did not find any works covering all three aspects.

After introducing the basic phase space variables, Sect. 2 deals with the
“good states”. Using the terminology introduced later, these are the ones with
continuous characteristic functions (Thm. 4), and turn out to be the state
space of a C*-algebra without unit, denoted ${\mathrm{C}}^{*}(\Xi,\sigma)$.
Observables are treated in Sect. 3. This is the most technical part of the
paper. Here we get different choices of spaces of observables. Each is
represented as a class of functions from the classical parameter space to the
bounded operators of the quantum subsystem with varying degrees of regularity,
e.g., weak measurability, strong*-continuity, and norm continuity. The aim is
to show later that these properties are preserved under arbitrary quasifree
channels. A direct approach would be fraught with case distinctions concerning
the separation and recombination of classical and quantum parts under the
channel. Therefore we go another way (Sect. 3.1), namely giving a
characterization of these properties in a general operator algebraic setting,
which applies to classical, quantum, and hybrid systems alike. The
preservation of properties is then almost trivial to show (Prop. 26).

Quasifree channels are defined and characterized in Sect. 4. In order to get a
feeling for the large variety of operations covered by that definition, we
recommend skipping ahead to Sect. 5 for examples. Meanwhile, Sect. 4 focuses
on general constructions. Basic ways to combine them are described in Sect.
4.6. A recurring theme is a correspondence between the quasifree channel and a
state that we call its noise state. The complete positivity condition is
exactly equivalent to the positivity condition for this state, which is a
state on an explicitly given hybrid system. This may have a classical part, or
may be entirely classical, even for channels between purely quantum systems.
Thus the hybrid work needs to be done even if one wants to study only quantum
channels. A key result is the factorization of any quasifree channel into
preparing the noise state on some environment and then executing a noiseless
quasifree operation, for which the noise state is pure and classical.
Noiseless operations (Sect. 4.7) are homomorphisms, so this is a variant of
the Stinespring dilation.

Sect. 5 sketches some of the possibilities of combining classical and quantum
information in input and output. In particular, we parameterize covariant
phase space instruments and their characteristic tradeoff between measurement
accuracy and disturbance. Some aspects, like optimal cloning, the classical
limit, and dynamical semigroup evolutions, are only sketched because they are
treated in past or future articles of their own. In these cases, we merely
indicate how these subjects fit into the framework.

###  1.1 The basic problem: Good spaces for states and observables

The basic statistical interpretation of quantum theory has two primitives,
states and observables, which operationally stand for preparations and
measurements. Mathematically they are represented in appropriate ordered
Banach spaces, and the basic interpretation demands that there is a bilinear
form allowing to evaluate the expectation value of any observable in any given
state. In all the usual theories, however, this symmetric view is broken, and
either the states or the observables are taken as primary and the dual objects
as secondary, derived quantities. This section is about that chicken-and-egg
situation. It is of special interest for hybrids since, in the end, we will
settle for a third option.

The standard view of quantum mechanics has states described as density
operators (positive trace class operators of trace $1$ on a Hilbert space
${\mathcal{H}}$). The maximal set of observables for which expectation value
evaluations can be defined is the set of all bounded linear functionals on the
trace class ${\mathcal{T}}({\mathcal{H}})$, i.e., the Banach space dual, which
in this case is equal to ${\mathcal{B}}({\mathcal{H}})$, the space of bounded
operators on ${\mathcal{H}}$, where the expectation values are expressed by
$\operatorname{tr}(\rho A)$ for $\rho\in{\mathcal{T}}({\mathcal{H}})$ and
$A\in{\mathcal{B}}({\mathcal{H}})$. For classical systems, two different
choices are common. One can either take the algebra ${\mathcal{C}}(X)$ of
continuous complex-valued functions on a compact space $X$ as the observables.
The dual then consists of all finite Borel measures on $X$. Or else, one can
take the space $L^{1}(X,\mu)$ of probability densities with respect to some
reference measure as the space of states, of which the dual is
$L^{\infty}(X,\mu)$. These basic options are visualized in Fig. 1.

statesobservablesClassical${\mathcal{C}}(X)^{*}$$L^{1}(X,\mu)$$L^{\infty}(X,\mu)$${\mathcal{C}}(X)$$\subset$$\subset$statesobservablesQuantum${\mathcal{T}}({\mathcal{H}})$${\mathcal{B}}({\mathcal{H}})$
Figure 1:  Dualities of spaces of states (left column of each diagram) and
observables (right column). A line indicates a dual pairing, i.e., a way to
compute the expectation of any observable in the right space with any state on
the left. The W*-approach (blue line) starts from the states, and allows the
full dual space as observables. The C*-approach (red line) makes the opposite
choice. In the classical case the combination (black line) is also well
defined. Traditional quantum mechanics has only the W*-approach.

In the classical case, the structure of ${\mathcal{C}}(X)$ is determined by
the topology of $X$, whereas the structure of $L^{1}$ is based on the measure
theory of $(X,\mu)$. In the non-classical setting of operator algebraic
quantum theory [4, 5, 6, 7, 8] the distinction appears as the contrast between
the C*-algebraic approach, which generalizes the topological side, and the
W*-algebraic approach, which generalizes the measure-theoretic side. As
indicated by the letters C and W, the distinction is one between the abstract
versions of norm closed *-subalgebras of ${\mathcal{B}}({\mathcal{H}})$ versus
weakly closed ones (also called von Neumann algebras). Then a famous
characterization theorem [9] links this distinction to the chicken-and-egg
problem mentioned in the first paragraph: The W*-algebras are precisely those
C*-algebras that are dual Banach spaces, i.e., coincide with all bounded
linear functionals on some state space.

As the classical case shows, neither view is in any sense more “correct” than
the other: Topology and measure theory just capture different aspects of the
system description. Curiously, one of the founding papers of the algebraic
quantum field theory (AQFT) school [10] tries to make the point that the
C*-algebras should be taken as primary. However, the arguments presented there
can be straightforwardly dualized to give the dual conclusion, namely, that
the states should be taken as primary. Luckily, the AQFT school has largely
ignored the advice of [10], and did substantial work singling out subsets of
“physically relevant states” and separating the local part of the problem from
the global aspects by defining canonical W*-algebras for local regions, while
analyzing the structure at infinity (superselection sectors) in terms of the
representation theory of a quasilocal C*-algebra. So it can be said that the
mature version of AQFT [5] takes a rather refined and complex view of chicken
and egg.

In axiomatic quantum mechanics, many schools have focused on the finite-
dimensional case, where the problem does not arise: In that case, the spaces
are reflexive, i.e., equal to their second duals. One traditional school
allowing infinite dimension from the outset is the approach by Ludwig [11,
12]. According to him, a complete picture of quantum mechanics should be
symmetric, even though his axiomatic reconstruction just returns the
asymmetric standard view shown in Fig. 1. The problem of choosing an
appropriate space of physically realizable observables
(${\mathcal{D}}\subset{\mathcal{B}}({\mathcal{H}})$ in Ludwig’s notation) is
left as an open problem. It is not one to be solved once and for all. A
solution will have to depend on further specifics of the system [13]. This is
analogous to the classical case: All $L^{1}$-spaces are isomorphic (if $X$ has
no “atoms”, i.e., points of positive measure, [14, Thm. III.1.22]), so the
measure theory does not even see the dimension of $X$. Similarly,
${\mathcal{T}}({\mathcal{H}})$ depends only on $\dim{\mathcal{H}}$, but any
reasonable ${\mathcal{D}}$ will contain more structure.

A minimal condition on a space of observables is that there are sufficiently
many observables to distinguish the states. With only this minimal condition,
it is guaranteed that the weak limits of observables span the full dual of the
state space. That is, all choices are weak* dense in each other. One might use
this to justify an asymmetric scenario including all these limits as idealized
observable elements. However, what gets lost in this asymmetric picture is a
description of the physical distinguishability of states. Therefore one loses
the idealized states one would similarly find in the dual of the observable
space. This is, in fact, the basis of the argument in [10], only that it works
both ways (see [15] for a formal parallel development of both aspects from the
Ludwig point of view). This symmetric view can be seen very clearly in the
classical case (Fig. 1): Starting from just the bottom duality of probability
densities and continuous functions, we can approximate arbitrary measures
weakly by densities, including point measures. Dually we can take weak* limits
in $L^{\infty}$ to approximate arbitrary bounded measurable functions by
continuous ones. Again we gain additional extremal elements, like indicator
functions, which are typically not continuous. However, there is no natural
evaluation of a point probability measure on an indicator function in
$L^{\infty}$: The elements of $L^{\infty}$ are classes with respect to almost
everywhere equality, and so the value at a point (typically of measure zero)
is not defined. Thus there is no natural pairing at the top level of the
diagram of dualities (see, however, [16] for the existence of some non-
constructive pairing). In the non-commutative case, the story is similar: By
going to the full dual spaces, one gains idealized elements in the form of
pure states and projections, whose abundance is guaranteed by the
Banach–Alaoglu and Krein–Milman Theorems. These extremal elements are often
the building blocks of an analysis, as in the decomposition of arbitrary
states as an integral over pure states (Choquet Theory, [4, Ch. 4]) and,
dually, in the spectral resolution of normal observables.

The explanation for the existence of pure states does not account for the many
pure states ${|\psi\rangle\\!\langle\psi|}\in{\mathcal{T}}({\mathcal{H}})$,
i.e., standard quantum mechanics, which we characterized as an instance of the
W*-view. Indeed for W*-algebras of Murray–von Neumann type II and III, there
are no normal pure states (see also Lem. 10). However, the pure states can be
related to another duality: The trace class ${\mathcal{T}}({\mathcal{H}})$ is
the Banach space dual of ${\mathcal{K}}({\mathcal{H}})$, the space of compact
operators. The unit ball of ${\mathcal{T}}({\mathcal{H}})$, therefore, has to
have many extreme points. It would be impractical, however, to use the compact
operators as the basic observable algebra: It does not contain a unit, so we
cannot express the normalization condition for POVM observables.

That
${\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}\notin{\mathcal{K}}({\mathcal{H}})$
means that the normalization functional is not continuous in the weak*
topology, so the states are not a weak*-closed subset of the unit ball and not
compact. This is a crucial observation for limits in this topology. For
example, consider a sequence of states averaged over an expanding range of
spatial translations. Any cluster point would have to be translation
invariant; but no such density operator exists, because translations have
continuous spectrum. On the other hand, by the weak* compactness of the unit
ball, cluster points must exist. Indeed the weak* limit is the zero
functional. So it does happen that the limit of states fails to be a state. In
close analogy, we can consider classical states represented by probability
measures on ${\mathbb{R}}^{n}$ and their shifts or averages to infinity. These
are in the dual of ${\mathcal{C}}_{0}({\mathbb{R}}^{n})$, the space of
continuous functions vanishing at $\infty$, and weak limits of shifted
sequences are zero.

So in both cases, the natural state space can be located in the dual of an
algebra ${\mathcal{A}}$ without identity, namely
${\mathcal{A}}={\mathcal{K}}({\mathcal{H}})$ in the quantum case, and
${\mathcal{A}}={\mathcal{C}}_{0}({\mathbb{R}}^{n})$ in the classical case. But
this does not mean going back to an asymmetric picture with observables taken
as primary, i.e., another turn in the chicken-and-egg conundrum because
${\mathcal{A}}$ is not itself the algebra of observables. The full set of
observables will be some set of functionals on the states in
${\mathcal{A}}^{*}$, so a subspace of ${\mathcal{A}}^{**}$. All such
observables can be approximated weakly by elements of ${\mathcal{A}}$. In
particular, there is an approximate unit, so
${\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$ is recovered. This scheme is
outlined in Fig. 2. We propose to use it for hybrids, as well, and with the
particular choice
${\mathcal{A}}={\mathcal{K}}({\mathcal{H}})\otimes{\mathcal{C}}_{0}(X)$ as the
C*-tensor product of quantum and classical parts.

Classicalstatesobservables${\mathcal{C}}_{0}(X)^{*}$${\mathcal{C}}_{0}(X)^{**}$${\mathcal{C}}_{0}(X)$Quantumstatesobservables${\mathcal{T}}({\mathcal{H}})$${\mathcal{B}}({\mathcal{H}})$${\mathcal{K}}({\mathcal{H}})$Hybridstatesobservables${\mathcal{A}}^{*}$${\mathcal{A}}^{**}$${\mathcal{A}}={\mathcal{K}}({\mathcal{H}})\otimes{\mathcal{C}}_{0}(X)$
Figure 2: Extended dualities suitable for a joint generalization of the
classical and the quantum case to hybrids. The states are here functionals on
an underlying non-unital algebra ${\mathcal{A}}$, namely $\mathcal{C}_{0}(X)$
resp. ${\mathcal{K}}({\mathcal{H}})$ in the classical resp. quantum case. The
biduals ${\mathcal{C}}_{0}(X)^{**}$ and, in general, ${\mathcal{A}}^{**}$ are
relatively wild objects, whose elements are not easily characterized.
Selecting more managable subspaces is the subject of Sect. 3.

It turns out that this is equivalent to another way of identifying a good
hybrid state space for systems of canonical variables. One takes, for each
state, the expectation values of the Weyl operators (displacement operators).
This is called the characteristic function of the state. Then the good states
in ${\mathcal{A}}^{*}$ are precisely (Thm. 4 and 5) those with a continuous
characteristic function (usually called “regular”). Intuitively, these states
live essentially in bounded regions of phase space, so the expectation of Weyl
operators can be expected to depend continuously on its argument. This is in
contrast to the observation that distinct Weyl operators have maximal norm
distance because somewhere far out, even periodic functions with very similar
periods will differ maximally. This regularity condition has an equally simple
analog for quasifree channels, which is equivalent to the channel mapping
regular states to regular states. We include this in our definition of
quasifree channels. On the side of observables, things are more complex,
however. On the maximal choice of observable algebra, the bidual
${\mathcal{A}}^{**}$ the Heisenberg picture of a channel is defined as the
Banach space adjoint. However, the bidual is a monster, even in the classical
case. We devote Sect. 3 to the task of finding smaller, more manageable
subalgebras, which nevertheless support a Heisenberg picture for arbitrary
quasifree channels. That there is some choice here is a strength of our theory
rather than a weakness: It amounts to a variety of analytic conditions (weak
measurability, strong*-continuity, norm continuity, and in some sense many
more), which are all automatically preserved by the Heisenberg actions of
quasifree channels.

Let us contrast this with an approach that naively takes inspiration from the
finite case, where $\dim{\mathcal{H}}<\infty$, and the classical variables lie
in a compact set $X$, and the appropriate observable algebra is the C*-tensor
product ${\mathcal{A}}={\mathcal{B}}({\mathcal{H}})\otimes{\mathcal{C}}(X)$.
When $X\cong{\mathbb{R}}^{s}$ locally compact and $\dim{\mathcal{H}}=\infty$,
we could just replace ${\mathcal{C}}(X)$ by the bounded continuous functions
${{\mathcal{C}}_{\mathrm{b}}}(X)$, and consider the C*-algebra
$\widetilde{\mathcal{A}}={\mathcal{B}}({\mathcal{H}})\otimes{{\mathcal{C}}_{\mathrm{b}}}(X)$.
This algebra contains the Weyl operators as well, so we can associate with
every state on $\widetilde{\mathcal{A}}$ a characteristic function. In fact,
we started our investigation of quasifree channels with this choice, running
into more and more difficulties: Since the Weyl operators do not span a norm
dense subalgebra of $\widetilde{\mathcal{A}}$, the characteristic function
will fail to characterize a state. Moreover, many states on
$\widetilde{\mathcal{A}}$ are singular in the following sense: The classical
marginal will be a state on ${{\mathcal{C}}_{\mathrm{b}}}(X)$, which is
isomorphic to ${\mathcal{C}}(\beta X)$, where the $\beta X$ denotes the
Stone–Čech compactification of $X$. But if such a state has any weight on the
infinite points $\beta X\setminus X$, it will not be a probability measure on
$X$. Similarly, on the quantum side, a state on ${\mathcal{B}}({\mathcal{H}})$
may fail to be given by a density operator. For a usable calculus of quasifree
channels, this is bad news because it would be unclear how to even define such
channels on all of $\widetilde{\mathcal{A}}$, how to exclude that the output
of any such operation is singular, and even if all that is worked out, whether
the Heisenberg picture would map into the input counterpart of
$\widetilde{\mathcal{A}}$.

###  1.2 Previous works on Canonical Commutation Relations

The description of physical systems in terms of canonical commutation
relations constitutes a core part of modern mathematical physics. The
tradition begins with Weyl and von Neumann and extends to the community that
founded the journal Communications in Mathematical Physics in 1965. It was
felt then that this structure, and more generally a C*-algebraic view of
physics, was a key element of both quantum field theory in the approach of
Haag, Kastler [17, 10], and Araki [18], and of statistical mechanics in the
school of Verbeure and others (see [4, 19] for a textbook expounding these
ideas). We are obviously building on this tradition and can hardly give due
credit, not even to the major contributors.

Notable expositions are [19] for a Fock space based view with statistical
mechanics in mind and [20] for a field theoretical one. An encyclopedic work
aiming at quantum optics is [21]. Irreversible operations, i.e., quasifree
channels as developed in Sect. 4, came up in [3, 22]. These are, of course,
also the focus of works on quantum information. In that community, the
canonical systems are called systems with “continuous variables” as opposed to
those composed of discrete quantum bits. The main interest has been in the
Gaussian case because the vacuum at the empty port of a beam splitter, laser
light, and cooled oscillators are all Gaussian. A collection of review
articles is [23]. Systematic expositions of the Gaussian structure are [24,
Ch. 12] and [25].

Hybrids did show up occasionally in this literature, but usually not as the
main focus. For example, some core results in [21] are formulated without
assuming the commutation form to be non-degenerate (see also [26]). This
assumes the observable algebra on the classical side to be CCR-like, i.e., the
almost periodic functions, which is not a good choice by the considerations of
the previous section, but by far the most frequent choice in the literature
described here.

###  1.3 Previous works on hybrid systems

In the following list, we have collected some of the appearances of quantum-
classical hybrids in the literature. The motivations are rather different, and
after each brief description, we point out how the respective research project
differs from the present study. l

In quantum field theory, the algebra of canonical commutation relations
provided a way to deal with the commutation relations of field operators in
QFT without discussing tricky domain questions of these unbounded operators.
In this way, quantum fields could be included in the newly forming
C*-algebraic approach to quantum mechanics, which is crucially based on
bounded operators. However, these works were not interested in going beyond
the CCR-algebra, i.e., the C*-algebra generated by the Weyl operators. The
technical difficulties coming from an infinite-dimensional test function space
(phase space) and the resulting failure of von Neumann’s uniqueness theorem
(closely related to “Haag’s Theorem”) seemed more relevant.

It was noted only later that even for field theory, the CCR-algebra has its
drawbacks. Classically it corresponds [27] to the almost periodic functions,
which by definition depend very sensitively on infinite values of the fields.
This is reflected by the structure of the pure states on the algebra of almost
periodic functions, which form the so-called Bohr compactification of
${\mathbb{R}}$. Along with the points in ${\mathbb{R}}$ it contains further
limit points. But these new points are in no sense “at infinity” and
themselves dense, so that the finite and the infinite are highly intertwined.
Of course, this is related to the observation that almost periodic classical
variables are almost impossible to measure, so the topology of the Bohr
compactification is unrelated to physical distinguishability. The same
criticism applies to the use of the CCR-algebra. In particular, it is
difficult to implement physical dynamics as a C*-dynamical system on this
algebra [19, p. 345]. A more regular approach using resolvents rather than
exponentials of the fields has therefore been proposed [28] (cf. Example 28).

In contrast to the QFT literature, we consider only finite-dimensional phase
spaces in this paper. In other aspects, our approach is more general,
particularly by including irreversible operations. QFT focuses on reversible
dynamics, so irreversible operations play no role (but see [29]).

Both classical and quantum systems employ Hamiltonians to generate the
dynamics, and it appears almost like a minor difference that one uses
commutators while the other uses Poisson brackets. Consequently, there have
been many attempts to fuse these structures into one common framework (see
[30] for a review). This gains further plausibility from systems with
quadratic Hamiltonians, which generate isomorphic Lie algebras in the quantum
and classical worlds. On the whole, however, this basic idea has proved to be
a failure [31, 32]. Typical problems include dynamics, which do not preserve
the positivity of states or allow non-local signaling due to some obscure non-
linearity imported into the quantum system from the classical side. The core
of these problems is actually a very familiar No-Go Theorem from quantum
information theory: There is no information gain without disturbance [33].
That is, whenever an interaction leads to the possibility of measuring a
variable of the classical subsystem and thereby gaining information about the
initial quantum state, some irreversibility must be involved. Thinking of
dynamics in terms of Hamiltonians and canonical structure is, however, so tied
up with reversible dynamics that any approach based on canonical structures is
bound to fail, at the latest, when there is a non-trivial interaction.

For our paper, this has the consequence that we do not even assume a
symplectic structure on the classical system, i.e., the classical phase space
is a real vector space without further structure.

The No-Go Theorem strongly suggests the use of dissipative time evolutions to
express the measurement interaction [34, 35, 36]. The quasifree case [35]
benefits especially from the clarification of the complete-positivity
conditions for channels ([22], our Sect. 4).

In [37], we show that, even without quasifreeness, this leads to a fusion of
the classical theory of diffusion generators on one hand and Lindblad
generators on the other, with a full understanding of the additional
interaction terms that describe the information transfer from the quantum to
the classical subsystem.

In the quantum information community, many researchers think of the
observables of a classical system as the diagonal matrices embedded into a
larger full matrix algebra. Similarly, for a classical particle described by
position variables in ${\mathbb{R}}^{n}$, one can get a quantum extension by
including the generators of the spatial shifts, i.e., conjugate momenta, in a
crossed-product [14] construction. This construction can be done at the von
Neumann algebra level so that the enlarged quantum system has the full algebra
of bounded operators over $L^{2}({\mathbb{R}}^{n},dx)$ as observables. This is
the approach to hybrids chosen, for example, in [35]. In this setting, the
distribution of the classical variables in a normal state always has an
$L^{1}$-density, which excludes the pure states of a hybrid. We will see later
that the pure states of a modified hybrid also correspond to extremal quantum
channels, so this approach excludes the optimal, e.g., minimal noise channels
for some tasks.

In our approach, pure states are included from the outset, and the von Neumann
algebraic crossed-product embedding is characterized as a special case for
which states are norm continuous under translations (Sect. 2.7).

A common point of view is that the classical world is merely emergent as a
limiting case of large quantum systems. In this spirit, a hybrid would always
be a large quantum system with one subsystem close to a thermodynamic limit.
There is no problem then writing down Hamiltonian interactions between the
almost classical and the quantum part. However, this does not resolve the No-
Go Theorem for hybrid dynamics. The classical variables in such a system will
generally evolve into some combination involving their conjugates, or as [38]
phrases this, the classical variables lose their “integrity”. The required
physical discussion at this point would be that effectively, and to good
approximation, the classical integrity is preserved. But often, the models of
quantized classical systems are so simple (e.g., one degree of freedom [39])
that a physical discussion of a thermodynamic limit is not really possible. It
should be noted that the classical limit is very closely related to the mean-
field limit, and the latter has indeed been proposed as a model for
measurement processes involving large quantum systems [40]. In this case, in
spite of an infinite range mean field interaction, the measurement result
becomes definite only in the infinite time limit.

The many-body aspect of the classical system will not come into play in our
paper or even enter the formalism. Conceptually, this is because we consider
that limit already being done, and we work with a much-reduced set of
classical variables, a finite set of reals, such as a measurement record in a
continuing observation process.

In the mean-field limit, one gets dynamical equations for quantum states,
which are of canonical Hamiltonian form and allow strong non-linearities
forbidden in standard quantum theory. This is not paradoxical if one realizes
that the “quantum states” here are not states of a quantum system at all but
distribution parameters for a many-body system. Nevertheless, the resulting
kind of non-linear Hamiltonian evolution [41, 42] has been proposed as a
testable generalization of quantum mechanics [43, 44]. In the simplest case,
i.e., a qubit, it is classical mechanics with the surface of the Bloch sphere
as a phase space manifold and the surface $2$-form as symplectic form. The
standard quantum evolutions are driven by Hamiltonians which are linear in the
state. These are just rotations of the Bloch sphere and result from non-
interacting mean-field systems. The symplectic structure on the pure states
has been variously noted, but it is quite misleading to conclude that using it
somehow unifies classical and quantum theory. Indeed the classical Hamiltonian
structure goes against a basic impossibility claim of standard quantum
mechanics, namely that all mixtures giving the same density operator (e.g.,
unpolarized light) are indistinguishable. To summarize: As an approach to
hybrids, the theories starting from the Hamiltonian structure of quantum
states resolve the tension between classical and quantum theories by turning
the quantum part into a classical system.

In this paper, we stick to the “minimal statistical interpretation” for
quantum, classical, and hybrid systems alike. The states and observables then
operationally represent preparations and measurements. This interpretation
implies linearity of all meaningful operations on states and observables, so
linearity is not an accidental feature of the theory that can easily be
dropped, and we will assume it throughout.

A recent discussion of hybrids for quantum fields coupled to gravity
illustrates several of the options mentioned above. In [45, 46] we find an
approach making the dissipative nature of the interaction implicit. In [47,
48] it is argued that gravitationally induced entanglement would serve as
proof of the non-classical nature of gravity. This is contradicted by [49],
where the authors emphasize that this will depend on the notion of hybrids and
that the non-linear variant, in particular, would allow for entanglement via a
classical intermediary.

Our motivation for hybrid structures is practical and comes from continuous
observation and other measurement processes. Whether the resulting structures
are also helpful for some fundamental theory is far beyond the scope of this
paper. However, we hope that a sharper understanding of the mathematical
structures will also be helpful in such projects.

##  2 Hybrid states

###  2.1 Setup

In this section, we fix the basic structure of the systems we consider and the
basic notations relating to phase spaces. For those who are familiar with
phase space quantum mechanics, this amounts to applying a remarkably simple
principle, namely just eliminating the assumption that the commutation form
should be symplectic, i.e., non-degenerate.

We consider systems of $n$ quantum canonical degrees of freedom and $s$
classical ones. This means that we have a position variable
$q\in{\mathbb{R}}^{n}$, and its momentum counterpart, which lies in the dual
space ${\mathbb{R}}^{n}$. This only means that a scalar product $q{\cdot}p$ is
defined, and the phase space of the system, the set of pairs $(q,p)$ carries a
natural symplectic form
$\sigma((q,p),(q^{\prime},p^{\prime}))=q{\cdot}p^{\prime}-p{\cdot}q^{\prime}$.
More abstractly one needs only to demand that $\sigma$ is antisymmetric and
non-degenerate, i.e., the only pair $(q,p)$ such that
$\sigma((q,p),(q^{\prime},p^{\prime}))=0$ for all $(q^{\prime},p^{\prime})$ is
$p=q=0$. Then with a suitable choice of “canonical coordinates” $\sigma$ will
take the given form.

We now drop the assumption of non-degeneracy, i.e., we allow non-zero null
vectors for $\sigma$. In a basis this means that the $2n$ variables $p,q$ can
be augmented by $0\leq s<\infty$ unpaired classical variables
$x\in{\mathbb{R}}^{s}$, which can be thought of as position variables without
corresponding momenta. So, as an extended phase space
$\Xi={\mathbb{R}}^{2n+s}$ we consider the set of triples $\xi=(q,p,x)$. The
extended symplectic form will be defined

$\sigma((q,p,x),(q^{\prime},p^{\prime},x^{\prime}))=q{\cdot}p^{\prime}-p{\cdot}q^{\prime}=\sum_{ij}\xi_{i}\sigma_{ij}\xi^{\prime}_{j},$
(1)

which is still antisymmetric and bilinear. When we want to emphasize the
generalization, we call $\Xi$ a “hybrid phase space”. But since this is the
normal case in our paper, we will often drop the adjective. Since we later
consider arbitrary linear maps on phase spaces, we usually adopt a convenient
basis free view, where the (hybrid) phase space is just a real vector space
$\Xi$ with antisymmetric form $\sigma$, so the type of system is given by the
pair $(\Xi,\sigma)$. The classical part is always singled out as the space of
null vectors:

$\Xi_{0}=\\{\xi\in\Xi\mid\forall\eta:\,\sigma(\xi,\eta)=0\\}.$ (2)

Thus we can split $\Xi=\Xi_{1}\oplus\Xi_{0}$, where $\Xi_{1}$ is a suitable
subspace on which $\sigma$ is non-degenerate, i.e., a standard quantum system.
The direct sum symbol here indicates a unique decomposition
$\xi=\xi_{1}+\xi_{0}$ with $\xi_{i}\in\Xi_{i}$ for any vector $\xi$, and that
the form $\sigma$ also has a block structure, as in the coordinatization (1).
However, other than an orthogonal complement, the quantum part $\Xi_{1}$ is
not uniquely defined, i.e., there are $\sigma$-preserving linear maps changing
the decomposition. Some of our constructions depend on the decomposition
$\Xi=\Xi_{1}\oplus\Xi_{0}$, but we usually do not show explicitly that this
dependence is harmless. In fact, such proofs become trivial exercises once our
full theory is established. The necessary isomorphisms will be noiseless
quasifree in the terminology of Sect. 4.

Going quantum means that the components of these tuples are turned into
operators

$R=(R_{1},\ldots,R_{2n+s})=(Q_{1},\ldots,Q_{n},P_{1},\ldots,P_{n},X_{1},\ldots,X_{s})$
(3)

with the commutation relations

$[R_{j},R_{k}]=i\sigma_{jk}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}.$ (4)

Again, we refer to the classical $X_{j}$ as “operators” out of convenience,
although classical “random variables” might perhaps be more appropriate. These
are the $R_{j}$ that commute with all others.

A basic symmetry of the theory are the phase space translations, which add a
constant, i.e., a multiple of the identity to each $R_{j}$. We denote this
transformation by

$\alpha_{\xi}(R_{j})=R_{j}+\xi_{j}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}},$
(5)

where $\xi_{j}$ are the components of $\xi$. Clearly, $\alpha_{\xi}$ preserves
the commutation relations and will always be a homomorphism (preserve operator
products).

There are many subtleties in the task of finding all operators satisfying (4),
related to domain questions of these unbounded operators [50]. The main
regularity condition singling out the usual case is that the operators are
essentially selfadjoint on their common domain so that they generate unitary
groups. These should satisfy an integrated version of (4), and rather than
diving into the details, we will make that our starting point. In fact, it can
be argued [51, Sect.4.1] that the integrated version (7) is historically a bit
older, and due to Weyl, who proposed it to Max Born, even before the latter
published (4). Hence, following a strong tradition, we pass to the operators

$W(\xi)=\exp(i\xi{\cdot}R).$ (6)

We refer to the $W(\xi)$ as Weyl operators, even though with classical
arguments, they may sometimes be more like functions. In (6) the expression
$\xi{\cdot}R:=\sum_{j}\xi_{j}R_{j}$ means a mixed vector/operator scalar
product, by which the commutation relations become

$\displaystyle W(\xi)W(\eta)$
$\displaystyle=e^{(-i/2)\xi{\cdot}\sigma\eta}W(\xi+\eta)$ (7)
$\displaystyle=e^{-i\xi{\cdot}\sigma\eta}W(\eta)W(\xi).$ (8)

We will refer to (7) as the Weyl relation, while (8) are called the canonical
commutation relations (CCR) in Weyl form. The CCR-algebra over $(\Xi,\sigma)$,
denoted by ${\mathrm{CCR}}(\Xi,\sigma)$, is the universal C*-algebra of these
generators and relations. That is, every realization of the relations by
unitary operators $W(\xi)$ on a Hilbert space ${\mathcal{H}}$ is given by a
representation of ${\mathrm{CCR}}(\Xi,\sigma)$ which takes the abstract
generators to the $W(\xi)$.

There are some notational choices here that we should comment on. We have not
included $\sigma$ in (6), simply because this would set the classical
contribution to zero. In phase spaces with a proper symplectic form, this form
is often used to identify the space with its dual (e.g., [27]). Any
constructions using this will not work in our context. This means that in a
coordinate-free spirit, the variable $\xi$ in (6) does not lie in the phase
space $\Xi$ but in its dual $\widehat{\Xi}$. We will keep the notation simple
by nevertheless identifying both spaces with ${\mathbb{R}}^{2n+s}$ and using a
dot for the standard scalar product.

This convention will suffice for almost all of the paper, that is, unless we
explicitly distinguish some components as position-like and others as
momentum-like. In those rare cases, mainly the instrument in Sect. 5.7, we try
to help readers keeping track by using corresponding letters: For the phase
space $\Xi$, and therefore also for the arguments of $\alpha_{\xi}$ we already
introduced in the ordering $(q,p,x)$ for the groups of $n+n+s$ variables. For
the dual space $\widehat{\Xi}$, i.e., in the arguments of Weyl operators and
characteristic functions it is then suggestive to use the ordering
$(\hat{p},\hat{q},k)$. Here we take into account that position space and
momentum space are dual vector spaces and $k$ is the wave-number variable dual
to classical shifts, as customary in $e^{ik{\cdot}x}$. In order to keep all
appearances of the symplectic matrix explicit, we do not change one of the
signs for elements in $\widehat{\Xi}$.

###  2.2 Standard Hilbert space representations and von Neumann algebras

In this section we investigate the kind of hybrid theory suggested by the
Hilbert space tensor product of a classical and a quantum subsystem.

In the quantum case, there is no choice: von Neumann showed [1] that
Schrödinger’s operators $P$ and $Q$ are the only solution of the commutation
relations. As taught in every course on quantum mechanics, these live in the
Hilbert space ${\mathcal{H}}_{1}=L^{2}({\mathbb{R}}^{n},dq)$, with $Q_{i}$
acting by multiplication with the $i^{\rm th}$ coordinate, and $P_{i}$ by
differentiating with respect to it (and a factor $i$). Equivalently, the Weyl
operators are given by

$\bigl{(}W_{1}(a,b)\psi\bigr{)}(r)=e^{\frac{ia{\cdot}b}{2}+ia{\cdot}r}\psi(r+b),\quad\mbox{for
}(a,b)\in\Xi_{1},\ \psi\in{\mathcal{H}}_{1}=L^{2}({\mathbb{R}}^{n},dr).$ (9)

In contrast, there is no such uniqueness for the classical case, basically
because there are uncountably many inequivalent irreducible representations of
the classical observable algebra (labelled by the points of $\Xi_{0}$). This
non-uniqueness forces the choice of a measure $\mu$ on the classical subspace
$\Xi_{0}$, so that the classical algebra is represented as the multiplication
operators in $L^{2}(\Xi_{0},\mu)$. A hybrid system can thus be set up in the
tensor product, as defined in the following Def. 1. This kind of
representation is the analog of the Schrödinger representation: An explicit
choice of Weyl operators satisfying the relations (7), initially without the
claim that all good representations look like that. Indeed, it will be the
next step to establish that claim, and hence the hybrid analog of von
Neumann’s result (Thm. 3 below).

######  Definition 1.

Let $\Xi=\Xi_{1}\oplus\Xi_{0}$ be a hybrid phase space with antisymmetric form
$\sigma=\sigma_{1}\oplus 0$. Then a standard representation is a
representation of the Weyl relations in the Hilbert space
${\mathcal{H}}={\mathcal{H}}_{1}\otimes L^{2}(\Xi_{0},\mu)$, where $\mu$ is
some regular Borel measure on $\Xi_{0}$, and ${\mathcal{H}}_{1}$ is the
Hilbert space of the Schrödinger representation
$W_{1}:\Xi_{1}\to{\mathcal{B}}({\mathcal{H}}_{1})$ for $(\Xi_{1},\sigma_{1})$.
The Weyl operators are given by

$W(\xi_{1}\oplus\xi_{0})=W_{1}(\xi_{1})\otimes W_{0}(\xi_{0}),$ (10)

where $W_{0}(\xi_{0})$ is the multiplication operator

$\bigl{(}W_{0}(\xi_{0})\phi\bigr{)}(x)=e^{i\xi_{0}{\cdot}x}\phi(x)$ (11)

for $\phi\in L^{2}(\Xi_{0},\mu)$ and $x\in\Xi_{0}$. A state on the CCR-algebra
is called standard if it is given by a density operator on ${\mathcal{H}}$ in
a standard representation.

We remark that the standard representation depends on $\mu$ only up to
equivalence. That is, when two measures $\mu$ and $\mu^{\prime}$ have the same
null sets, the Hilbert spaces $L^{2}(\Xi_{0},\mu)$ and
$L^{2}(\Xi_{0},\mu^{\prime})$ are the same by a unitary transformation that
acts by multiplication (with $\sqrt{d\mu/d\mu^{\prime}}$) and, in particular,
intertwines the multiplication operators (11). We can, therefore, always
choose $\mu$ to be a probability measure, typically the classical marginal of
a state under consideration. Note that the translate of a standard state is
again standard, but generally not in the same representation, unless $\mu$ is
quasi-invariant (equivalent to its translates). This is the case just for the
Lebesgue measure and will be discussed further in Sect. 2.7.

The von Neumann algebra generated by a standard representation is

${\mathcal{M}}_{\mu}={\mathcal{B}}({\mathcal{H}}_{1}){\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}L^{\infty}(\Xi_{0},\mu),$
(12)

where ${\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}$ denotes the tensor product
of von Neumann algebras. Indeed, since the $W_{1}(\xi_{1})$ are irreducible on
${\mathcal{H}}_{1}$, they generate ${\mathcal{B}}({\mathcal{H}}_{1})$ as a von
Neumann algebra, and similarly the Weyl multiplication operators generate the
maximal abelian algebra of all multiplication operators $M_{f}$ with $f\in
L^{\infty}(\Xi_{0},\mu)$, which is isomorphic to $L^{\infty}(\Xi_{0},\mu)$.
Putting this together, and using the commutation theorem for tensor products
[14, Thm. IV.5.9] gives (12). Note that this algebra still depends on $\mu$
because in $L^{\infty}(\Xi_{0},\mu)$ functions, which only agree $\mu$-almost
everywhere, are identified. By identifying $A\otimes f$ with the function
$x\mapsto f(x)A$ we can think of the elements of ${\mathcal{M}}_{\mu}$ as
measurable ${\mathcal{B}}({\mathcal{H}}_{1})$-valued functions on $\Xi_{0}$.
We will later strive to get rid of the $\mu$-dependence in the definition of
observable algebras, as is motivated at the beginning of Sect. 3.

Standard states are thus normal states on some ${\mathcal{M}}_{\mu}$, hence
elements of the predual ${\mathcal{T}}^{1}({\mathcal{H}}_{1})\otimes
L^{1}(\Xi_{0},\mu)$, where ${\mathcal{T}}^{1}({\mathcal{H}}_{1})$ denotes the
trace class. They can hence be decomposed as

$\omega(A\otimes f)=\langle\omega,A\otimes f\rangle=\int\mu(dx)c(x)\
f(x)\operatorname{tr}(\rho_{x}A),$ (13)

where $c\mu$ is the probability measure determining the classical marginal,
i.e., the expectations of multiplication operators, and $x\mapsto\rho_{x}$ is
a measurable family of density operators. The factor $c(x)$ is introduced to
allow that $\operatorname{tr}\rho_{x}=1$ for all $x$. When we consider a
particular state and its GNS-representation, we usually take $\mu$ directly as
the classical marginal of that state, i.e., set $c(x)\equiv 1$. The required
measurability conditions for the family of states $\rho_{x}$ are spelled out
in [14, Sect. IV.7].

The definition of standard states brings in a dependence on $\mu$, so that it
is not a priori clear that convex combinations of standard states are
standard. However, the integral decomposition (13) makes clear that for a
countable convex combination $\rho=\sum_{j}\lambda_{j}\rho_{j}$ we can take
$\mu=\sum_{j}\lambda_{j}c_{j}\mu_{j}$, and, set $h_{j}$ to be the
Radon–Nikodym derivative of $\lambda_{j}c_{j}\mu_{j}$ with respect to $\mu$.
Note that $0\leq h_{j}(x)\leq 1$, and $\sum_{j}h_{j}=1$. Then $\rho_{0}=1$ and
$\rho_{x}=\sum_{j}h_{j}\rho_{x,j}$. In particular, a normal state in a direct
sum of standard representations can be rewritten as a state using just a
single summand, i.e., is also standard in the sense of the above definition.

This argument also shows that the von Neumann algebra approach to hybrids can
be made to work on larger and larger sets of states: If needed, one can
consider any countable (and thereby any norm separable) family of states as
absolutely continuous with respect to a common reference measure. However, the
set of measures on $\Xi_{0}$ is not norm separable, so there is no single
standard representation which can be used for all practical purposes. One
could represent a single observable $F$ by a net of functions
$F_{\mu}\in{\mathcal{B}}({\mathcal{H}})\otimes L^{\infty}(\Xi_{0},\mu)$, each
defined up to $\mu$-a.e. equality. Then indices are ordered by absolute
continuity $\mu\ll\nu$, i.e., $\nu$ has fewer null sets than $\mu$, and in
this case, $F_{\nu}$ is more sharply defined than $F_{\mu}$. There is no
natural limit to such nets because we cannot include all the uncountably many
point measures. However, the notion of universally measurable sets and
functions (see Sect. 3) does allow us to get rid of the Lebesgue completions.

Since standard states thus form a convex set, it makes sense to ask for the
extreme points, i.e., the pure states. These are readily characterized:

######  Lemma 2.

A standard state $\omega$ on the CCR-algebra is extremal iff there is a point
$x\in\Xi_{0}$ and a unit vector $\phi\in{\mathcal{H}}_{1}$ such that in the
decomposition (13) $\mu=\delta_{x}$ is a point measure and
$\rho_{x}={|\phi\rangle\\!\langle\phi|}$.

###### Proof.

Suppose that $\omega$ is extremal. Then let $f\in L^{\infty}(\Xi_{0})$ with
$\varepsilon<f<{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}-\varepsilon$ for
some $\varepsilon>0$. $\omega$ is then decomposed into the sum of two positive
functionals

$\omega(X)=\omega(f)\,\frac{\omega(fX)}{\omega(f)}+\omega(1-f)\,\frac{\omega((1-f)X)}{\omega(1-f)}.$
(14)

This is a convex combination of states, so by extremality, the two states have
to be proportional, i.e., $\omega(fX)=\lambda\omega(X)$ for all $X$. This
forces $\lambda=\omega(f)$, by putting
$X={\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$, and hence we conclude that
$f=\omega(f){\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$ almost everywhere
with respect to $\mu$. Hence, $\mu$ is a point measure at some point $x$, say.
The choice of $\rho_{y}$ for $y\neq x$ is irrelevant because the whole
complement of $\\{x\\}$ has measure zero. The state $\rho_{x}$ is now given by
a density operator, which clearly has to be extremal as well, so
$\rho_{x}={|\phi\rangle\\!\langle\phi|}$. ∎

Note that a state $\omega$ may have no extremal components, i.e., no extreme
points $\omega^{\prime}$ such that $\omega\geq\lambda\omega^{\prime}$ with
$\lambda>0$. Indeed this will be the case whenever the measure $\mu$ has no
atoms (points of non-zero measure). It is therefore not a priori clear in
which sense standard states can be decomposed into extreme points. This will
be clarified in Sect. 2.5, where it will be seen that the standard states are
the state space of a certain C*-algebra, so the convex combinations of extreme
points are dense in a suitable weak* topology.

###  2.3 Hybrid Uniqueness Theorem

It is straightforward to check that in the standard representation,
$\xi\mapsto W(\xi)$ is continuous with respect to the strong operator
topology. It turns out that this characterizes standard representations. This
is the main content of the following theorem, which is very close in its
formulation and its proof to von Neumann’s famous result [1].

######  Theorem 3 (Hybrid Uniqueness Theorem).

Every representation of the Weyl relations on a Hilbert space, for which the
mapping $\xi\mapsto W(\xi)$ is continuous in the strong operator topology, is
unitarily equivalent to a direct sum of standard representations.

In the literature, it is traditional [52] to use a weaker continuity
condition, which does not demand joint continuity of $W$ in all $2n+s$
variables in $\xi\in\Xi$, but only along one-dimensional subspaces. This is
the minimum required to get self-adjoint canonical operators and is usually
called “regularity” [19, 21]. This weaker version avoids some of the
topological subtleties of infinite-dimensional $\Xi$. In the finite-
dimensional case, there is no difference.

###### Proof.

Consider a strongly continuous representation $W$ on a Hilbert space
${\mathcal{H}}$. For the most part, we will only need to use the
representation $\xi_{1}\mapsto W(\xi_{1}\oplus 0)$ of the subgroup $\Xi_{1}$.
Following von Neumann, and even his notation up to a factor $2\pi$, we
introduce a Gaussian function $a:\Xi_{1}\to{\mathbb{C}}$ and the operator

$A=\int d\xi_{1}\ a(\xi_{1})W(\xi_{1}\oplus 0).$ (15)

The integral exists as a strong integral because $W$ is continuous. Because
$a$ is integrable, $A$ is clearly a bounded operator. With von Neumann’s
choice, it is even a projection, and in the Schrödinger representation, it is
just the one-dimensional projection ${|\Omega\rangle\\!\langle\Omega|}$ onto
the harmonic oscillator ground state vector $\Omega\in{\mathcal{H}}_{1}$.
Since algebraic relations between $A$ and anything in
${\mathrm{CCR}}(\Xi_{1},\sigma_{1})$ are the same in any representation, it is
hardly a surprise that we have

$AW(\xi_{1}\oplus
0)A=\langle\Omega|W_{1}(\xi_{1})|\Omega\rangle\,A=:\chi(\xi_{1})A,$ (16)

where $W_{1}$ is the Schrödinger representation and
$\chi(\xi_{1})=\exp(-1/4\,\xi_{1}^{2})$ is a Gaussian, the characteristic
function of the oscillator ground state. But, of course, one can also show
this (as von Neumann does) by explicit computation based on the Weyl
relations.

It is a key part of von Neumann’s argument that $A$ cannot vanish for any
continuous representation of $\Xi_{1}$. Indeed, in such a representation also
$W(\eta\oplus 0)AW(\eta\oplus 0)^{*}$ would vanish for all $\eta$, which is
exactly of the form (15), with a kernel function
$a_{\eta}(\xi_{1})=\exp(i\eta{\cdot}\sigma\xi_{1})a(\xi_{1})$. As a function
of $\eta$, the integral of the modified (15) is thus the Fourier transform of
an operator-valued $L^{1}$-function, hence vanishes only if $a$ does, which is
false.

Consider now the subspace ${\mathcal{H}}_{0}:=A{\mathcal{H}}$ and the set $M$
of vectors of the form $W(\xi_{1}\oplus 0)\psi_{0}$ for $\xi_{1}\in\Xi_{1}$
and $\psi_{0}=A\psi_{0}\in{\mathcal{H}}_{0}$. We claim that its linear span is
dense. If $\Psi\in{\mathcal{H}}$ were orthogonal to $M$, we would have that
$\langle AW(\xi_{1}\oplus 0)\Psi,A\psi_{0}\rangle=0$ for all $\psi_{0}$, so
$A$ would vanish on the cyclic sub-representation space of $\Xi_{1}$ generated
by $\Psi$, contradicting von Neumann’s result $A\neq 0$.

We define a function $U:M\to{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{0}$ by

$UW(\xi_{1}\oplus 0)\psi_{0}=W_{1}(\xi_{1})\Omega\otimes\psi_{0}.$ (17)

Scalar products between different vectors on $M$ are preserved, so, in
particular, it sends a linear combination representing the null vector again
to a linear combination with vanishing norm. That is, it extends to a linear
operator on the algebraic linear span of $M$. Clearly, this extension is
isometric as well, so extends by continuity to ${\mathcal{H}}$. Hence (17)
defines an isometry
$U:{\mathcal{H}}\to{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{0}$. It is also
onto, because the vectors $W_{1}(\xi_{1})\Omega$ span ${\mathcal{H}}_{1}$. To
summarize: $U$ is unitary. Now since $A$ commutes with $W(0\oplus\eta_{0})$ by
virtue of the hybrid commutation relations, we can replace $\psi_{0}$ in (17)
by $W(0\oplus\eta_{0})\psi_{0}$, and upgrade that equation to a full
intertwining relation on $M$:

$\displaystyle UW(\eta_{1}\oplus\eta_{0})W(\xi_{1}\oplus 0)\psi_{0}$
$\displaystyle=UW(\eta_{1}\oplus 0)W(\xi_{1}\oplus
0)\bigl{(}W(0\oplus\eta_{0})\psi_{0}\bigr{)}$
$\displaystyle=W_{1}(\eta_{1})W_{1}(\xi_{1})\Omega\otimes
W(0\oplus\eta_{0})\psi_{0}$ $\displaystyle=W_{1}(\eta_{1})\otimes
W(0\oplus\xi_{0})UW(\xi_{1}\oplus 0)\psi_{0}.$ (18)

Hence $W(\eta_{1}\oplus\eta_{0})=U^{*}W_{1}(\eta_{1})\otimes
W(0\oplus\eta_{0})U$.

To complete the proof, it suffices to observe that the strongly continuous
representation $\xi_{0}\mapsto W(0\oplus\xi_{0})$ of the group
$\Xi_{0}\cong{\mathbb{R}}^{s}$ on ${\mathcal{H}}_{0}$ can be decomposed into a
direct sum of cyclic ones, and the cyclic representations are of the form
given in Def. 1. To see that this decomposition works together correctly with
von Neumann’s construction for $\Xi_{1}$ was the main reason to include an
abridged version of his argument. ∎

###  2.4 Bochner’s Theorem

A state on the CCR-algebra is completely determined by its expectations on
Weyl operators, hence by the function

$\chi(\xi)=\omega(W(\xi)),$ (19)

which we call the characteristic function of $\omega$. Thus it is a natural
question which functions exactly arise in this way. This demands unifying two
well-known results: The purely classical case of this is known as Bochner’s
Theorem (sometimes: Bochner–Khintchine Theorem [53]). Its quantum analog was
apparently first formulated by Araki [18], with further relevant work by [17,
54, 55, 56, 22]. Its hybrid version (also in [21]) is the following.

######  Theorem 4 (Hybrid Bochner Theorem).

Let $\Xi$ be a vector space with antisymmetric form $\sigma$. Then a function
$\chi:\Xi\to{\mathbb{C}}$ is the characteristic function of a standard state
on ${\mathrm{CCR}}(\Xi,\sigma)$ if and only if it is

  * (1)

continuous,

  * (2)

normalized, $\chi(0)=1$, and

  * (3)

$\sigma$-twisted positive definite, which means that, for any choice
$\xi_{1},\ldots,\xi_{N}$, the $N\times N$-matrix

$M_{k\ell}=\chi(-\xi_{k}+\xi_{\ell})\,e^{\textstyle-\frac{i}{2}\sigma(\xi_{k},\xi_{\ell})}$
(20)

is positive semi-definite.

###### Proof.

Just conditions (2) and (3) are equivalent to $\omega$ being a state on the
CCR-algebra. Indeed, the positive definiteness condition is precisely
equivalent to $\omega(A^{*}A)\geq 0$, where $A=\sum_{i}c_{i}W(\xi_{i})$, and
the Weyl relations are used. By the GNS-construction, every positive linear
functional comes from a Hilbert space representation, and by definition of the
CCR-algebra as the universal C*-algebra of the Weyl relations, the state thus
extends to the whole algebra.

Continuity of $\chi$ for a standard $\omega$ is obvious because a standard
representation is strongly continuous. Conversely, suppose that $\chi$ is
continuous, and let $\Omega\in{\mathcal{H}}_{\omega}$ denote the cyclic vector
of the GNS-representation of $\omega$. Then

$\displaystyle\xi\to$
$\displaystyle\langle\pi_{\omega}\bigl{(}W(\eta_{1})\bigr{)}\Omega,\pi_{\omega}\bigl{(}W(\xi)\bigr{)}\
\pi_{\omega}\bigl{(}W(\eta_{2})\bigr{)}\Omega\rangle=\langle\Omega,\pi_{\omega}\bigl{(}W(-\eta_{1})W(\xi)W(\eta_{2})\bigr{)}\
\Omega\rangle$
$\displaystyle=\exp\left(-\frac{i}{2}\bigl{(}\sigma(\xi,\eta_{2})+\sigma(-\eta_{1},\xi+\eta_{2})\bigr{)}\right)\langle\Omega,\pi_{\omega}\bigl{(}W(\xi-\eta_{1}+\eta_{2})\bigr{)}\
\Omega\rangle$
$\displaystyle=\chi(\xi-\eta_{1}+\eta_{2})\exp\left(-\frac{i}{2}\bigl{(}\sigma(\xi,\eta_{2})-\sigma(\eta_{1},\xi)-\sigma(\eta_{1},\eta_{2})\bigr{)}\right)$
(21)

is continuous. Since the Weyl operators are bounded, this extends to the norm
limits of linear combinations of
$\pi_{\omega}\bigl{(}W(\eta_{2})\bigr{)}\Omega$ which is, by definition, all
of ${\mathcal{H}}_{\omega}$. Hence $\pi_{\omega}(W({\cdot}))$ is weakly
continuous, but for unitary operators, this is the same as strong continuity.
Hence $\omega$ is normal in a strongly continuous representation.

By Thm. 3 this is a direct sum of standard representations, and by the
argument preceding it, we conclude that $\omega$ itself is standard. ∎

To see the power of the continuity condition, it may be useful to point out
some rather wild states of the CCR-algebra. Indeed, this algebra is just the
hybrid version of the almost periodic functions, in the precise sense of Prop.
21. Pure states on the almost periodic functions form the Bohr
compactification of $\Xi_{0}$ [57, Sect. 4.7], among which the points of
$\Xi_{0}$ (i.e., their point evaluations) are just a small part, and not even
an open subset. This expresses the observation that almost periodic functions
cannot distinguish a point from many others that are arbitrarily far away, so
the finite and the infinite are intertwined more intimately than “observables”
would ever distinguish. An algebra whose states are better behaved may be more
adequate for physics. Even for quantum field theory, where the CCR-algebra has
been used extensively (with infinite-dimensional $\Xi$), this need has been
felt, and a recent proposal by Buchholz to consider not the exponentials of
field operators but their resolvents [28, 58] (see Example 28), can be seen in
this light. Certainly, this eliminates the extreme sensitivity to infinite
values and, for example, allows the algebra to be invariant under typical
quantum mechanical time evolutions [28, 58].

This suggests finding a C*-algebra whose states are just the “good” ones
described by Bochner’s Theorem. This will be done in the next section.

###  2.5 The standard states as a C*-state space

The CCR-algebra is constructed so that its representations exactly correspond
to the representations of the Weyl relations. In this correspondence, the
topology of $\Xi$ plays no role at all. The way to set up a similar
correspondence for just the continuous unitary representations is well known
from the theory of locally compact groups: One goes to the convolution algebra
over the group. In fact, the term “group algebra” of a group is usually
reserved for the C*-envelope of the convolution algebra $L^{1}(G)$, and not
for the topology-free analog of the CCR-algebra [59, Ch. 13]. Von Neumann’s
proof uses the same idea by introducing the operator $A$ as an integral. In
this section, we follow this lead.

This will require a twisted version of the group algebra construction [60]. An
alternative construction would be via the group C*-algebra of a related non-
abelian group, a central extension [61, Ch. VII] of the additive group $\Xi$,
called the Heisenberg group. The approach used below is a bit more direct in
that it avoids the introduction of the central phase parameter, which is, in
the end, integrated out anyhow. After we finished this work, we realized that
the idea had already been followed through by Grundling [62, 63], with much
the same motivation of getting a C*-description of continuous representations,
and even extended to more general groups, also beyond locally compact ones. To
keep this paper self-contained, we nevertheless include our version.

For $h\in L^{1}(\Xi,d\xi)$ and any given measurable representation $W$ of the
Weyl relations we write the Bochner integral

$W[h]=\int d\xi\ h(\xi)W(\xi).$ (22)

The bracket notation indicates that $h\mapsto W[h]$ is closely related to the
representation $W$. We can then describe the algebra of such operators
directly in terms of operations on $L^{1}(\Xi,d\xi)$. The multiplication rule
and adjoints for such operators follows directly from the Weyl relations,
namely $W[h]W[g]=W[h{\ast_{\sigma}}g]$ and $W[h]^{*}=W[h^{*}]$ with

$\displaystyle\bigl{(}h{\ast_{\sigma}}g\bigr{)}(\xi)$ $\displaystyle=\int
d\eta\ h(\xi-\eta)g(\eta)e^{-\frac{i}{2}\xi{\cdot}\sigma\eta},$ (23)
$\displaystyle h^{*}(\xi)$ $\displaystyle=\overline{h(-\xi)}.$ (24)

These operations turn $L^{1}(\Xi)$ into a Banach *-algebra, which we call the
$\sigma$-twisted convolution algebra of $\Xi$. Any set of elements
$h_{\varepsilon}$ such that $h_{\varepsilon}\geq 0$, $\int
d\xi\,h_{\varepsilon}(\xi)=1$, and $h_{\varepsilon}(\xi)=0$ for $\xi$ outside
a ball of radius $\varepsilon$ around the origin is an approximate unit. As in
the untwisted case, this follows from the strong continuity of translations on
$L_{1}(\Xi)$.

The enveloping C*-algebra of the convolution algebra will be called the
twisted group algebra of $(\Xi,\sigma)$ and denoted by
${\mathrm{C}}^{*}(\Xi,\sigma)$. This is defined [59, Ch. 2, §7] as the
completion in the norm

$\|h\|=\sup_{\pi}\|\pi(h)\|=\sup_{\omega}\omega\bigl{(}h^{*}\ast_{\sigma}h\bigr{)}^{1/2},$
(25)

where the supremum over $\pi$ runs over all *-representations of the algebra
by Hilbert space operators, and $\omega$ runs over all positive linear
functionals of norm $\leq 1$. By slight abuse of notation, we denote the
element in ${\mathrm{C}}^{*}(\Xi,\sigma)$ associated with $h\in L^{1}(\Xi)$ by
the completion process again by $h$. This is justified by the observation that
the canonical embedding
$L^{1}(\Xi,\sigma)\hookrightarrow{\mathrm{C}}^{*}(\Xi,\sigma)$ is injective.

######  Proposition 5.

Let $(\Xi,\sigma)$ be a hybrid phase space. Then every state $\omega$ on
${\mathrm{C}}^{*}(\Xi,\sigma)$ is given by a unique standard state
$\omega^{\prime}$ on ${\mathrm{CCR}}(\Xi,\sigma)$ and conversely, such that

$\omega\bigl{(}h\bigr{)}=\int d\xi\
h(\xi)\,\omega^{\prime}\bigl{(}W(\xi)\bigr{)}.$ (26)

This proposition gives us the third way of looking at Weyl elements $W(\xi)$.
At first, they were defined as explicit operators in any standard
representation. Secondly, they appeared as the abstract generators of a CCR-
algebra. These two views are equivalent by virtue of Bochner’s Theorem, which
identifies standard states with linear functionals $\omega^{\prime}$ on
${\mathrm{CCR}}(\Xi,\sigma)$. The above proposition allows us to further
introduce, for each $\xi$, the linear functional
$\omega\mapsto\omega^{\prime}(W(\xi))$. This element of the bidual
${\mathrm{C}}^{*}(\Xi,\sigma)^{**}$ is yet another version of the Weyl
element, which we will also denote by $W(\xi)$. It is clear that we cannot
realize such an element in ${\mathrm{C}}^{*}(\Xi,\sigma)$, since this algebra
has no unit and hence contains no unitary elements. However, we can get close
in the same sense the expression (22) can be close to $W(\xi)$, if $h$ is
concentrated near $\xi$. The idea of the following proof is to do this limit
in the GNS representation of $\omega$.

###### Proof.

It is a general feature of the enveloping C*-algebra construction [59, Prop.
2.7.5] that the states $\omega$ on ${\mathrm{C}}^{*}(\Xi,\sigma)$ are in
bijective correspondence to the positive linear functionals
$\widetilde{\omega}$ on the convolution algebra with norm $1$. Here the norm
is taken as a linear functional on the Banach space $L^{1}(\Xi)$. That is,
there is a function $\chi\in L^{\infty}(\Xi)$ with $\|\chi\|_{\infty}=1$ such
that

$\widetilde{\omega}(h)=\int d\xi\ h(\xi)\chi(\xi)$ (27)

The main task of the proof is to show that the functions $\chi$ arising in
this way are exactly the characteristic functions characterized by the Bochner
Theorem, and in particular continuous. The uniqueness of the correspondence is
clear from this equation since, on the one hand, it gives an explicit formula
for $\omega$ (resp. $\widetilde{\omega}$) in terms of $\omega^{\prime}$, and,
on the other, two states $\omega^{\prime},\omega^{\prime\prime}$ satisfying it
for the same $\widetilde{\omega}$ would have to be equal as elements of
$L^{\infty}(\Xi)$, hence equal almost everywhere, and hence equal by
continuity.

We begin by defining a version of the Weyl operators acting on $L^{1}(\Xi)$,
namely

$(\widetilde{W}(\xi)h)(\eta):=e^{-\frac{i}{2}\xi{\cdot}\sigma\eta}h(\eta-\xi).$
(28)

It is constructed so that

$h{\ast_{\sigma}}g=\int d\xi\ h(\xi)\,\widetilde{W}(\xi)g.$ (29)

Intuitively, we can think of $\widetilde{W}(\xi)$ as the operator of
convolution with $\delta_{\xi}$, the limit of probability densities
concentrated near the point $\xi$. While this is not an element of the
algebra, its operation is defined analogously to the approximate unit
$\delta_{0}$. It is easy to check that the operators $\widetilde{W}(\xi)$
satisfy the Weyl multiplication rules (7). However, unitarity does not make
sense since $L^{1}(\Xi)$ is not a Hilbert space. The crucial observation is
that $\xi\mapsto\widetilde{W}(\xi)g$ is continuous in the norm of
$L^{1}(\Xi)$. Indeed, it is a product of a translation and a multiplication
operator, which are both strongly continuous on $L^{1}$.

Consider now a positive linear functional $\widetilde{\omega}$ on
$L^{1}(\Xi)$. Its GNS representation space $\widetilde{\mathcal{H}}$ is the
unique Hilbert space generated by vectors $v(h)$, $h\in L^{1}(\Xi)$, with the
scalar product $\langle
v(h)|v(k)\rangle=\widetilde{\omega}(h^{*}\ast_{\sigma}k)$. On these, the
representation $W:L^{1}(\Xi)\to{\mathcal{B}}(\widetilde{\mathcal{H}})$ acts by
left multiplication in the convolution algebra, i.e., according to the formula

$W[h]v(g)=v\bigl{(}h\ast_{\sigma}g\bigr{)}.$ (30)

According to [14, I.9.14] the GNS space has a cyclic vector $\Omega$ and a
representation $W:L^{1}(\Xi)\to{\mathcal{B}}(\widetilde{\mathcal{H}})$ such
that $v(h)=W[h]\Omega$, and
$\langle\Omega,W[h]\Omega\rangle=\widetilde{\omega}(h)$. Indeed, one has
$\Omega=\lim_{\varepsilon\to 0}v(h_{\varepsilon})$, where $h_{\varepsilon}$ is
a bounded approximate unit.

Our next aim is to show that $W$ arises exactly as in (22) from the
integration of a representation $W$ of the Weyl relations (recall that the two
functions will be typographically distinguished by their argument brackets).
The obvious candidate for the Weyl operators $W(\cdot)$ are the operators
$\widetilde{W}(\xi)$ from (28), represented on $\widetilde{\mathcal{H}}$ in
GNS style. That is, in analogy to (30) we set

$W(\xi)v(g)=v\bigl{(}\widetilde{W}(\xi)g\bigr{)}.$ (31)

Then it is elementary to check that $W(\xi)$ is unitary, and these operators
satisfy the Weyl relations. Moreover, the $L^{1}$-norm continuity of
$\xi\mapsto\widetilde{W}(\xi)g$ established earlier implies that $\xi\mapsto
W(\xi)$ is continuous in the strong operator topology. Finally, (29) implies

$W[h]v(g)=v\bigl{(}h\ast_{\sigma}g\bigr{)}=\int d\xi\
h(\xi)v\bigl{(}\widetilde{W}(\xi)g\bigr{)}=\int d\xi\
h(\xi)W(\xi)v\bigl{(}g\bigr{)},$

i.e., the GNS-representation $W[h]$ is related to the continuous
representation $W$ by (22). In particular

$\widetilde{\omega}(h)=\int d\xi\ h(\xi)\langle\Omega,W(\xi)\Omega\rangle,$

so (27) holds with $\chi(\xi)=\langle\Omega,W(\xi)\Omega\rangle$, which is
clearly a normalized twisted positive definite function, and continuous
because $W(\xi)$ is strongly continuous.

Conversely, given a state on the CCR-algebra, we can define its characteristic
function $\chi(\xi)=\omega^{\prime}(W(\xi))$. In general, that might fail to
be even measurable, so the formula (27) might make no sense. For standard
states, however, $\chi$ is continuous, and the integral is well defined. ∎

We now determine the algebras ${\mathrm{C}}^{*}(\Xi,\sigma)$ concretely. It
turns out that this is best done by splitting into a purely classical and a
purely quantum part.

######  Proposition 6.

Let $(\Xi,\sigma)$ be a hybrid phase space, split as
$\Xi=\Xi_{1}\oplus\Xi_{0}$ with $\sigma=\sigma_{1}\oplus 0$. Then

${\mathrm{C}}^{*}(\Xi,\sigma)={\mathrm{C}}^{*}(\Xi_{1},\sigma_{1})\otimes{\mathrm{C}}^{*}(\Xi_{0},0)\cong{\mathcal{K}}({\mathcal{H}}_{1})\otimes{\mathcal{C}}_{0}(\Xi_{0}),$
(32)

where ${\mathcal{K}}({\mathcal{H}}_{1})$ denotes the compact operators on the
representation space ${\mathcal{H}}_{1}$ of the irreducible quantum system
$(\Xi_{1},\sigma_{1})$, and ${\mathcal{C}}_{0}(\Xi_{0})$ denotes the
continuous functions on $\Xi_{0}$ vanishing at infinity.

###### Proof.

We first observe that for the underlying $L^{1}$-spaces, the direct sum
naturally coincides with the projective product, which is predual to the
tensor product of von Neumann algebras. That is

$L^{1}(\Xi_{1}\oplus\Xi_{0})=L^{1}(\Xi_{1})\otimes L^{1}(\Xi_{0}).$ (33)

Indeed, the tensor products $f\otimes g$ on the right hand side can be
identified with the product functions
$fg(\xi_{0}\oplus\xi_{1})=f(\xi_{1})g(\xi_{0})$, and this embedding is clearly
isometric on step functions. Since the measurable structure of
$\Xi_{1}\oplus\Xi_{0}$, which is $\Xi_{1}\times\Xi_{0}$ as a set, is defined
as generated by rectangles, the product functions span a dense subspace of
$L^{1}(\Xi_{1}\oplus\Xi_{0})$. It is elementary to verify that the isomorphism
(33) is also consistent with the definitions of adjoint operation and
convolution product.

The completion in the construction of the enveloping C*-algebra also works
out: As one side of the tensor product is abelian, and the maximal and the
minimal tensor product coincide, the tensor product is uniquely determined, as
is the algebra in tensor product form.

In the second step, we need to show the claimed isomorphisms: Beginning with
the classical case, $L^{1}(\Xi_{0})$ is the convolution algebra of
$\Xi_{0}\cong{\mathbb{R}}^{s}$. Its irreducible representations are described
by the Gelfand isomorphism for abelian Banach algebras: In this case they are
given by the point evaluations of the Fourier transform. Hence the C*-norm of
the enveloping algebra, $\|h\|=\sup_{\pi}\|\pi[h]\|$ is equal to the supremum
norm of the Fourier transform of $h$. Now by the Riemann–Lebegue Lemma, the
Fourier transforms of $L^{1}$-functions are continuous and go to zero at
infinity. On the other hand, by the Stone–Weierstraß Theorem, these Fourier
transforms separate points and are hence uniformly dense. Hence
${\mathrm{C}}^{*}(\Xi_{0})={\mathcal{C}}_{0}(\Xi_{0})$.

For quantum systems, note that every continuous representation of the Weyl
relations is isomorphic to the Schrödinger representation on
${\mathcal{H}}_{1}$ by von Neumann’s Uniqueness Theorem. Hence we only need to
show that in that representation, the operators of the form $W[h]$ with $h\in
L^{1}(\Xi_{1})$ are compact, and these operators form a dense subalgebra of
${\mathcal{K}}({\mathcal{H}}_{1})$. This follows immediately by the
correspondence theory [27, Cor. 5.1.(4)].

An alternative approach using better known facts goes via first showing that
operators $h\mapsto W[h]$ are not only continuous from $L^{1}$ to
${\mathcal{B}}({\mathcal{H}}_{1})$ but also an isometry for the $2$-norms,
i.e. $\|W[h]\|_{2}=\|h\|_{2}$, for $h\in L^{1}(\Xi_{1})\cap
L^{2}(\Xi_{1})\equiv{\mathcal{A}}$, and the Schatten $2$-norm (Schmidt norm)
on the operator side. Continuous extension via these $2$-norms is even
unitary. Hence $W[{\mathcal{A}}]$ consists of Hilbert–Schmidt operators, which
are compact, and by taking limits in $2$-norm, we find that $W[{\mathcal{A}}]$
is operator norm dense in the Hilbert–Schmidt class, hence in
${\mathcal{K}}({\mathcal{H}}_{1})$. ∎

We note that as a consequence of this characterization, we find that there are
many extremal standard states since the state space of
${\mathrm{C}}^{*}(\Xi,\sigma)$ is the weak*-closed convex hull of its extreme
points, by the Banach–Alaoglu and Krein–Milman Theorems. Of course, these were
already identified in Lem. 2.

###  2.6 Restoring translation symmetry

Translations were part of our basic setup from the outset since the phase
space $\Xi$ is a vector space. The notion of standard representations (Def. 1)
breaks the translation symmetry. However, it is restored in the twisted
convolution construction. Indeed, combining (5) and (6) we get
$\alpha_{\eta}(W(\xi))=\exp(i\eta{\cdot}\xi)W(\xi)$. Although Weyl operators
are not themselves in ${\mathrm{C}}^{*}(\Xi,\sigma)$ we think of this algebra
as generated by integrated Weyl operators (22), and so we must define

$(\alpha_{\eta}h)(\xi)=e^{i\eta{\cdot}\xi}h(\xi).$ (34)

Of course, this extends to the enveloping algebra
${\mathrm{C}}^{*}(\Xi,\sigma)$. In the tensor product structure of Prop. 6 we
can apply this separately to the classical parts, so
$\alpha_{\eta_{1}\oplus\eta_{0}}=\alpha_{\eta_{1}}\alpha_{\eta_{0}}$. On the
classical part ${\mathcal{C}}_{0}(\Xi_{0})$ the action becomes the shift
$(\alpha_{\eta_{0}}f)(\xi_{0})=f(\xi_{0}+\eta_{0})$. Similarly, we can compute
the action on the quantum part, finding

$\alpha_{\eta}(X)=W(\sigma\eta)^{*}XW(\sigma\eta).$ (35)

In this expression, we use $\sigma$ as a matrix acting on the vector
$\eta\in\Xi$, which is possible because we choose a fixed basis in
$\Xi\cong{\mathbb{R}}^{2n+s}$ (cp. Sect. 2.1). Since $\sigma$ vanishes on the
classical part, the component $\eta_{0}$ of the translation argument
automatically drops out, and only the quantum Weyl operators are used.

The tensor product
${\mathcal{K}}({\mathcal{H}}_{1})\otimes{\mathcal{C}}_{0}(\Xi_{0})$ can be
considered as the algebra of norm continuous functions
$F:\Xi_{0}\to{\mathcal{K}}({\mathcal{H}}_{1})$ vanishing at infinity, by
identifying $K\otimes f$ with the function $F(\xi)=f(\xi)K$. In this “function
form”, which will later extend to certain subspaces of
${\mathrm{C}}^{*}(\Xi,\sigma)^{**}$ the action of translations becomes, for
$\eta=\eta_{1}\oplus\eta_{0}$,

$\bigl{(}\alpha_{\eta}(F)\bigr{)}(\xi_{0})=W(\sigma\eta)^{*}F(\xi_{0}+\eta_{0})W(\sigma\eta).$
(36)

###  2.7 Continuity of state translations

We note that $\alpha_{\eta}^{*}$ is not strongly continuous on the Banach
space of states, i.e., the function $\xi\mapsto\alpha_{\eta}^{*}\rho$ is not
continuous in norm. Indeed, an arbitrarily small shift applied to a point
measure moves it as far away as possible in the natural norm on states. Since
translations are strongly continuous on $L^{1}({\mathbb{R}}^{n},dx)$, this is
different for probability measures with absolutely continuous densities. We
can use this to single out one particular standard representation, namely that
using the Lebesgue measure for $\mu$.

######  Proposition 7.

Let $\omega\in{\mathrm{C}}^{*}(\Xi,\sigma)^{*}$ be a state with characteristic
function $\chi$, and let $\mu$ be its marginal probability measure on
$\Xi_{0}$. Then the following are equivalent:

  * (1)

$\omega$ is norm continuous under phase space translations, i.e.,
$\lim_{\eta\to 0}\|\omega-\alpha_{\eta}^{*}(\omega)\|=0$.

  * (2)

$\mu$ is absolutely continuous with respect to the Lebesgue measure.

  * (3)

$\omega$ is the restriction of a standard state $\widehat{\omega}$ on a purely
quantum system, in which the classical variables in $\Xi_{0}$ also have
conjugate momenta.

In this case $\chi\in\mathcal{C}_{0}(\Xi)$. As a partial converse, if $\chi\in
L^{p}(\Xi,d\xi)$ for some $p\in[1,2]$ then the above conditions hold.

###### Proof.

(1)$\Rightarrow$(2): (2) only depends on the restriction of $\omega$ to the
classical algebra. So this is a purely classical observation, which is valid
for any locally compact group. Let $\alpha_{x}^{*}$, $x\in{\mathbb{R}}^{n}$
denote the action of translations on measures over ${\mathbb{R}}^{n}$. Then
(1) says $\lim_{x\to 0}\|\mu-\alpha_{x}^{*}(\mu)\|=0$, where the norm is the
dual norm of the supremum norm on ${\mathcal{C}}_{0}({\mathbb{R}}^{n})$, i.e.,
the total variation norm on measures. For any $\varepsilon>0$, there is thus a
neighbourhood $U_{\varepsilon}\ni 0$ such that
$\|\mu-\alpha_{x}^{*}(\mu)\|\leq\varepsilon$ for $x\in U_{\varepsilon}$. We
pick a positive measurable function $h\in L^{1}({\mathbb{R}}^{n},dx)$ with
integral $1$ and support in $U_{\varepsilon}$, and set

$\alpha^{*}_{h}(\mu)=\int dx\ h(x)\alpha^{*}_{x}(\mu),$ (37)

which is to be read as a weak* integral. By the triangle inequality
$\|\mu-\alpha^{*}_{h}(\mu)\|\leq\varepsilon$. On the other hand,
$\alpha^{*}_{h}(\mu)$ is absolutely continuous, because (37) is the
convolution of the two measures $\mu$ and $h\,dx$. Explicitly, for an
arbitrary $f\in{\mathcal{C}}_{0}({\mathbb{R}}^{n})$, we get

$\bigl{(}\alpha^{*}_{h}\mu\bigr{)}(f)=\int dx\ h(x)\int\mu(dy)\
f(y-x)=\int\mu(dy)\int dx\ h(y-x)f(x),$ (38)

which is to say that $\alpha^{*}_{h}\mu$ has density
$\widetilde{h}(x)=\int\mu(dy)\ h(y-x)$ with respect to Lebesgue measure. We
also recall that the variation norm of absolutely continuous measures is just
the $L^{1}$ norm of their densities. Now since $\alpha^{*}_{h}(\mu)$ converges
in norm to $\mu$ as $\varepsilon\to 0$, the corresponding densities
$\widetilde{h}$ form a Cauchy net in $L^{1}({\mathbb{R}}^{n},dx)$. Its limit
$\widetilde{h}_{0}$ must be in $L^{1}$ by completeness of $L^{1}$. This is
then a density for $\mu$, so $\mu$ is itself absolutely continuous.

(2)$\Rightarrow$(3): Consider the standard representation for $\mu$ in the
Hilbert space ${\mathcal{H}}_{1}\otimes L^{2}(\Xi_{0},\mu)$. We can consider
this as a subspace of ${\mathcal{H}}_{1}\otimes L^{2}(\Xi_{0},dx)$ with the
Lebesgue measure. By Def. 1 $\omega$, as a state on the CCR-algebra, can be
represented as a normal state on this space. Apart from the canonical position
operators in $L^{2}(\Xi_{0},dx)$, which are already part of the hybrid setup,
we can take the shift generators in this tensor factor as further canonical
momentum operators. The full set of canonical operators is then clearly
irreducible, so the extended system is purely quantum.

(3)$\Rightarrow$(1): The Weyl translations in a standard representation are
strongly continuous, which implies that the action
$\alpha_{\xi}^{*}(\rho)=W(\sigma\xi)^{*}\rho W(\sigma\xi)$ is norm continuous
for every $\rho\in{\mathcal{T}}({\mathcal{H}})$ so we can just restrict the
continuity condition for the extended system to those translations $\eta$
which make sense in the original hybrid.

The final remark in the proposition is clear in one direction from the
Riemann–Lebesgue Lemma and its quantum version [27]. In the converse
direction, it follows that the Fourier transform of $\chi$ is the density with
respect to the Lebesgue measure, which is then even continuous and goes to
zero. ∎

###  2.8 $L^{p}$-spaces

If one is not interested in pure states, a good setting for hybrids is to
restrict consideration to the norm continuous states characterized by Prop. 7.
This leads to a purely von Neumann algebraic picture: Since all probability
measures $\mu$ are then absolutely continuous with respect to Lebesgue measure
$dx$, we can represent all states in ${\mathcal{H}}={\mathcal{H}}_{1}\otimes
L^{2}(\Xi_{0},dx)$, which leads to the spaces

$L^{1}(\Xi,\sigma):={\mathcal{T}}^{1}({\mathcal{H}}_{1})\otimes
L^{1}(\Xi_{0},dx)\qquad\mbox{and}\quad
L^{\infty}(\Xi,\sigma):={\mathcal{B}}({\mathcal{H}}_{1}){\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}L^{\infty}(\Xi_{0},dx).$
(39)

Here the tensor product on the right uses the $1$-norm completion, and
${\mathcal{T}}^{p}$ denotes the Schatten classes for $1\leq p<\infty$, so
${\mathcal{T}}^{1}$ is the trace class. ${\mathcal{T}}^{\infty}$ would be
ambiguous, meaning either all bounded operators ${\mathcal{B}}({\mathcal{H}})$
or just the compact operators ${\mathcal{K}}({\mathcal{H}}_{1})$, so we prefer
to specify explicitly. The tensor product on the right is then the von Neumann
algebra version, constructed as the completion of the product operators in the
weak (or similar) operator topology. This choice was adopted, for example, in
[35, 36, 64], see Sect. 1.3. Note that the von Neumann algebra
$L^{\infty}(\Xi,\sigma)$ is not a subalgebra of ${\mathcal{U}}(\Xi,\sigma)$
but a quotient, because the $(L^{1},L^{\infty})$-duality is defined by
selecting a subspace of states (cf. Fig. 3). In fact, it does not make sense
to evaluate an element $F\in L^{\infty}(\Xi,\sigma)$ on a pure state because
functions differing at one point, e.g., the point $x\in\Xi_{0}$ on which the
pure state lives, are identified.

The combined Schatten/Lebesgue spaces $L^{p}(\Xi,\sigma)$ can be obtained by a
purely von Neumann algebraic construction using the semifinite trace
$\widehat{\operatorname{tr}}(f\otimes A)=\int dx\ f(x)\operatorname{tr}A$ on
$L^{\infty}(\Xi,\sigma)$. Then $L^{p}(\Xi,\sigma)$ comes out as the $p$-norm
completion of the elements $F\in L^{\infty}$ so that
$\|F\|_{p}^{p}:=\widehat{\operatorname{tr}}|F|^{p}<\infty$. These spaces are
also connected by interpolation [65]. The only case of interest to us,
however, is $p=2$ because of a fact, which is well–known in both the classical
and the quantum case, so its generalization to hybrids is unsurprising: The
Fourier-Weyl transform111Note that the definition given here differs from that
in [27] by a symplectic matrix in the argument, which would not make sense in
the classical case.

$({\mathcal{F}}F)(\xi)=\widehat{\operatorname{tr}}(FW(\xi))$ (40)

is a unitary isomorphism from $L^{2}(\Xi,\sigma)$ onto $L^{2}(\Xi,0)$, i.e.,
for elements $F\in L^{p}(\Xi,\sigma)\cap L^{2}(\Xi,\sigma)$ and $G\in
L^{q}(\Xi,\sigma)\cap L^{2}(\Xi,\sigma)$ with $p^{-1}+q^{-1}=1$, $1\leq
p\leq\infty$ we have

$\widehat{\operatorname{tr}}F^{*}G=\int
dx\,\operatorname{tr}(F(x)^{*}G(x))=(2\pi)^{-(n+s)}\int d\xi\
\overline{\operatorname{tr}(FW(\xi))}\operatorname{tr}(GW(\xi)),$ (41)

where $s$ and $n$ are the numbers of degrees of classical and quantum freedom,
respectively (cp. Sect. 2.1). This extends by continuity to all of
$L^{2}(\Xi,\sigma)$, and the map is clearly onto. Its inverse is an instance
of Weyl quantization, in our case, a partial one acting only on the quantum
part of the system.

A similar useful formula concerns an integral over translates: For $F,G\in
L^{1}(\Xi,\sigma)\cap L^{\infty}(\Xi,\sigma)$ we have

$\int d\xi\
\widehat{\operatorname{tr}}\bigl{(}F\alpha_{\xi}(G)\bigr{)}=(2\pi)^{n}\
\widehat{\operatorname{tr}}(F)\widehat{\operatorname{tr}}(G).$ (42)

The proof is immediate for the classical part, and the quantum part is
essentially the square integrability of the quantum Weyl operators [27, Lem.
3.1.].

###  2.9 The squeezed limit

This subsection harks back to the classic paper [66] by Einstein, Podolsky and
Rosen. Think of a standard quantum system with phase space $\Xi$ and non-
degenerate $\sigma$. We then consider an “opposite” system over
$(\Xi,-\sigma)$, which can be realized in the same Hilbert space, and the Weyl
system $\overline{W}$ with $\overline{W}(a,b)=W(-a,b)$. That is in the
position representation (6) we can write $\overline{W}(\xi)=\Theta
W(\xi)\Theta$ with $\Theta$ the complex conjugation in that representation.
Then the operators $\widetilde{W}(\xi)=W(\xi)\otimes\overline{W}(\xi)$
commute, and so the characteristic function
$\chi_{2}(\xi)=\omega_{2}(\widetilde{W}(\xi))$ is not subject to uncertainty
constraints. That is, it may correspond to a probability distribution sharply
concentrated at the origin. In the EPR paper the state is to have sharp
distribution for the canonical operators $Q_{1}-Q_{2}$ and $P_{1}+P_{2}$. What
they actually write down is an unnormalizable wave function, but the intention
is clearly to have “pretty sharp” distributions for these operators, of
course, at the expense of their canonical conjugates $P_{1}-P_{2}$ and
$Q_{1}+Q_{2}$. The limit they suggest is really not necessary for their
argument. In fact, they are perfectly aware that a maximally entangled state
can be written out equivalently in complementary bases, so their argument
would go through literally on finite phase spaces with sharp distributions,
just as in the version for spins that was later championed by Bohm [67,
Sect.22.17].

Taking here the limit and to go to really sharp distributions is an
interesting exercise in singular states [68], which may serve here to
highlight the importance of the continuity condition in Bochner’s Theorem.
First of all, by weak* compactness the limit of states with sharper and
sharper distributions for $Q_{1}-Q_{2}$ and $P_{1}+P_{2}$ exists (at least
along a subnet) as a state on
${\mathcal{B}}({\mathcal{H}}\otimes{\mathcal{H}})$. Since we have not
specified any details of the sequence there are very many states that might
arise as a weak* cluster point of such a sequence, and since we are implicitly
invoking the axiom of choice, there is no way to give a finite specification
ensuring convergence on all observables. However, for some simple operators,
like the Weyl operators themselves, the scant characterization is sufficient
to determine the limit. Thus we get a “characteristic function”, namely

$\langle\omega_{\mathrm{EPR}},W(\xi)\otimes\overline{W}(\eta)\rangle=\left\\{\begin{array}[]{rl}1&\xi=\eta\\\
0&\hbox{otherwise.}\end{array}\right.$ (43)

This is clearly discontinuous, and so corresponds to no density operator. It
is also insufficient to specify the state on observables not expressed as
linear combinations of Weyl operators. But some things can be read off easily.
For example, the marginal for the first party ($\eta=0$ in the above) will
have the “characteristic function” which vanishes everywhere except for
$\xi=0$. This means that the probability for finding a value in some finite
interval $[a,b]$ for the spectral resolution of any canonical operator is
zero. Although the joint probability for $Q_{1}$ and $Q_{2}$ may be in some
sense concentrated on $x_{1}=x_{2}$, the values themselves are infinite with
probability one, and can only be specified as infinite points in some (non-
constructive) compactification of position space. This is hardly what the
authors intended, so it is much more useful to stick with “pretty sharp”
distributions.

This is also a practical issue for quantum optics, where squeezed states play
an important role. The following is the prototype:

######  Example 8.

The two-mode squeezed state

$\triangleright$ Consider a standard Gaussian product state of the doubled
system, i.e.
$\chi_{0}(\xi_{1}\oplus\xi_{2})=\exp\frac{-1}{2}\bigl{(}p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\bigr{)}$.
Now apply a hyperbolic rotation to the momenta, i.e.,
$(p_{1},p_{2})\mapsto(cp_{1}+sp_{2},sp_{1}+cp_{2})$ with $c=\cosh\lambda$,
$s=\sinh\lambda$, and the inverse to the positions, so that the symplectic
form $\xi\cdot\sigma\xi^{\prime}=p\cdot q^{\prime}-q\cdot p^{\prime}$ remains
invariant. Then, taking into account the inversion of one of the arguments in
$\overline{W}$, we get a state $\omega$ with

$\langle\omega_{\lambda},W(\xi)\otimes\overline{W}(\eta)\rangle=\exp\frac{-1}{4}\Bigl{(}e^{2\lambda}(\xi-\eta)^{2}+e^{-2\lambda}(\xi+\eta)^{2}\Bigr{)},$
(44)

where $\xi^{2}=(p,q)^{2}=p^{2}+q^{2}$. When $\eta=\xi$, this converges
pointwise to (43), but since the limit is not continuous, Lévy’s convergence
theorem (Prop. 16) does not apply. $\triangleleft$

Taking just the case $\xi=\eta$, we see that (44) converges pointwise to $1$
as $\lambda\to\infty$. This property will be useful in the proof of Cor. 25,
as well as in the analysis of teleportation (Sect. 5.8). We therefore
generalize it to arbitrary hybrids:

######  Lemma 9.

For any hybrid system $(\Xi,\sigma)$ there is a family of states
$\omega^{\varepsilon}$ for the system

$(\Xi\oplus\Xi,\sigma\oplus(-\sigma))$ such that

$\lim_{\varepsilon\to
0}\langle\omega^{\varepsilon},W(\xi)\otimes\overline{W}(\xi)\rangle\geq\exp\Bigl{(}\frac{-\varepsilon}{2}\,\xi{\cdot}A\xi\Bigr{)}\qquad\text{for
all}\ \xi\in\Xi,$ (45)

where $A$ is the covariance matrix of a quantum state on $(\Xi,\sigma)$, i.e.,
$A+i\sigma\geq 0$. In particular, the left hand side goes to $1$ as
$\varepsilon\to 0$.

###### Proof.

When $(\Xi,\sigma)$ is a direct sum of other spaces, we can reduce the proof
to each summand by taking tensor products. We need only two types of summands:
single degree of freedom quantum systems for which Example 8 provides just the
required states with $\varepsilon=e^{-2\lambda}$, $\lambda\to\infty$, and one
dimensional classical systems. For these we can take any probability measure
on the doubled system, which is concentrated on the diagonal. This will
satisfy the condition even with $\varepsilon=0$. ∎

##  3 Observables as functions

The aim of this section is to complement the description of hybrid states by
spaces of observables. In contrast to the states there are different options
to consider. The two basic possibilities are the following: l

Any standard representation in the sense of Def. 1 gives us a natural
observable algebra, the von Neumann algebra
${\mathcal{B}}({\mathcal{H}}_{1}){\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}L^{\infty}(\Xi_{0},\mu)$,
in which observables are ${\mathcal{B}}({\mathcal{H}}_{1})$-valued functions
on $\Xi_{0}$. The catch is that this depends on $\mu$, and any choice of $\mu$
excludes some states. For example, with the Lebesgue measure for $\mu$, we
exclude all pure states.

This is the point of view based on the C*-algebra
${\mathcal{A}}={\mathrm{C}}^{*}(\Xi,\sigma)$, allowing all states of
${\mathcal{A}}$. The maximal space of observables, for which these states
provide probability distributions, is, by definition, the bidual
${\mathcal{A}}^{**}$. This, however, is a rather wild non-separable object,
with many elements that owe their existence to the axiom of choice, and are
impossible to describe explicitly. So instead we will be interested in
intermediate algebras ${\mathcal{M}}$ with
${\mathcal{A}}\subset{\mathcal{M}}\subset{\mathcal{A}}^{**}$.

We approach this choice from a very pragmatic point of view. The channels we
want to study in Sect. 4 have a very simple description not involving the
measure $\mu$. Our aim will be to establish also a straightforward action on
observables, i.e., the corresponding Heisenberg picture. This makes the
$\mu$-dependent approach cumbersome, since it requires the specification of a
measure both for the input and the output system, and conditions on the
channel to ensure that these choices match up for the given channel. Since the
measure is characteristic of only classical part of the hybrid, and classical
and quantum information can be reshuffled arbitrarily by a quasifree channel,
such conditions have to be stated carefully, and we would like to avoid this
in the general theory.

In the $\mu$-free approach one has to choose appropriate algebras
${\mathcal{M}}$, but we will be looking for choices with an automatic
Heisenberg picture for all quasifree channels without the need to check any
further conditions. For that we will describe the candidate algebras
${\mathcal{M}}$ by intrinsic properties in terms of ${\mathcal{A}}$, which
make sense for arbitrary C*-algebras, hybrid or not. Roughly speaking,
${\mathcal{M}}$ will describe a degree of regularity for observables, which is
preserved by all quasifree channels. That is, the adjoint of a channel
${\mathcal{T}}:{\mathcal{A}}^{*}_{\mathrm{in}}\to{\mathcal{A}}^{*}_{\mathrm{out}}$
defined in the Schrödinger picture, which by definition maps
${\mathcal{T}}^{*}:{\mathcal{A}}^{**}_{\mathrm{out}}\to{\mathcal{A}}^{**}_{\mathrm{in}}$,
satisfies
${\mathcal{T}}^{*}{\mathcal{M}}_{\mathrm{out}}\subset{\mathcal{M}}_{\mathrm{in}}$.
This is equivalent [69] to the continuity of ${\mathcal{T}}$ in the weak
topologies $\sigma({\mathcal{A}}^{*},{\mathcal{M}})$ induced by these spaces.
Thus rather than a weakness, the multitude of spaces ${\mathcal{M}}$ is a
strength: Without the need to check for any special properties of the given
channel it establishes a variety of continuity properties (see Prop. 26).

The appropriate spaces ${\mathcal{M}}$ will be introduced in the following
Sect. 3.1, without taking the hybrid structure of ${\mathcal{A}}$ into
account. This section is based on the seminal early work of a functional
analyst doing analysis, namely Gert Pedersen, whose slogan for his textbook
“Analysis NOW” [70] also stands for “Analysis based on Norms, Operators, and
Weak topologies”. In spite of its relative abstractness, this approach leads
painlessly to just the Heisenberg picture characterizations we wanted to see.
The following Sect. 3.2 specializes these notions to the hybrid structure.

###  3.1 Semicontinuity in C*-algebras

The constructions in this section are inspired by the commutative case [71],
but have been generalized to arbitrary C*-algebras in [72, 73, 74] (see also
[14, III, §6] for a textbook version). The commutation relations are not
needed for this so that we will consider first a general C*-algebra
${\mathcal{A}}$, typically without a unit, and only in the next section
specialize to hybrids, i.e.,
${\mathcal{A}}={\mathcal{C}}_{0}(X,{\mathcal{K}}({\mathcal{H}}))$. The
standard states are then elements of the dual, and we are interested in well-
behaved subalgebras of the bidual. A special role will be played by the pure
states of ${\mathcal{A}}$.

For the various dualities between spaces of states and spaces of observables,
we use the following notation: $\langle\omega,A\rangle$ will be the
expectation value of the observable $A$ in the state $\omega$, where
$\omega\in{\mathcal{A}}^{*}$ and $A\in{\mathcal{A}}$ or
$A\in{\mathcal{A}}^{**}$, and this is extended to the whole linear spaces. We
thereby identify ${\mathcal{A}}$ with a subspace of ${\mathcal{A}}^{**}$. By
definition, the state space of ${\mathcal{A}}$ is the set of positive linear
functionals of norm $1$. Even wenn ${\mathcal{A}}$ has no unit, this still can
be written as
$\langle\omega,{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}\rangle=1$, but
then ${\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}\in{\mathcal{A}}^{**}$. The
weak*-topology on ${\mathcal{A}}^{*}$ is the topology making all functionals
$\omega\mapsto\langle\omega,A\rangle$ with $A\in{\mathcal{A}}$ continuous,
whereas the weak-topology is similarly defined with $A\in{\mathcal{A}}^{**}$.
By the Banach–Alaoglu Theorem, the “quasi-state space”
$Q=\\{\omega\in{\mathcal{A}}^{*}|\omega\geq 0,\|\omega\|\leq 1\\}$ is
weak*-compact, so by the Krein–Milman Theorem, $Q$ is the closed convex hull
of its extreme points. Thus there are many extremal, that is “pure” states,
and one additional extreme point $0\in Q$: When
${\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}\notin{\mathcal{A}}$ the
normalization functional is not weak*-continuous, so the state space is not
weak*-compact, and there will be sequences of pure states converging to $0$.

The second dual ${\mathcal{A}}^{**}$ can be identified with its “enveloping
von Neumann algebra”, which is the von Neumann algebra generated by
${\mathcal{A}}$ in its universal representation [14, Ch. III.6]. This is
simply the direct sum of all GNS-representations of ${\mathcal{A}}$. Its
center is the natural arena for the representation theory of ${\mathcal{A}}$
in the following sense: For every representation
$\pi:{\mathcal{A}}\to{\mathcal{B}}({\mathcal{H}}_{\pi})$ there is a central
projection $z_{\pi}\in{\mathcal{A}}^{**}$ such that
$\pi({\mathcal{A}})^{\prime\prime}\cong z_{\pi}{\mathcal{A}}^{**}$ as a von
Neumann algebra. Representations $\pi_{1},\pi_{2}$ which can be connected to
each other by an isomorphism from $\pi_{1}({\mathcal{A}})^{\prime\prime}$ to
$\pi_{1}({\mathcal{A}})^{\prime\prime}$ are called quasi-equivalent. So the
central projections of ${\mathcal{A}}^{**}$ classify representations up to
quasi-equivalence [14, Thm. III.2.12].

Consider now a pure state $\omega\in{\mathcal{A}}^{*}$. Its GNS-representation
is irreducible, and hence the corresponding central projection $z_{\pi}$ is
minimal, meaning that there is no projection $p$ in the center of
${\mathcal{A}}^{**}$, other than $0$ and $z_{\pi}$, such that $0\leq p\leq
z_{\pi}$. Minimal projections are also called atoms of the projection lattice.
Then the minimal projections of ${\mathcal{A}}^{**}$ correspond exactly to the
pure states on ${\mathcal{A}}$. We record this simple observation as a
statement for arbitrary von Neumann algebras (cp. [75]).

######  Lemma 10.

Let ${\mathcal{M}}$ be a von Neumann algebra. Then there is a one-to-one
correspondence between minimal projections $p\in{\mathcal{M}}$ and extremal
normal states $\omega\in{\mathcal{M}}_{*}$, given by

$pxp=\omega(x)p,\quad\mbox{for all\ }x\in{\mathcal{M}}.$ (46)

###### Proof.

For any projection $p$ we consider the von Neumann subalgebra
$\widetilde{\mathcal{M}}=p{\mathcal{M}}p$. The crucial issue is whether this
algebra is one-dimensional. When that is the case, we must have
$pxp=\omega(x)p$, for some functional $\omega$, which is necessarily a normal
state. We will proceed by showing the implications “$p$ minimal”
$\Leftrightarrow$“$\dim\widetilde{\mathcal{M}}=1$”$\Rightarrow$ “$\omega$
extremal”$\Rightarrow$ “the support projection of $\omega$ satisfies
$\dim\widetilde{\mathcal{M}}=1$”.

Indeed, $p$ is minimal iff, for any projection $q$, $0\leq q\leq p$ implies
$q=0$ or $q=p$. This is equivalent to $0$ and $p$ being the only projections
in $\widetilde{\mathcal{M}}$, i.e., to $\dim\widetilde{\mathcal{M}}=1$.

In this case consider the state $\omega$. Then from
$\omega=\lambda\omega_{1}+(1-\lambda)\omega_{2}$ we conclude
$\omega_{1}\leq\lambda^{-1}\omega$, hence

$|\langle\omega_{1},x(1-p)\rangle|{{}^{2}}\leq\langle\omega_{1},(1-p)x^{*}x(1-p)\rangle\leq\lambda^{-1}\langle\omega,(1-p)x^{*}x(1-p)\rangle=0.$
(47)

But then
$\langle\omega_{1},x\rangle=\langle\omega_{1},pxp\rangle=\langle\omega,x\rangle\langle\omega_{1},p\rangle$
and $\langle\omega_{1},p\rangle=1$ by choosing
$x={\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$ in this equation. Hence
$\omega_{1}=\omega$, so $\omega$ is extremal.

Now let $\omega$ be extremal and normal, and let $p$ be its support, i.e., the
smallest projection such that $\langle\omega,p\rangle=1$. Then $\omega$
restricted to $\widetilde{\mathcal{M}}$ is also pure and, in addition,
faithful, i.e., $x\in\widetilde{\mathcal{M}}$ with
$\langle\omega,x^{*}x\rangle=0$ implies $x=0$. Indeed, the eigenprojection
$q\in\widetilde{\mathcal{M}}$ of $x^{*}x$ for the spectral set $\\{0\\}$ must
then satisfy $\langle\omega,q\rangle=1$. Hence on the one hand $q\leq p$,
because $q\in p{\mathcal{M}}p$, and $q\geq p$ by minimality of the support
projection $p$. Therefore $x^{*}x=0$.

So consider the GNS-representation $\pi_{\omega}$ of a faithful normal pure
state. Faithfulness implies that the representation is injective, and general
representation theorems [76, 1.16.2] imply that the image
$\pi_{\omega}(\widetilde{\mathcal{M}})$ is a von Neumann algebra. By purity,
$\pi_{\omega}(\widetilde{\mathcal{M}})$ is irreducible, so
$\pi_{\omega}(\widetilde{\mathcal{M}})={\mathcal{B}}({\mathcal{H}}_{\omega})$.
On the other hand, $\omega$ is given by a vector
$\Omega\in{\mathcal{H}}_{\omega}$. So, unless $\dim{\mathcal{H}}_{\omega}=1$,
there is a vector orthogonal to it and hence a non-zero element $x$ with
$\pi_{\omega}(x)\Omega=0$, and hence $\langle\omega,x^{*}x\rangle=0$,
contradicting faithfulness. Hence $\dim{\mathcal{H}}_{\omega}=1$, and
$\dim\widetilde{\mathcal{M}}=1$. ∎

For many von Neumann algebras, this is a statement about the empty set, namely
when ${\mathcal{M}}$ is of type II or III, or $L^{\infty}(X,\mu)$ when there
are no points with positive $\mu$-measure. However, for a second dual, there
are many extreme points. Their central cover is the smallest central
projection $z_{\mathfrak{a}}$, so that $p\leq z_{\mathfrak{a}}$ for all
minimal projections, and $z_{\mathfrak{a}}{\mathcal{A}}^{**}$ is isomorphic to
the von Neumann algebra generated by the uncountable direct sum of all normal
pure state representations, called the atomic representation of
${\mathcal{A}}^{**}$. This is the part that will be useful for a function
representation.

Indeed, consider for a moment the classical case
${\mathcal{A}}={\mathcal{C}}_{0}(X)$. Then there is a simple way to associate
with an element $A\in{\mathcal{A}}^{**}$ a function $\check{A}$ on $X$, namely
to evaluate $A$ in the pure state $\delta_{x}$, the point measure at $x\in X$,
setting $\check{A}(x)=\delta_{x}(A)$. However, when $A$ has support in the
complement of the atomic subspace (also called the diffuse subspace) we get
$\delta_{x}(A)p=pAp=0$, where $p$ is the projection associated with
$\delta_{x}$ via (46). Hence $\check{A}=0$, so the function $\check{A}$ has
nothing to say about $A$.

Nevertheless, for suitable subalgebras of ${\mathcal{A}}^{**}$ the atomic
representation, and hence the function representation $\check{A}$ contains
full information. The idea of [72] is to use monotone limits to construct
useful algebras with this property. Since these constructions work in the same
way in arbitrary C*-algebras, they also serve to provide a Heisenberg picture
for general dual channels.

In ${\mathcal{A}}^{**}$ bounded, increasing nets are automatically
weak*-convergent, and if this algebra is represented on a Hilbert space, the
limits exist in the strong operator topology. This makes most sense in the
hermitian part ${\mathcal{A}}^{**}_{h}$ of ${\mathcal{A}}^{**}$. For any
subset $M\subset{\mathcal{A}}^{**}_{h}$ we denote by $M_{\uparrow}$ the set of
limit points of such nets from $M$. Similarly,
$M_{\downarrow}=-(-M)_{\uparrow}$ represents the limits of decreasing nets
from $M$.

######  Definition 11.

Let ${\mathcal{A}}$ be a C*-algebra. Then

  * •

The multiplier algebra of ${\mathcal{A}}$, denoted by $M({\mathcal{A}})$, is
the set of elements $m\in{\mathcal{A}}^{**}$ such that, for all
$a\in{\mathcal{A}}$, $ma\in{\mathcal{A}}$ and $am\in{\mathcal{A}}$.

  * •

${\mathcal{A}}_{\uparrow}:=({\mathcal{A}}_{h}+{\mathbb{R}}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}})_{\uparrow}$
is called the lower semicontinuous cone of ${\mathcal{A}}_{h}^{**}$. The upper
semicontinuous cone is
${\mathcal{A}}_{\downarrow}=({\mathcal{A}}_{h}+{\mathbb{R}}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}})_{\downarrow}$

  * •

An element $a\in{\mathcal{A}}^{**}_{h}$ is called universally measurable if,
for every state $\omega\in{\mathcal{A}}^{*}$, and every $\varepsilon>0$, there
are $x\in{\mathcal{A}}_{\uparrow}$, $y\in{\mathcal{A}}_{\downarrow}$ such that
$x\leq a\leq y$, and $\omega(y-x)<\varepsilon$. The real vector space of
universally measurable elements is denoted by ${\mathcal{U}}({\mathcal{A}})$.

We remark that there are some subtle distinctions in defining the
semicontinuous cone, depending on whether the unit is adjoined first (as
above) and on whether a norm closure of the cone is taken. These are discussed
carefully in [73, 74]. The main observations for us are that
${\mathcal{A}}_{\uparrow}\cup{\mathcal{A}}_{\downarrow}=M({\mathcal{A}})_{h}$
[14, Thm. III.6.24], and that the atomic representation is isometric on
${\mathcal{U}}({\mathcal{A}})$ [14, Thm. III.6.37].

In the classical case, ${\mathcal{A}}={\mathcal{C}}_{0}(X)$ with $X$ locally
compact, the lower semicontinuous cone consists just of the bounded lower
semicontinuous functions $f$ in the sense of point set topology (lower level
sets $\\{x|f(x)\leq a\\}$ are closed). The multipliers are
$M({\mathcal{A}})={{\mathcal{C}}_{\mathrm{b}}}(X)$, all bounded continuous
functions. For the universally measurable functions, note that, for a fixed
measure $\mu\in{\mathcal{C}}_{0}(X)^{*}$, by definition, all bounded Borel
measurable functions can be integrated. However, one usually completes the
Borel algebra by including all $\mu$-null sets. The completion can be
understood by adding all sets which can be approximated from above and below
by Borel measurable sets, whose $\mu$-volume differs by arbitrarily little.
The completion construction depends on $\mu$, but some sets will be added for
all $\mu$, and these are called universally measurable [77]. The functions
that are measurable for the completed $\sigma$-algebra are called
$\mu$-measurable, and their classes up to $\mu$-a.e. equality form
$L^{\infty}(X,\mu)$. The approximation from above and below for defining
$\mu$-measurable sets has its counterpart for functions in the definition
given above, with fixed $\mu=\omega$. Hence the universally measurable
functions are those that are $\mu$-measurable for all $\mu$.

For these subsets of observables, there is an automatic Heisenberg picture for
channels defined on states:

######  Lemma 12.

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be C*-algebras, and
$T:{\mathcal{A}}^{*}\to{\mathcal{B}}^{*}$ a linear map taking states to
states. Let $T^{*}:{\mathcal{B}}^{**}\to{\mathcal{A}}^{**}$. Then the
inclusions $T^{*}M({\mathcal{B}})\subset M({\mathcal{A}})$,
$T^{*}{\mathcal{B}}_{\uparrow}\subset{\mathcal{A}}_{\uparrow}$ and
$T^{*}{\mathcal{U}}({\mathcal{B}})\subset{\mathcal{U}}({\mathcal{A}})$ hold.

###### Proof.

Dual channels $T^{*}:{\mathcal{B}}^{**}\to{\mathcal{A}}^{**}$ preserve
positivity and normalization. The latter condition can be written as
$T^{*}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}={\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$.
They map increasing nets to increasing nets and are continuous for the
respective limits. Hence
$T^{*}{\mathcal{B}}_{\uparrow}\subset{\mathcal{A}}_{\uparrow}$. Then the
characterization of multipliers as both upper and lower continuous shows
$T^{*}M({\mathcal{B}})\subset M({\mathcal{A}})$. This is actually not so
obvious just from the definition of multipliers.

For the universally measurable class we proceed directly: Fix
$b\in{\mathcal{U}}({\mathcal{B}})$ and $\varepsilon>0$. Then by definition we
can find $b_{i}\in{\mathcal{B}}_{i}$ for $i=\uparrow,\downarrow$ such that
$b_{\uparrow}\leq b\leq b_{\downarrow}$ and
$(T\omega)(b_{\downarrow}-b_{\uparrow})\leq\varepsilon$ dualizing $T$ in the
last inequality and applying $T^{*}$ to the inequality for $b$ gives the
required upper and lower bounds $T^{*}b_{i}$ for $T^{*}b$. ∎

We note that such inclusions are always equivalent to a continuity condition
for $T$. If we chose subspaces
$\widetilde{\mathcal{A}}\subset{\mathcal{A}}^{**}$ and
$\widetilde{\mathcal{B}}\subset{\mathcal{B}}^{**}$, the inclusion
$T^{*}(\widetilde{\mathcal{B}})\subset\widetilde{\mathcal{A}}$ is equivalent
[69, IV.2.1] to the continuity with respect to the weak topologies
$\sigma({\mathcal{A}}^{*},\widetilde{\mathcal{A}})$ and
$\sigma({\mathcal{B}}^{*},\widetilde{\mathcal{A}})$, which are defined to make
just those linear functionals ${\mathcal{A}}\to{\mathbb{C}}$ continuous, which
are given by elements of $\widetilde{\mathcal{A}}$ (and similarly for
${\mathcal{B}}$).

###  3.2 Hybrid observables as functions

Let us now apply the ideas of the previous section to hybrids. The symplectic
form and the group theoretical structure of the phase space does not play a
key role here, so we are looking at a slightly more general case, where the
underlying C*-algebra is
${\mathcal{A}}={\mathcal{K}}\otimes{\mathcal{C}}_{0}(X)$, where
${\mathcal{K}}$ denotes the compact operators on a separable Hilbert space
${\mathcal{H}}$, and $X$ is a locally compact metrizable space generalizing
$\Xi_{0}$. As a above we can identify ${\mathcal{A}}$ with
${\mathcal{C}}_{0}(X,{\mathcal{K}})$, the functions $X\mapsto{\mathcal{K}}$
which are continuous in norm and vanishing in norm at infinity. Under this
identification $A=f\otimes K$ becomes the function $A(x)=f(x)K$. In [74, end
of intro.], this algebra is actually suggested as the intuition-building model
case for the theory we outlined in the previous section.

Going to the first dual ${\mathcal{A}}^{*}$, we find the states, and their
disintegration as in (13): We can write each state $\omega$ as

$\omega(f\otimes A)=:\langle\omega,f\otimes A\rangle=\int\mu(dx)m(x)\
f(x)\,\operatorname{tr}(\rho_{x}A),$ (48)

where $\mu$ is a (not necessarily finite) measure on $X$, $m\in L^{1}(X,\mu)$
is a probability density, so that $\mu m$ is an arbitrary probability measure.
This splitting of the classical marginal merely emphasized that we may realize
states absolutely continuous with respect to the same $\mu$ in the same
Hilbert space. In (48) it is clear that the integrand at each point $x$ is a
linear functional in $A\in{\mathcal{K}}$, hence given by a density operator
$\rho_{x}$. By taking linear combinations and norm limits we can write this in
terms of the operator valued function $A(x)=\sum_{j}f_{j}(x)A_{j}$ as

$\omega(A)=:\langle\omega,A\rangle=\int\mu(dx)m(x)\
\operatorname{tr}(\rho_{x}A(x)),$ (49)

or, in a useful shorthand notation,

$\omega=\int^{\oplus}\\!\\!\mu(dx)m(x)\ \rho_{x}.$ (50)

Equation (49) is the relation we want to extend to a much larger class of
operator valued functions $x\mapsto A(x)$.

It is worth noting that $\rho_{x}$ could be modified on $\mu$-null sets
without change, and for the sake of this integral expression, $A(x)$ might be
similarly modified. This is the hallmark of the $\mu$-dependent approach. The
states obtained with fixed $\mu$ are in the tensor product
${\mathcal{T}}({\mathcal{H}})\otimes L^{1}(X,\mu)$, the norm completion of the
span of the product states $\rho\otimes m$ for which $\rho_{x}\equiv\rho$ is
constant. We can also express this by the map

$i_{\mu}:{\mathcal{T}}({\mathcal{H}})\otimes
L^{1}(X,\mu)\to{\mathcal{A}}^{*}:\qquad\langle i_{\mu}(\rho\otimes
m),A\rangle=\int\mu(dx)m(x)\ \operatorname{tr}(\rho A(x)).$ (51)

This provides the first (“$\mu$-dependent”) way of associating operator valued
functions to elements of ${\mathcal{A}}^{**}$: For every
$A\in{\mathcal{A}}^{**}$ we can consider
$i_{\mu}^{*}(A)\in{\mathcal{B}}({\mathcal{H}}){\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}L^{\infty}(X,\mu)$,
the von Neumann algebra tensor product [14, Ch. IV.5], which is the dual of
${\mathcal{T}}({\mathcal{H}})\otimes L^{1}(X,\mu)$, and also the von Neumann
algebra generated by ${\mathcal{A}}$ in its representation on
${\mathcal{H}}\otimes L^{2}(X,\mu)$. Note that while $i_{\mu}^{*}$ is onto, it
is not injective: all details of $A$ related to $\mu$-null sets are
obliterated.

The $\mu$-free alternative is given by the formula

$\langle\rho\otimes\delta_{x},A\rangle=\operatorname{tr}\rho\check{A}(x)\quad\text{for}\
A\in{\mathcal{A}}^{**}.$ (52)

This is based on the observation, that for fixed $A$ the evaluation on the
left hand side is a bounded linear functional with respect to $\rho$, and thus
of the given form with a unique $\check{A}(x)\in{\mathcal{B}}({\mathcal{H}})$.
For fixed $x$ this extends the point evaluation at $x$, i.e., $({\rm
id}\otimes\delta_{x}):{\mathcal{K}}\otimes{\mathcal{C}}_{0}(X)\to{\mathcal{K}}$,
to functions in ${\mathcal{A}}$. It is an extension by continuity for the weak
topology induced by ${\mathcal{A}}^{*}$ so $\check{A}(x)=({\rm
id}\otimes\delta_{x})^{**}(A)$.

To contrast these approaches, take $\mu$ to be Lebesgue measure, or any other
“diffuse” measure assigning $\mu(\\{x\\})=0$ to every singleton. Let $z_{\mu}$
be the common support projection of the states $i_{\mu}(\rho\otimes m)$, i.e.,
the smallest projection in the W*-algebra ${\mathcal{A}}^{**}$ which gives
probability $1$ to all such states. Then, since $\delta_{x}$ and $\mu$ are
disjoint, their support projections in ${\mathcal{C}}_{0}(X)^{**}$ are
orthogonal, so $\check{z}_{\mu}=0$, although $i_{\mu}^{*}(z_{\mu})=1$.
Similarly, we can consider $z_{\mathfrak{a}}$, the smallest projection in
${\mathcal{A}}^{**}$ giving $1$ on all pure states
${|\psi\rangle\\!\langle\psi|}\otimes\delta_{x}$. In this case
$i_{\mu}^{*}(z_{\mathfrak{a}})=0$, although $\check{z}_{\mathfrak{a}}(x)=1$
for all $x$. So the two ways of assigning a function can be diametrically
opposite. On the other hand, for $A\in{\mathcal{A}}$, both approaches give the
continuous function representation that we started from, although
$i_{\mu}^{*}(A)$ is strictly speaking an equivalence class up to $\mu$-a.e.
equality. The question is then how far we can extend this agreement, if we
avoid “wild” elements like $z_{\mu}$ and $z_{\mathfrak{a}}$. In the following
proposition this is answered by the notion of universal measurability, and
illustrated in Fig. 3.

######  Proposition 13.

Let $A\in{\mathcal{U}}({\mathcal{A}})$, and let
$\omega=\int^{\oplus}\\!\\!\mu(dx)m(x)\ \rho_{x}\in{\mathcal{A}}^{*}$ be a
state. Then the function $x\mapsto\operatorname{tr}\rho_{x}\check{A}(x)$ is
$\mu$-measurable, and $\mu$-almost everywhere equal to $i_{\mu}^{*}(A)$.
Moreover,

$\langle\omega,A\rangle=\int\mu(dx)m(x)\
\operatorname{tr}(\rho_{x}\check{A}(x)).$ (53)

and $\|A\|=\sup_{x}\|\check{A}(x)\|$.

###### Proof.

Fix a state $\omega$, and consider an increasing net
$A_{i}\in{\mathcal{A}}_{h}+{\mathbb{R}}{\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}}$
with limit $A\in{\mathcal{A}}_{\uparrow}$. Then from (52) we find
$\check{A}(x)=\sup_{i}A_{i}(x)$. Also in (49) the limit exists and by the
monotone convergence theorem the integrand is indeed given by the pointwise
supremum, i.e., $\operatorname{tr}(\rho_{x}\check{A}(x))$. This shows the
claim for $A\in{\mathcal{A}}_{\uparrow}$, and, of course, for
$A\in{\mathcal{A}}_{\downarrow}$.

Now suppose $A\in{\mathcal{U}}({\mathcal{A}})$, and $\varepsilon>0$. Then we
can find $X\in{\mathcal{A}}_{\uparrow}$, $Y\in{\mathcal{A}}_{\downarrow}$ such
that $X\leq A\leq Y$ and $\langle\omega,Y-X\rangle\leq\varepsilon$. Let us
first consider the case $\omega=\rho\otimes m$ with fixed $\rho$. Then the
function $a_{\rho}(x)=\operatorname{tr}(\rho\check{A})$, and the similarly
defined functions for $X,Y$ (which are lower/upper semicontinuous satisfy
$x_{\rho}\leq a_{\rho}\leq y_{\rho}$ and their integrals with $m\mu$ differ by
less than $\varepsilon$. By Def. 11 we conclude that $a_{\rho}$ is universally
measurable. This establishes the $\mu$-measurability of the integrand in (53)
for constant $\rho$, and the formula itself. Hence
$i_{\mu}^{*}(A)(x)=\check{A}(x)$, $\mu$-almost everywhere.

Clearly, this extends to linear combinations $\sum_{i}\rho_{i}\otimes m_{i}$.
These are norm dense in ${\mathcal{T}}({\mathcal{H}})\otimes L^{1}(X,\mu)$.
Thus approximating a general state (50) by such step functions we find that
the functions $x\mapsto m(x)\,\operatorname{tr}(\rho_{x}\check{A}(x))$
converges in $L^{1}(X,\mu)$. So the limit is $\mu$-measurable and the formula
holds in general. We note that the function
$x\mapsto\operatorname{tr}(\rho_{x}\check{A}(x))$ is not universally
measurable, since the $\rho_{x}$ depend on $\mu$.

For the norm equality we invoke the result [14, Thm. III.6.37] that the atomic
representation is isometric on ${\mathcal{U}}({\mathcal{A}})$. Thus we only
have to show that $\sup_{x}\|\check{A}(x)\|=\sup_{\omega\,\rm
pure}\|\pi_{\omega}(A)\|$, where the supremum is over all pure states, and
$\pi_{\omega}$ denotes the associated GNS representation. Now the pure states
are of the form $\omega={|\psi\rangle\\!\langle\psi|}\otimes\delta_{x}$ with
$\psi\in{\mathcal{H}}$, so we only have to show that
$\|\check{A}(x)\|=\|\pi_{\omega}(A)\|$ for any pure state of this form, i.e.,
the right hand side does not depend on $\psi$. Indeed, we will show that even
$\pi_{\omega}(A)=\check{A}(x)$ up to the usual isomorphism around the GNS
representation: By the discussion after the definition (52) of $\check{A}(x)$,
$({\rm id}\otimes\delta_{x})^{**}$ is a normal representation of
${\mathcal{A}}^{**}$ on ${\mathcal{B}}({\mathcal{H}})$, which is obviously
irreducible, hence cyclic for any vector. Hence if we identify $\psi$ as the
GNS vector, it exactly meets the description of $\pi_{\omega}$. ∎

$\mu$-free
settingstatesobservables${\mathrm{C}}^{*}(X,{\mathcal{K}})^{**}$$\subset$${\mathcal{U}}(X,{\mathcal{K}})$$\subset$$M(X,{\mathcal{K}})$$\subset$${\mathrm{C}}^{*}(X,{\mathcal{K}})^{*}$${\mathrm{C}}^{*}(X,{\mathcal{K}})$$\mu$-dependent
settingstatesobservables${\mathcal{B}}\bigl{(}{\mathcal{H}}_{1}\otimes
L^{2}(X,\mu)\bigr{)}$$\subset$${\mathcal{B}}({\mathcal{H}}_{1}){\overset{{\rule{5.0pt}{1.0pt}}}{\otimes}}L^{\infty}(X,\mu)$$\subset$$M(X,{\mathcal{K}})$$\subset$${\mathcal{T}}^{1}({\mathcal{H}}_{1})\otimes
L^{1}(X,\mu)$${\mathcal{K}}({\mathcal{H}})\otimes\mathcal{C}_{0}(X)$$i_{\mu}^{*}$$\cong$$\check{}$
Figure 3: The two hybrid settings: The $\mu$-free setting is based only on the
C*-algebra ${\mathrm{C}}^{*}(X,{\mathcal{K}})$. In the $\mu$-dependent setting
this algebra is represented on the Hilbert space
${\mathcal{H}}_{\mu}={\mathcal{H}}_{1}\otimes L^{2}(X,\mu)$. On the level of
states the connection is given by the map $i_{\mu}$ taking the state space on
the far right to the one on the left (not shown). Its adjoint $i_{\mu}^{*}$
maps all the $\mu$-free observable spaces are mapped to their $\mu$-dependent
counterparts. $i_{\mu}^{*}$ is surjective from
${\mathcal{U}}(X,{\mathcal{K}})$, indicated by a double arrow tip. This map is
also realized by the function $A\mapsto\check{A}$ from (52) (cf. Prop. 13). It
is injective on $M(X,{\mathcal{K}})$ if $\mu$ has full support.

######  Proposition 14.

Let $X$ be locally compact, ${\mathcal{K}}$ the algebra of compact operators
on a separable Hilbert space ${\mathcal{H}}_{1}$, and set
${\mathcal{A}}={\mathcal{C}}_{0}(X,{\mathcal{K}})$. Then for $A\in
M({\mathcal{A}})$ the function $\check{A}$ is strong*-continuous, i.e.,
$x\mapsto\check{A}(x)\psi$ and $x\mapsto\check{A}(x)^{*}\psi$ are both
continuous for all $\psi\in{\mathcal{H}}$. Conversely, every uniformly bounded
function $\check{A}$ with this property defines a multiplier $A\in
M({\mathcal{A}})$.

###### Sketch of proof:.

This is found in [78, Cor 3.4]. We nevertheless sketch the basic ideas of an
approach not using the full-fledged theory.

According to Def. 11, multipliers can be thought of in terms of the operators

$L_{m},R_{m}:{\mathcal{A}}\rightarrow{\mathcal{A}},\,L_{m}(A)=mA,\quad
R_{m}(A)=Am.$ (54)

Their characteristic feature is $L_{m}(AB)=L_{m}(A)B$, and similarly for
$R_{m}$. We note that the whole concept is redundant for C*-algebras with a
unit, since then we just get multiplication with
$L_{m}({\leavevmode\mathrm{1}\mkern-5.4mu\mathrm{I}})=m\in{\mathcal{A}}$. This
suggests that the application of $L_{m}(U_{\lambda})$ for a bounded
approximate unit $U_{\lambda}$ may help to characterize the action of $L_{m}$.

The first observation in the hybrid context is that these operators act
pointwise, i.e., $\bigl{(}L_{m}(A)\bigr{)}(x)$ depends only on $A(x)$, or,
equivalently: $A(x)=0\Rightarrow(L_{m}A)(x)=0$.

To show this, assume $A(x)=0$, and take a bounded approximate unit
$U_{\lambda}\in{\mathcal{A}}$ with $\|U_{\lambda}A-A\|\rightarrow 0$ for all
$A\in{\mathcal{A}}$. Then, because $L_{m}$ and $U_{\lambda}$ are bounded, and
the product in ${\mathcal{A}}$ is defined pointwise:

$\displaystyle\|(L_{m}A)(x)\|$
$\displaystyle\leq\|(L_{m}(A-U_{\lambda}A))(x)\|+\|L_{m}(U_{\lambda})(x){A(x)}\|$
(55)
$\displaystyle\leq\|A-U_{\lambda}A\|+\|(L_{m}(U_{\lambda}))(x)\|\,\|A(x)\|.$
(56)

Then the second term is equal to zero, and the first goes to zero as
$\lambda\to 0$.

It follows that $L_{m}(A)(x)=L_{m}^{x}(A(x))$, where $L_{m}^{x}$ is a
multiplier of ${\mathcal{K}}$ in the sense that it satisfies the basic
relation $L_{m}^{x}(AB)=L_{m}^{x}(A)B$ for $A,B\in{\mathcal{K}}$. Now the
multiplier algebra of ${\mathcal{K}}$ is known to be
$M({\mathcal{K}})={\mathcal{B}}({\mathcal{H}})$. Indeed, for
$A={|\Phi\rangle\\!\langle\Psi|}$ with a unit vector $\Psi$, we get