# Asymptotic implementation of multipartite quantum channels and other quantum
instruments using local operations and classical communication

Scott M. Cohen<EMAIL_ADDRESS>Department of Physics, Portland State
University, Portland Oregon, USA 97201

###### Abstract

We prove a necessary condition that a quantum channel on a multipartite system
may be approximated arbitrarily closely using local operations and classical
communication (LOCC). We then extend those arguments to obtain a condition
that applies to all quantum instruments, which range from the most refined
case, a generalized measurement, to the most coarse-grained, which is a
quantum channel. We illustrate these results by a detailed analysis of a
quantum instrument that is known not to be implementable by LOCC, but which
can be arbitrarily closely approximated within that framework. As one
outgrowth of this analysis, we find a quantum measurement that falls into the
same category: it cannot be implemented exactly by LOCC, but can be
approximated by LOCC arbitrarily closely. This measurement has an infinite
number of outcomes, leaving open the question as to whether or not there
exists a measurement within this same category but having only a finite number
of outcomes.

###### pacs:

03.65.Ta, 03.67.Ac

##  I Introduction

In two recent papers [1, 2], we developed a new approach to the study of
quantum measurements implemented by multiple parties utilizing _local
operations_ in their respective laboratories and _classical communication_ to
their partners to share the results of those local operations, a fundamental
paradigm in quantum information processing commonly known as LOCC. We believe
our approach provides valuable insights and tools for analyzing what can be
accomplished under these circumstances, where quantum systems are not (under
practical conditions, often cannot be) transmitted from one party to another.
In the first of these papers [1], we identified certain geometric conditions
that must hold in order that a measurement can be approximated arbitrarily
closely by LOCC, and then in the second paper [2], we refined that approach
for use in studying local quantum state discrimination, showing how one can
often determine whether success with such a task is impossible, even while
knowing only the set of states to be discriminated but nothing about which
measurements, if any, might be possible candidates for successfully
accomplishing such a task. Here, we generalize our approach to address the
question of whether or not a given quantum channel can be closely approximated
using LOCC. We then extend the ideas further to apply to any quantum
instrument [3].

The paradigm of LOCC has been widely studied [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 13] since publication of the seminal paper of Peres
and Wootters [19]. While much progress has been made, and many important and
sometimes surprising results have been obtained, LOCC has also been widely
recognized as being quite difficult to characterize. One particularly
challenging aspect arises when one is interested in whether or not a task can
be accomplished while allowing an arbitrarily small error to be incurred in
doing so. In studying such questions, one must allow the parties to share
their outcomes back and forth for as many rounds of communication as needed in
order to reduce the error as much as possible. With some tasks that can be
easily accomplished by acting globally on the entire multipartite system, it
is known that when restricted to LOCC, even allowing for an infinite number of
rounds is not enough to reduce the error to zero. One such celebrated example
involves perfect discrimination of the so-called domino states [6], which
demonstrated that separable measurements—ones that involve measurement
operators that are simple tensor products on a multipartite system and can be
easily implemented globally—cannot always be implemented, or even closely
approximated, by LOCC, even with the use of an infinite number of
communication rounds. This discovery gave birth to a wide-ranging effort at
understanding the surprisingly difficult, and rich, questions concerning the
difference between global actions and those restricted to LOCC.

While LOCC has a rich and difficult to characterize structure, one commonly
discusses it in terms of a relatively simple picture, that of a rooted tree
graph, or _LOCC tree_. In the case of a finite LOCC tree and starting at the
root node, one may follow one of the tree’s many branches to its endpoint,
which is known as a leaf node. Each node along any given branch represents an
outcome of a (local) measurement performed by one of the parties, and the next
node along the branch represents an outcome obtained in a subsequent (local)
measurement by a different party. These nodes are connected by edges, and it
is easy to envision a path following this branch along these successive edges
from the root node all the way to its leaf. We recognized [1] that this path
is not just an abstract visualization of the sequence of outcomes, otherwise
having no useful meaning, but rather that it corresponds directly to a
particular type of path lying within a specific geometric object.
Significantly, that object is determined uniquely by the outcomes associated
with the leaf nodes appearing at the ends of the collection of all branches in
the tree. That is to say the measurement ${\cal M}$ implemented by the LOCC
protocol represented by this tree uniquely identifies a geometric object
$Z_{\cal M}$, within which lies each and every path associated with the
branches in the tree. We have characterized these paths [1] and used that
characterization to obtain a powerful condition, reproduced below as Theorem
1, which must hold if it is possible for ${\cal M}$ to be implemented
arbitrarily closely by LOCC.

Here, we extend that analysis beyond measurements to encompass the much more
general notion of quantum channels and indeed, any quantum instrument [3]. For
channels, we find a condition, see Theorem 2 below, which must hold if quantum
channel ${\cal E}$ can be closely approximated by LOCC. The condition is a
direct generalization of that for measurements in Theorem 1. It also involves
characterization of the paths discussed in the preceding paragraph, which in
similar fashion must lie within a more general geometric object $Z_{\cal E}$,
which itself is uniquely determined by ${\cal E}$. The difference in
structures of $Z_{\cal E}$ and $Z_{\cal M}$ reflects the fact that in
implementing a quantum channel, no classical information about outcomes can be
available, in contrast to measurements, where complete (classical) knowledge
of those outcomes must be known. Both channels and measurements are special
cases of quantum instruments, which make allowance for that classical
information to vary from all to none.

We have found that dealing with $Z_{\cal E}$ is more difficult as compared to
$Z_{\cal M}$. As we shall see, the latter is a special case of the former, and
it is this greater generality of $Z_{\cal E}$ which makes it harder to
analyze. Furthermore, whereas it was quite easy to find examples of previously
studied measurements where we could apply, and learn from, our condition for
measurements, Theorem 1, examples of quantum channels to which we can
fruitfully apply Theorem 2 are much more difficult to find. Perhaps the
explanation for this dearth of examples is that quantum channels—in
particular, multipartite quantum channels—are just more difficult to study, in
general, so comparatively little work has been done in this direction. One may
hope that the results presented here will stimulate further efforts in that
direction.

A dearth is not the same as a complete absence, however. In fact, there exists
a two-qubit example of a quantum instrument for which it is known that it
cannot be implemented exactly but can be approximated arbitrarily closely by
LOCC [3]. The inspiration for this instrument came from ideas developed in a
series of earlier papers on the random distillation of entanglement from a
three-qubit system onto two of those qubits [20, 21, 22, 23]. We study the
two-qubit example in detail below, first as a quantum channel, then as a
general instrument, and finally as a quantum measurement. This analysis leads
to insights regarding how the critical property of our paths, that of being
_piecewise local_ , see below, disappears in the limit as the number of rounds
tends to infinity. It also illustrates other properties of these paths, such
as the fact that certain matrices must have a block diagonal form determined
by the structure of the given instrument in terms of its constituent
completely positive (CP) maps. We also find from this example an interesting
result regarding the closure of LOCC measurements.

The relationship between LOCC and its closure, $\overline{\textrm{LOCC}}$, has
long been a point of interest to researchers. An early result along these
lines was the demonstration, in the context of random distillation of two-
party entanglement starting from the three qubit $W$ state, that a convergent
sequence of transformations is not achievable by LOCC in the limit [23].
Later, utilizing a two-party generalization of those arguments, it was shown
that LOCC is also not closed for two-qubit systems [3]. In each of these
cases, the parties must rely upon classical information obtained in these
protocols, leaving open the question of whether or not LOCC is closed when
considering quantum channels, for which no classical information is retained.
It has also been an open question whether LOCC is closed as applied to quantum
measurements, where the maximum amount of classical information is needed to
uniquely identify each and every measurement outcome. Here, in the context of
the two-qubit example just mentioned, we discover a measurement which is not
in LOCC but is within its closure. As far as we are aware, this is the first
example of a measurement exhibiting such a property.

The remainder of the paper is organized as follows. In the next section, we
provide an overview of the tools we will use in the subsequent sections,
including a discussion of quantum channels and instruments and their
mathematical representation, finite-round LOCC and its extension to infinite
rounds, and the geometric objects, $Z_{\cal M}$ and $Z_{\cal E}$, mentioned
above. In Section III, we present our main result, Theorem 2. Section IV
provides a proof of Theorem 2, which utilizes a result linking convergence of
a sequence of quantum channels to convergence of a sequence of the geometric
objects, $Z_{\cal E}$. The latter result is proved in Appendix A. In Section V
the reader will find an extensive analysis, in the context of Theorem 2, of
the two-qubit example channel from Ref. [3]. Section VI provides a
generalization of Theorem 2 to the case of arbitrary quantum instruments,
encompassing and interpolating between Theorem 1 and Theorem 2. There, the
reader will also find a brief discussion in this context of the two-qubit
example of Section IV. The important question of whether or not LOCC is a
closed set, a question that may depend on the type of circumstances to which
it is being applied, is addressed in Section VII where it is shown that LOCC
is not closed when applied to quantum measurements. Section VIII presents a
generalization of the example of Section IV to any number of qubits, and then
finally, we offer our conclusions in Section IX.

##  II Preliminaries

In this section, we describe basic ideas underlying the various tools we will
use for the analysis presented in this paper. We begin with a discussion of
quantum channels and instruments, continue with local operations and classical
communication (LOCC), followed by a discussion of infinite-round LOCC, and
finally, the geometric objects we will need, known as zonoids.

###  II.1 Quantum Channels, Measurements, and Instruments

Let ${\cal H}$ be a Hilbert space of finite dimension $d$. Denote the set of
bounded linear operators acting on ${\cal H}$ as ${\cal B}({\cal H})$. A
quantum channel ${\cal E}$ is a completely positive, trace-preserving (CPTP)
linear map which sends density operators $\rho\in{\cal B}({\cal H})$ to
density operators ${\cal E}(\rho)\in{\cal B}({\cal H}_{o})$, where a density
operator $\rho$ is positive semidefinite and has trace equal to unity. Any
channel ${\cal E}$ may be represented as follows. Consider a trio of Hilbert
spaces, ${\cal H},{\cal H}_{o},{\cal H}_{e}$, and an isometry $V$, so that
$V^{\dagger}V=I_{{\cal H}}$, $V:{\cal H}\to{\cal H}_{o}\otimes{\cal H}_{e}$,
with $I_{{\cal H}}$ the identity operator on ${\cal H}$. This maps operator
$\rho\in{\cal B}({\cal H})$ to operator $V\rho V^{\dagger}\in{\cal B}({\cal
H}_{o})\otimes{\cal B}({\cal H}_{e})$. ${\cal H}$ describes the input to the
channel, ${\cal H}_{o}$ the output, and ${\cal H}_{e}$ is commonly referred to
as the environment, which can be viewed as introducing noise into the
evolution of the input system as it evolves to the output. Taking partial
traces, we obtain two separate channels,

$\displaystyle{\cal E}(\rho)$ $\displaystyle=\textrm{Tr}_{e}\left(V\rho
V^{\dagger}\right)$ $\displaystyle{\cal F}(\rho)$
$\displaystyle=\textrm{Tr}_{o}\left(V\rho V^{\dagger}\right).$ (1)

Channel ${\cal F}$, which maps density operators on ${\cal B}({\cal H})$ to
those on ${\cal B}({\cal H}_{e})$, is called the complementary channel with
reference to ${\cal E}$, and vice-versa. Note that output and environment
spaces for ${\cal F}$ are exchanged relative to those for ${\cal E}$, while
the inputs are the same. One can always write

$\displaystyle V=\sum_{j=1}^{d_{e}}|j\rangle_{e}\otimes
K_{j}=\sum_{l=1}^{d_{o}}|l\rangle_{o}\otimes Q_{l},$ (2)

where $|j\rangle_{e}\in{\cal H}_{e}$, while $K_{j}$ maps states on ${\cal H}$
to states on ${\cal H}_{o}$ and are referred to as Kraus operators [24] for
channel ${\cal E}$. Similarly, $|l\rangle_{o}\in{\cal H}_{o}$ while
$Q_{l}:{\cal H}\to{\cal H}_{e}$ and are Kraus operators for the complementary
channel, ${\cal F}$. From Eq. (2), it is easily seen that ${}_{o}\langle
l|K_{j}=~{}_{e}\langle j|Q_{l}$. This relation, along with the representation
of ${\cal E}$ and ${\cal F}$ by $V$, will be used in Appendix A.

In the main text of this paper, we will use an alternative representation for
a quantum channel, which may be obtained from Eq. (II.1) and Eq. (2) in terms
of Kraus operators as ${\cal E}(\rho)=\sum_{j}K_{j}\rho K_{j}^{\dagger}$ (and
${\cal F}(\rho)=\sum_{l}Q_{l}\rho Q_{l}^{\dagger}$). The choice of such a
representation is not unique. Rather, with $\kappa$ the Kraus rank, which
gives the minimum number of operators needed for a Kraus representation of
${\cal E}$, if $\\{\hat{K}_{m}\\}_{m=1}^{\kappa}$ is a minimal representation,
then the set $\\{K_{j}\\}$ is also a representation of ${\cal E}$ as long as
$K_{j}=\sum_{j}W_{jm}\hat{K}_{m}$ for every $j$, where $W$ is any isometry,
$W^{\dagger}W=I_{\kappa}$ and $I_{\kappa}$ is the $\kappa\times\kappa$
identity matrix. There are no constraints on individual Kraus operators, but
for channels, as a collection they must satisfy the condition
$\sum_{j}K_{j}^{\dagger}K_{j}=V^{\dagger}V=I_{{\cal H}}$. Any given set of
Kraus operators also represents a specific measurement, each $K_{j}$
transforming state $\rho\to K_{j}\rho K_{j}^{\dagger}$ with probability
$p_{j}=\textrm{Tr}\left(K_{j}^{\dagger}K_{j}\rho\right)$. When there is a
classical record providing index $j$, we will refer to the set $\\{K_{j}\\}$
as a _generalized measurement_. The collection of operators
$\\{K_{j}^{\dagger}K_{j}\\}$ constitutes a positive-operator valued measure
(POVM), which arises from a measurement after which the final state of the
system is discarded, and each $E_{j}=K_{j}^{\dagger}K_{j}$ is called a POVM
element.

A _quantum instrument_ is a set of completely positive (but generally not
trace-preserving) maps, ${\cal E}=\\{{\cal J}_{r}\\}$, so that ${\cal
J}_{r}(\rho)=\sum_{j\in S_{r}}K_{j}\rho K_{j}^{\dagger}$, and $S_{r}$ is a
subset of the full index set identifying those $K_{j}$ that are associated
with CP map ${\cal J}_{r}$. The collection of these CP maps is trace-
preserving, $\sum_{r}\sum_{j\in S_{r}}K_{j}^{\dagger}K_{j}=I_{\cal H}$. A
generalized measurement is equivalent to a fine-grained quantum instrument
having one and only one Kraus operator for each CP map, whereas more general
instruments may involve multiple Kraus operators for one or more of its CP
maps. For example, from a measurement involving three operators,
$K_{1},K_{2},K_{3}$, a fine-grained quantum instrument (generalized
measurement) is the collection ${\cal J}_{r}(\rho)=K_{r}\rho
K_{r}^{\dagger},~{}r=1,2,3$. On the other hand, the pair of CP maps, ${\cal
J}_{1}^{\prime}(\rho)=K_{1}\rho K_{1}^{\dagger}+K_{2}\rho K_{2}^{\dagger}$ and
${\cal J}_{2}^{\prime}(\rho)=K_{3}\rho K_{3}^{\dagger}$ together also
constitute a quantum instrument, being a coarse-grained version of the former.
When the sum over terms is unrestricted, the instrument involves only a single
CP map, which is then trace-preserving and is therefore, itself, a quantum
channel. On the other hand, every quantum instrument is uniquely associated
with a quantum channel through the identification,

$\displaystyle{\cal E}(\rho)=\sum_{r}{\cal J}_{r}(\rho)\otimes[r]_{c}.$ (3)

The projector $[r]_{c}$ is to be understood as a register holding the
classical information about the various CP maps, since measuring this register
in the $|r\rangle_{c}$ basis provides knowledge of which of these CP maps was
implemented. These channels are referred to as _quantum-classical_ maps.

###  II.2 Local Operations and Classical Communication

In this paper, we are concerned with quantum instruments on multipartite
systems, where a multipartite system on $P$ parties is described by a tensor
product Hilbert space, ${\cal H}=\bigotimes_{\alpha}{\cal H}_{\alpha}$,
$\alpha\in\\{A,B,\ldots\\}$, and we shall refer to ${\cal H}_{\alpha}$ as the
_local_ Hilbert space of party $\alpha$. We wish to study the question of
whether or not channel (or instrument) ${\cal E}$, acting on multipartite
system ${\cal H}$ can be closely approximated when the parties are restricted
to using local operations and classical communication. That is, can parties
$A,B,\ldots$ closely simulate ${\cal E}$ by making local measurements in their
individual laboratories, communicating the outcomes of those measurements to
the other parties, continuing in this way for as many rounds of measurement
and communication as they wish, and then at the very end, forgetting the
outcomes of those local measurements? To analyze these questions, it is common
to represent an LOCC protocol by a tree graph. The root of the tree
corresponds to the situation before any party has done anything, so we label
this node by the identity operator, $I_{\cal H}$. For each local measurement
by any given party, each outcome is represented by a node with an edge
extending upward in the tree (in the direction of the root) to the node
representing the situation just before that measurement was performed. The
latter node is known as the _parent_ of its collection of _child_ nodes. A
finite branch stretches from the root node to a leaf node, which has no
children, but a protocol may also have infinite branches, which since they do
not end, have no leaf nodes. Since these trees represent LOCC protocols, for
which only one party measures at a time, the difference between parent and
child is _local_. To see this explicitly suppose, as we have found it
productive to do [25, 17, 18, 1, 2], that each node in the tree is labeled by
the POVM element corresponding to the action of all parties up to that point
in the protocol. That is, if the parent node is labeled as the POVM element
${\cal A}\otimes{\cal B}$ and Alice measures next, for example, then the child
node will be ${\cal A}^{\prime}\otimes{\cal B}$; only Alice’s local part has
changed. Finally, we recall a lemma from [26], which applies to all finite-
round protocols where each node is labeled by a positive semidefinite
operator, as just described.

######  Lemma 1.

[26]Each node $n$ in a finite-round LOCC tree is equal to the sum of the
collection of all leaf nodes that are descendant from that node $n$.

The foregoing discussion may give the appearance that only POVMs are being
considered, but one can show that any generalized measurement consisting of
operators $K_{j}^{\prime}=U_{j}\sqrt{K_{j}^{\dagger}K_{j}}$, with
$U_{j}^{\dagger}U_{j}=I_{\cal H}$, can be implemented by essentially the same
LOCC protocol as that implementing POVM elements $K_{j}^{\dagger}K_{j}$, as
long as each $U_{j}$ is a product operator of the form $U_{jA}\otimes
U_{jB}\otimes\ldots$. This is because product operators are the only ones LOCC
protocols are capable of implementing, so since $K_{j}$ and $K_{j}^{\prime}$
must both be product operators, then $U_{j}$ must be one, as well. In order to
implement the $K_{j}^{\prime}$, the parties simply perform the same protocol
as that implementing the $K_{j}$, after which they add one last step at the
end, each party $\alpha$ performing $U_{j\alpha}$ after
$\sqrt{K_{j}^{\dagger}K_{j}}$ has been obtained (here viewing the original
protocol as implementing a generalized measurement rather than a POVM). Since
the reverse implication is obvious, that implementation of the $K_{j}$ implies
that the $K_{j}^{\dagger}K_{j}$ can also be implemented, we see that as long
as all the $U_{j}$ are product unitaries, then the generalized measurement can
be implemented by LOCC if and only if the corresponding POVM can be so
implemented. A couple of other points are worth noting. First, even though a
quantum channel can be represented by many alternative sets of Kraus
operators, any given LOCC protocol implements a specific such set, so may
equally well be viewed as having implemented the particular measurement
corresponding to that set. We will use this observation below and will refer
to each corresponding measurement ${\cal M}$ as being _compatible_ with the
given channel (or instrument) ${\cal E}$. Secondly, we note that the outcome
of each local measurement in an LOCC protocol must be communicated to the
other parties, and any given implementation of the protocol will follow a
single branch with a string of local outcomes leading to a given final
outcome, where these outcomes have all been communicated and therefore,
recorded. Technically, however, when the outcome is known, one has failed to
implement the channel. This is why, in discussing LOCC implementation of
quantum channels, we imagine that at the end of the protocol, the parties
_forget_ the collection of outcomes obtained in each of the individual local
measurements. Similarly, for implementing quantum instruments, other than the
fully fine-grained measurements, one forgets all classical information other
than index $r$ identifying CP map ${\cal J}_{r}$.

###  II.3 Asymptotic LOCC and convergence of quantum channels

To study what happens with _asymptotic LOCC_ , where an infinite number of
rounds is allowed, one considers the infinite-round protocol to be the limit
of a given sequence of finite-round protocols. As discussed in Ref. [3], there
are two distinct kinds of such sequences. In the first case, each protocol in
the sequence is obtained from the protocol immediately preceding it by adding
one (or more) additional rounds. Such protocols are still LOCC. Let us now
discuss the other case, in which the next protocol in the sequence again has
additional rounds but is also allowed to differ from its preceding protocol
even in earlier rounds. As such, apart from the assumption of convergence,
there need not be a specific relationship between successive protocols of the
sequence. With either type of asymptotic LOCC, if the instruments implemented
by the sequence of protocols have a limiting instrument, then we say that the
latter is in, or can be implemented by, the closure of LOCC, denoted
$\overline{\textrm{LOCC}}$. That is, if protocol ${\cal P}^{\nu}$ implements
instrument ${\cal E}^{\nu}$ and $\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$,
then we shall say that instrument ${\cal E}$ can be implemented by
$\overline{\textrm{LOCC}}$: ${\cal E}\in\overline{\textrm{LOCC}}$. What these
limits mean is that for every real $\epsilon>0$ there exists natural number
$\mu$ such that for every natural number $\nu>\mu$,

$\displaystyle\left\|{\cal E}-{\cal E}^{\nu}\right\|_{\diamond}<\epsilon,$ (4)

where the diamond norm distance is defined as

$\displaystyle\left\|{\cal E}-{\cal
E}^{\nu}\right\|_{\diamond}=\max_{\left\|X\right\|_{1}\leq 1}\left\|({\cal
I}\otimes{\cal E})X-({\cal I}\otimes{\cal E}^{\nu})X\right\|_{1},$ (5)

with ${\cal I}$ the identity channel on ${\cal B}({\cal H})$, $X$ is an
operator on ${\cal B}({\cal H})\otimes{\cal B}({\cal H})$, and
$\left\|\cdot\right\|_{1}$ is the trace norm,
$\left\|X\right\|_{1}=\textrm{Tr}\left(\sqrt{X^{\dagger}X}\right)$. A lower
bound for this distance may be obtained in terms of these channels’ Choi-
Jamiolkowski operators, $\Phi$ and $\Phi^{\nu}$, respectively. This is done by
choosing a maximally entangled state on ${\cal B}({\cal H})\otimes{\cal
B}({\cal H})$ in Eq. (5),
$X=|\Omega\rangle\langle\Omega|=\sum_{ij}|i\rangle\langle
j|\otimes|i\rangle\langle j|/d$, and noting the definition of the Choi-
Jamiolkowski operator for ${\cal E}$ as $\Phi=({\cal I}\otimes{\cal
E})|\Omega\rangle\langle\Omega|$, with an identical relation between
$\Phi^{\nu}$ and ${\cal E}^{\nu}$ (${\cal E}$ and ${\cal E}^{\nu}$ must act on
the same input Hilbert space if a sequence of the latter is to converge to the
former). Inserting this in Eq. (4) yields,

$\displaystyle\left\|\Phi-\Phi^{\nu}\right\|_{1}<\epsilon.$ (6)

Therefore, convergence of the channels to ${\cal E}$ implies convergence of
the Choi-Jamiolkowski operators to $\Phi$.

###  II.4 Zonotopes and Zonoids

We will make use of certain geometric objects known as zonoids, which are
limits of objects known as zonotopes. A zonotope is a closed, convex set, the
Minkowski sum of a finite number, $N$, of line segments, $[0,E_{j}]$, written
$Z=\sum_{j}[0,E_{j}]$. An alternative definition is
$Z=\left\\{z\left|z=\sum_{j}c_{j}E_{j},0\leq c_{j}\leq 1\right.\right\\}$.
Zonoids are limits of sequences of zonotopes as $N\to\infty$. We have utilized
zonotopes, $Z_{\cal M}$, generated by POVM elements $E_{j}\in{\cal M}$, so
that the $E_{j}$ are positive semidefinite operators in a Hilbert space,
${\cal B}({\cal H})$, the sum of which is equal to $I_{\cal H}$ [1, 2]. The
zonoids, $Z_{\cal E}$, used here will be generated by a minimal Kraus
representation, $\\{\hat{K}_{m}\\}$, of a quantum channel ${\cal E}$ in the
sense that

$\displaystyle Z_{\cal
E}=\left\\{z\left|z=\sum_{mm^{\prime}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}},I_{\kappa}\geq\hat{C}\geq
0\right.\right\\},$ (7)

with $\hat{C}$ a positive semidefinite $\kappa\times\kappa$ matrix, $\kappa$
is the Kraus rank of channel ${\cal E}$, and $I_{\kappa}$ is the
$\kappa\times\kappa$ identity matrix. Since positive semidefinite matrices can
always be diagonalized, we can write
$\hat{C}_{mm^{\prime}}=\sum_{j}W_{jm}^{\ast}c_{j}W_{jm^{\prime}}$ for some
isometry, $W$, so that $z$ in the above definition of $Z_{\cal E}$ becomes

$\displaystyle z$
$\displaystyle=\sum_{j}c_{j}\left(\sum_{m}W_{jm}^{\ast}\hat{K}_{m}^{\dagger}\right)\left(\sum_{m^{\prime}}W_{jm^{\prime}}\hat{K}_{m^{\prime}}\right)=\sum_{j}c_{j}K_{j}^{\dagger}K_{j}=\sum_{j}c_{j}E_{j},$
(8)

Since Eq. (8) holds for every set of coefficients, $\\{c_{j}\\}$ and every
isometry $W$, we see that if $z\in Z_{\cal M}$, then $z\in Z_{\cal E}$ as
well, and we have thus proved the following lemma.

######  Lemma 2.

Given zonoid $Z_{{\cal E}}$ as defined in Eq. (7) and zonotope $Z_{{\cal M}}$
for any ${\cal M}$ compatible with ${\cal E}$ in the sense that measurement
operators $K_{j}$ of ${\cal M}$ constitute a Kraus representation of ${\cal
E}$, then $Z_{\cal M}\subseteq Z_{\cal E}$.

The following two examples may provide the reader with a bit of intuition
about these geometric objects. First, consider a simple quantum channel on a
single qubit, $d=2$, represented by Kraus operators that are rank-$1$
projectors, $\\{[0],[1]\\}$, with $[\psi]=|\psi\rangle\langle\psi|$. Operators
$K_{l}=V_{l0}[0]+V_{l1}[1]$, with $l=1,\ldots,N$ for any $N$, also represent
this same quantum channel, as long as $V^{\dagger}V=I_{2}$, the $2\times 2$
identity matrix. Then, for any $z\in Z_{\cal E}$, we have that

$\displaystyle z=C_{00}[0]+C_{11}[1].$ (9)

Since $[0][1]$ vanishes, there is no contribution from the off-diagonal
entries of $C$. Thus, $Z_{\cal E}$ is generated by a finite number (two) of
line segments, these segments stretching from the zero operator to $[0]$ and
$[1]$, respectively. Therefore, $Z_{\cal E}$ is in fact a zonotope, rather
than a zonoid, and has the shape of a square.

The second example is again on a single qubit, in this case represented by the
two Kraus operators $|0\rangle\langle 0|,|0\rangle\langle 1|$. In this case,
any set of operators, $|0\rangle\langle\psi_{l}|$, satisfying
$\sum_{l}[\psi_{l}]=I_{2}$ also represents this same channel. Any $z\in
Z_{\cal E}$ is of the form

$\displaystyle z=C_{00}[0]+C_{01}|0\rangle\langle 1|+C_{10}|1\rangle\langle
0|+C_{11}[1],$ (10)

showing that similarly to $C$, the only constraint on $z$ is that $0\leq z\leq
I_{\cal H}$. Then, $Z_{\cal E}$ is a four-dimensional cone-like object which,
denoting the fourth dimension as $r=\textrm{Tr}\left(z\right)$, stretches from
$0\leq r\leq 2$, and every three-dimensional slice through $Z_{\cal E}$ and
perpendicular to the $r$-axis is a (Bloch-like) sphere of radius $r$. Given
these spherical cross-sections, $Z_{\cal E}$ can only be generated by $N$ line
segments in the limit of $N\to\infty$, so this object is a true zonoid.

###  II.5 Notation

A heads-up for the reader about notation: superscripts that are Greek letters,
such as $\mu,\nu$, are not exponents, but are labels denoting position in a
sequence; see for example, superscript $\nu$ in the various equations of
Section II.3, above. Those superscripts that are lowercase letters from the
Roman/Latin alphabet, such as $n$, represent exponents. Subscripts are always
just labels, and all lowercase Roman/Latin labels may be assumed to be natural
numbers.

##  III Approximating quantum channels by LOCC

In this section we present our main result, a necessary condition that ${\cal
E}\in\overline{\textrm{LOCC}}$, Theorem 2, the proof of which is given in the
subsequent section. We begin by recalling the main theorem of Ref. [1]. There,
we proved that if measurement ${\cal M}=\\{E_{j}\\}$ can be approximated by
LOCC arbitrarily closely, then for each $j$, there exists a continuous,
monotonic path, $\Pi_{j}(s)$, of positive semidefinite product operators from
$I_{\cal H}$ to each outcome $E_{j}$ of that measurement and lying entirely
within the zonotope, $Z_{{\cal M}}$, generated by those POVM elements,
$\\{E_{j}\\}$. Here, monotonicity means that the trace of $\Pi_{j}(s)$ is non-
increasing along the path. The proof of that result relied on the observation
that every finite-round LOCC protocol implements a measurement for which there
exists a continuous, monotonic, _piecewise local_ 111The notion of a piecewise
local path is borrowed from the piecewise constant curves we all learned about
when studying calculus. Instead of being constant, however, the pieces here
correspond to _local_ measurement outcomes and correspond to the edges between
a parent node and one of its children in an LOCC tree. Along this edge, only
one party’s part of the full measurement operator changes, justifying use of
the word local. See Ref. [1] for a more detailed explanation of this concept.
path of positive semidefinite product operators to each of the outcomes of
that measurement. Using this observation, it was then argued that from any
sequence of finite-round LOCC protocols implementing measurements ${\cal
M}^{\nu}$ for which $\lim_{\nu\to\infty}{\cal M}^{\nu}={\cal M}$, there exists
a sequence of piecewise local paths whose limit is a (not necessarily
piecewise local) path lying within $Z_{{\cal M}}$ and ending at one of the
outcomes of ${\cal M}$, and that this holds for each such outcome. The
resulting theorem reads as follows.

######  Theorem 1.

[1] If ${\cal M}\in\overline{\textrm{LOCC}}$, with measurement ${\cal M}$
consisting of POVM elements $E_{j}$, then for each $j$, there exists a
continuous, monotonic path of product operators from ${\cal I}_{\cal H}$ to a
point on the (half-open) line segment $(0,E_{j}]$, and this path lies entirely
within zonotope $Z_{\cal M}=\sum_{j}[0,E_{j}]$.

As argued above, the theorem holds whether one views ${\cal M}$ as a POVM or
as a generalized measurement. In either case, it is the POVM elements $E_{j}$
that determine zonotope $Z_{\cal M}$ and the endpoints of the consequent
paths.

In this section, we are concerned with quantum channels rather than
measurements. Since any LOCC protocol implementing channel ${\cal E}$ may just
as well be viewed as implementing the generalized measurement consisting of a
specific set of Kraus operators which represents ${\cal E}$, Theorem 1
provides a solid foundation for the following theorem, which is our main
result, providing a necessary condition that a quantum channel may be
simulated arbitrarily closely by LOCC.

######  Theorem 2.

Given quantum channel ${\cal E}$ acting on input space ${\cal B}({\cal H})$
and represented by the minimal set of $\kappa$ Kraus operators,
$\\{\hat{K}_{m}\\}$, then if ${\cal E}$ can be implemented by LOCC with
vanishingly small error, the following conditions must hold:

  1. 1.

There exists a set, $\\{\Pi_{j}(s)\\}$, of continuous, monotonic paths of
positive semidefinite product operators, each of which begins at $I_{\cal H}$,
the identity operator on ${\cal H}$, and ends at a positive semidefinite
product operator of the form,
$E_{j}=\sum_{mm^{\prime}}(\hat{C}_{j})_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$.

  2. 2.

Each $\hat{C}_{j}$ is a positive semidefinite matrix, $\hat{C}_{j}\geq 0$, and
may be chosen to have rank equal to unity, with the collection satisfying
$\sum_{j}\hat{C}_{j}=I_{\kappa}$, where $I_{\kappa}$ is the
$\kappa\times\kappa$ identity matrix and $\kappa$ is the Kraus rank of ${\cal
E}$.

  3. 3.

Each of these paths of product operators lies entirely within the zonoid,
$Z_{{\cal E}}$, consisting of all positive semidefinite linear combinations of
the operators, $\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$: $Z_{{\cal
E}}=\left\\{z\left|z=\sum_{mm^{\prime}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}},I_{\kappa}\geq\hat{C}\geq
0\right.\right\\}$, with $1\leq\textrm{rank}(\hat{C})\leq\kappa$.

Note that a _positive semidefinite linear combination_ simply means a linear
combination of the form given in the above definition of $Z_{{\cal E}}$, that
is, a linear combination where the coefficients constitute a positive
semidefinite matrix. In addition, the paths are parametrized by
$s=\textrm{Tr}\left(\Pi_{j}(s)\right)$, whose range is $d\geq s\geq 0$ with
$d$ the dimension of input Hilbert space ${\cal H}$; see [1] for a more
detailed explanation of this parametrization.

The condition of Theorem 2 that ${\cal E}$ can be implemented by LOCC with
vanishingly small error means there exists a sequence of finite-round
protocols ${\cal P}^{\nu}$ implementing quantum channel ${\cal E}^{\nu}$ such
that $\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$. Since as discussed in
Section II.2, each of these finite-round protocols may equally well be viewed
as implementing a measurement ${\cal M}^{\nu}$, which implies the existence of
the concomitant paths of product operators in Theorem 1 [1], this (at least)
suggests the existence of the set of limiting paths alluded to in Theorem 2.
Thus, one should not be surprised to find that the results for approximating
measurements by LOCC [1] can be generalized, as we do here, to apply in
similar fashion to approximating quantum channels by LOCC. One notable
difference was discussed in Section II.4, that while the approximation of
measurements involves paths lying entirely within a zonotope, $Z_{{\cal M}}$,
for the approximation of channels one must utilize the zonoid, $Z_{{\cal E}}$.

Recall from Section II.4 that in order to implement a quantum channel, say
${\cal E}^{\nu}$, by LOCC, one may implement any measurement ${\cal M}^{\nu}$
compatible with ${\cal E}^{\nu}$. This means that simply knowing one can
implement a given channel leaves one without knowledge of which compatible
measurement(s) can be implemented. Thus, we also do not know the set of Kraus
operators implemented, whereas that set is what determines the endpoints of
the paths introduced in Theorem 1 [1]. Therefore, we are left with the
question, how do we know what the endpoints of all the paths of Theorem 2 are?
The answer is, we do not. Instead, as is indicated in Item 2 of Theorem 2, we
only know that each of these endpoints corresponds to a rank-$1$ matrix
$\hat{C}_{j}$ and the collection of these matrices constitutes a rank-$1$
decomposition of the $\kappa\times\kappa$ identity matrix. This is the extent
to which we are able to characterize those endpoints without knowing in
advance what specific set of Kraus operators is to be implemented.
Nonetheless, this is sufficient to ensure that the resulting set of Kraus
operators are a representation of ${\cal E}$.

We now turn to the proof of Theorem 2.

##  IV Proof of Theorem 2

We begin the proof of Theorem 2 with the following lemma showing that
convergence of a sequence of quantum channels implies convergence of the
corresponding sequence of zonoids.

######  Lemma 3.

If there exists a sequence of quantum channels, ${\cal E}^{\nu}$ of Kraus rank
$\kappa^{\nu}$ and with minimal Kraus representation
$\\{\hat{K}_{m}^{\nu}\\}$, converging to channel ${\cal E}$ of Kraus rank
$\kappa$ and with minimal Kraus representation
$\\{\hat{K}_{m}\\}$—$\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$—then there
also exists a sequence of zonoids, $Z_{{\cal
E}^{\nu}}=\left\\{z^{\nu}\left|z^{\nu}=\sum_{mm^{\prime}}\hat{C}^{\nu}_{mm^{\prime}}\hat{K}_{m}^{\nu{\dagger}}\hat{K}_{m^{\prime}}^{\nu},I_{\kappa^{\nu}}\geq\hat{C}^{\nu}\geq
0\right.\right\\}$ associated with channel ${\cal E}^{\nu}$, which converges
to zonoid $Z_{\cal
E}=\left\\{z\left|z=\sum_{mm^{\prime}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{{\dagger}}\hat{K}_{m^{\prime}},I_{\kappa}\geq\hat{C}\geq
0\right.\right\\}$ associated with channel ${\cal E}$. That is,
$\lim_{\nu\to\infty}Z_{{\cal E}^{\nu}}=Z_{\cal E}$.

The proof is given in Appendix A.

The next step is to recall that each LOCC protocol ${\cal P}^{\nu}$, which
implements channel ${\cal E}^{\nu}$, may also be thought of as implementing a
POVM, ${\cal M}^{\nu}$, consisting of elements
$E_{j}^{\nu}=K_{j}^{\nu{\dagger}}K_{j}^{\nu}$, where with $W^{\nu}$ an
isometry, $K_{j}^{\nu}=\sum_{m}W_{jm}^{\nu}\hat{K}_{m}^{\nu}$ is a Kraus
operator in a collection representing channel ${\cal E}^{\nu}$. Therefore, by
Theorem 1, we know that there exists a continuous, monotonic path of product
operators from $I_{\cal H}$ to a point on the half-open line segment
$(0,E_{j}^{\nu}]$, for each outcome, $j$, and that these paths lie entirely
within the zonotope $Z_{{\cal M}^{\nu}}$ generated by the collection of these
operators, ${E_{j}^{\nu}}$. In fact, by Theorem $2$ of Ref. [1], these paths
are piecewise local. As discussed at the end of the preceding section, we do
not know ${\cal M}^{\nu}$. Therefore, we consider all possible ${\cal
M}^{\nu}$ compatible with channel ${\cal E}^{\nu}$, which means that instead
of considering our continuous paths as lying within zonotope $Z_{{\cal
M}^{\nu}}$, we instead view these paths as lying within zonoid $Z_{{\cal
E}^{\nu}}$. According to Lemma 2, $Z_{{\cal M}^{\nu}}\subseteq Z_{{\cal
E}^{\nu}}$ for every such ${\cal M}^{\nu}$, so this is always possible. We
could, of course, restrict to measurements that are implementable by LOCC, so
that such paths of product operators actually exist, but it is sufficient to
follow the easier route and include _all_ compatible measurements, as we do
here.

Thus, we have a sequence of zonoids, $\\{Z_{{\cal E}^{\nu}}\\}$, which
converges to zonoid $Z_{\cal E}$, and within each $Z_{{\cal E}^{\nu}}$, we
have a collection of continuous paths of product operators starting at
$I_{\cal H}$. This situation is very similar to the one encountered in Ref.
[1], including that the paths are all Lipschitz continuous with Lipschitz
constant equal to unity [1]. By the Arzéla-Ascoli theorem [27], this tells us
that there exists a subsequence of these paths which converges uniformly on
the range, $s\in[d,0]$, to a continuous limiting path. However, to this point,
we have many paths for each $\nu$ and have not actually identified a sequence
of paths to begin with. We now turn to the task of identifying such a
sequence. In fact we will identify, and characterize, many such sequences.

Whereas in our earlier work, we had a sequence of known measurements, ${\cal
M}^{\nu}$, whose limit was another known measurement, ${\cal M}$, here we do
not know what the ${\cal M}^{\nu}$ or ${\cal M}$ are. Instead, we only know
that each ${\cal M}^{\nu}$ is compatible with channel ${\cal E}^{\nu}$, the
sequence of which has the limit ${\cal E}$. In the previous case, knowing
${\cal M}^{\nu}$ meant that we also knew the final outcomes, $E_{j}^{\nu}$, of
the corresponding LOCC protocol, and also that there was an ordering of these
outcomes, provided by labels $\nu$ for fixed $j$, such that for each $j$, the
sequence $\\{E_{j}^{\nu}\\}_{\nu}$ satisfied
$\lim_{\nu\to\infty}E_{j}^{\nu}=E_{j}$, with $E_{j}$ an outcome of the
limiting measurement ${\cal M}$. This allowed us to identify sequences of
paths—i.e., those ending at $E_{j}^{\nu}$ with the aforementioned fixed label,
$j$—from which, according to Arzéla-Ascoli, it followed that there was a
limiting path from $I_{\cal H}$ to a point along $(0,E_{j}]$. In our present
case, with $\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$, we do not know
$E_{j}^{\nu}$ or the limiting outcomes $E_{j}$. Thus, we need a way to
partition the various paths we’ve identified as lying within the $Z_{{\cal
E}^{\nu}}$ so that each subset provides a sequence of paths whose limit we can
characterize. In particular, we wish to characterize the endpoints of the
limiting paths. We will discuss how to partition the paths forthwith. The
following lemma concerning the Kraus ranks of the ${\cal E}^{\nu}$ will be
useful, and it may be of interest in other contexts, as well.

######  Lemma 4.

Consider the sequence of quantum channels, $\\{{\cal E}^{\nu}\\}$, with the
Kraus rank of ${\cal E}^{\nu}$ being $\kappa^{\nu}$. If
$\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$ with Kraus rank of ${\cal E}$
equal to $\kappa$, then there exists $\mu$ such that $\kappa^{\nu}\geq\kappa$
for all $\nu>\mu$.

###### Proof.

We begin the proof by considering the Choi-Jamiolkowski operator $\Phi$
$(\Phi^{\nu})$ for ${\cal E}$ $({\cal E}^{\nu})$, defined below Eq. (5). It is
well known that the rank of $\Phi$ is equal to the Kraus rank $\kappa$ of
${\cal E}$, and similarly for $\Phi^{\nu}$. By the Eckart-Young-Mirsky low
rank approximation theorem [28], which holds for all unitarily invariant norms
[29], we have that when rank$(\Phi^{\nu})<\kappa$, then
$\left\|\Phi-\Phi^{\nu}\right\|_{1}\geq\sigma_{\kappa}$, where
$\sigma_{\kappa}$ is the $\kappa$th largest singular value of $\Phi$. Note
that since ${\cal E}$, and thus $\Phi$, is known, $\sigma_{\kappa}$ is some
fixed, positive number. Choose $\epsilon$ to satisfy
$\sigma_{\kappa}>\epsilon>0$ and then identify $\mu$ such that for every
$\nu>\mu$, $\epsilon>\left\|{\cal E}^{\nu}-{\cal E}\right\|_{\diamond}$. Since
$\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$, this is always possible.

The proof of the lemma now proceeds by contradiction. Suppose for some
$\nu>\mu$, $\kappa^{\nu}<\kappa$. Recalling from the discussion around Eqs.
(4) and (6) that $\left\|{\cal E}-{\cal
E}^{\nu}\right\|_{\diamond}\geq\left\|\Phi-\Phi^{\nu}\right\|_{1}$, then for
every $\epsilon<\sigma_{\kappa}$, we have

$\displaystyle\sigma_{\kappa}>\epsilon>\left\|{\cal E}-{\cal
E}^{\nu}\right\|_{\diamond}\geq\left\|\Phi-\Phi^{\nu}\right\|_{1}\geq\sigma_{\kappa},$
(11)

a contradiction. Therefore, either $\nu<\mu$ or $\kappa^{\nu}\geq\kappa$. That
is to say, for each $\epsilon>0$, there exists $\mu$ such that
$\kappa^{\nu}\geq\kappa$ for all $\nu>\mu$, and the proof is
complete.$\blacksquare$

The meaning of this result is clear: a sequence of low rank operators cannot
become arbitrarily close to an operator of higher rank. On the other hand, it
is clearly possible for higher rank operators to become closer and closer to
an operator of lower rank.

We now turn to a characterization of endpoints of our paths. To begin, notice
that each branch in finite-round LOCC protocol ${\cal P}^{\nu}$ ends at a leaf
node corresponding to POVM element $E_{j}^{\nu}$, or equivalently at
measurement operator,
$K_{j}^{\nu\prime}=V^{\nu}\sqrt{E_{j}^{\nu}}=\sum_{l}W_{jl}^{\nu}\hat{K}_{l}^{\nu}$,
where $V^{\nu}$ and $W^{\nu}$ are isometries (acting on different spaces, of
course) and the $\hat{K}_{l}^{\nu}$ constitute a minimal Kraus representation
of the quantum channel ${\cal E}^{\nu}$ implemented by ${\cal P}^{\nu}$. Then,

$\displaystyle E_{j}^{\nu}$
$\displaystyle=\sum_{ll^{\prime}}W_{jl}^{\nu\ast}W_{jl^{\prime}}^{\nu}\hat{K}_{l}^{\nu{\dagger}}\hat{K}_{l^{\prime}}^{\nu}$
$\displaystyle=\sum_{ll^{\prime}}(\hat{C}_{j}^{\nu})_{ll^{\prime}}\hat{K}_{l}^{\nu{\dagger}}\hat{K}_{l^{\prime}}^{\nu},$
(12)

with $\hat{C}_{j}^{\nu}=\vec{W}_{j}^{\nu}\vec{W}_{j}^{\nu{\dagger}}$ a
rank-$1$ matrix, and $\vec{W}_{j}^{\nu}$ is the $j$th column of isometry
$W^{\nu}$. Note that from Ref. [1], we know that each branch corresponds
directly to a continuous monotonic path of product operators lying entirely
within $Z_{{\cal M}^{\nu}}$, and thus within $Z_{{\cal E}^{\nu}}$. We see here
that each of these paths ends at a point corresponding to a rank-$1$ matrix,
$\hat{C}_{j}^{\nu}$.

Let us consider the collection of these endpoints and the paths
$\Pi_{j}^{\nu}(s)$ to each of them. Following Ref. [1], we parametrize these
paths by $s=\textrm{Tr}\left(\Pi_{j}^{\nu}(s)\right)$. Define $N^{\nu}$ to be
the number of these paths, equal to the number of leaf nodes,
$j=1,\ldots,N^{\nu},$ for each protocol ${\cal P}^{\nu}$. By the results of
Ref. [18], we can assume that $N^{\nu}$ is finite for all $\nu<\infty$.
Consider the first $\lambda$ of these protocols, $\nu=1,\ldots,\lambda$, and
set $N_{0}(\lambda)=\max_{\nu\leq\lambda}{N^{\nu}}$. For each $\nu$ such that
$N^{\nu}<N_{0}(\lambda)$, split up some of the paths into $p_{j}^{\nu}$
duplicates. We can do this splitting at the very final (local) measurement of
each path that is split. For example, if the final outcome of
$\Pi_{j}^{\nu}(s)$ is $E_{j}^{\nu}$, replace that single outcome by
$p_{j}^{\nu}$ individual outcomes, each of which is $E_{j}^{\nu}/p_{j}^{\nu}$.
This increases the number of outcomes of that final local measurement by
$p_{j}^{\nu}-1$. It doesn’t matter which final outcomes are duplicated, or how
many parts we choose to split each of them into, as long as we end up with a
new number of paths such that $N^{\nu\prime}=N_{0}(\lambda)$ for each $\nu$.
Note that each of these paths correspond to a branch in an LOCC protocol,
${\cal P}^{\nu\prime}$, which is only slightly modified from a branch in
${\cal P}^{\nu}$ by splitting a subset of the final outcomes as just
described. Redefine each of these final segments of the duplicated paths as
$\Pi_{j}^{\nu\prime}(s)=\Pi_{j}^{\nu}(s)/p_{j}^{\nu}$, for $s$ in the
appropriate range for that final local measurement (the paths are not altered
for the remainder of the range of $s$). This maintains the condition that the
sum of POVM elements corresponding to the entire collection of leaf nodes for
protocol ${\cal P}^{\nu\prime}$ is equal to $I_{\cal H}$, as it must be. We
now have that each protocol ${\cal P}^{\nu\prime}$ has $N_{0}(\lambda)$ final
outcomes, with a continuous path $\Pi_{j}^{\nu\prime}(s)$ to each one, and
each of these final outcomes corresponds to a positive semidefinite operator
satisfying Eq. (IV) with matrix $\hat{C}_{j}^{\nu}$ also positive semidefinite
and having rank equal to unity. Since the sum of all final outcomes is equal
to $I_{\cal H}$, there is always a choice of these matrices
$\hat{C}_{j}^{\nu}$ such that they are a rank-$1$ decomposition of the
identity, $I_{\kappa^{\nu}}$ (if, for example, the
$K_{l}^{\nu{\dagger}}K_{l^{\prime}}^{\nu}$ are linearly dependent, then there
can be other possible choices for the $\hat{C}$-matrices, but we are
specifically looking for those that are rank-$1$).

For each $j=1,\ldots,N_{0}(\lambda)$, we have $\lambda$ paths,
$\Pi_{j}^{\nu}(s),~{}\nu=1,\ldots,\lambda$ (dropping the primes, for
convenience). Since $N_{0}(\lambda)$ is a maximum over the first $\lambda$
protocols, then $N_{0}(\lambda^{\prime})\geq N_{0}(\lambda)$ whenever
$\lambda^{\prime}>\lambda$; $N_{0}(\lambda)$ is a non-decreasing function of
$\lambda$. When incrementing $\lambda\to\lambda+1$, one path is added for each
value of $j\leq N_{0}(\lambda)$. If $N_{0}(\lambda+1)>N_{0}(\lambda)$,
additional values of $j$ are then added (by splitting outcomes as discussed in
the preceding paragraph) up to $N_{0}(\lambda+1)$, along with $\lambda+1$
paths for each new value of $j$, one from each of the protocols ${\cal
P}^{\nu}$ for $\nu=1,\ldots,\lambda+1$. It is possible that
$\hat{N}_{0}=\lim_{\lambda\to\infty}N_{0}(\lambda)$ diverges in the limit
$\lambda\to\infty$ (see Section V for a detailed analysis of such a case) but
in any event, we end up with an infinite number of paths for each
$j=1,\ldots,\hat{N}_{0}$ in this limit. For each path, we have a rank-$1$
matrix $\hat{C}_{j}^{\nu}$ associated with its endpoint. We will now argue
that for each $j$, the sequence of sets of matrices
$\\{\hat{C}_{j}^{\nu}\\}_{\nu}$ has a subsequence with a limit
$\\{\hat{C}_{j}\\}$ the collection of which is also a rank-$1$ decomposition
of the identity. This argument is somewhat complicated by the fact that
$\kappa^{\nu}$ can vary as $\nu$ increases.

For each $j$, we apply the Arzelà-Ascoli theorem [27] (see Ref. [1] for
details applicable to the present context) to obtain a limiting path, but we
make these limits for different $j$ consistent with each other as follows. For
$j=1$ there exists a subsequence, $S_{1}=\\{\nu_{11},\nu_{12},\ldots\\}$, of
paths $\Pi_{1}^{\nu_{1j}}(s)$ for which, by Arzelá-Ascoli, a limiting path of
product operators exists, denote it $\Pi_{1}(s)$. Since the endpoint of each
path in this subsequence of paths is associated with a rank-$1$ matrix
$\hat{C}_{1}^{\nu_{1j}}$, then by the Eckart-Young-Mirsky theorem [28], the
endpoint of the limiting path must also be represented by a rank-$1$ matrix
$\hat{C}_{1}$. Next, for $j=2$, consider only those paths $\Pi_{2}^{\nu}(s)$
and matrices $\hat{C}_{2}^{\nu}$ such that $\nu\in S_{1}$. Here, again by
Arzelá-Ascoli, there exists a subsequence
$S_{2}=\\{\nu_{21},\nu_{22},\ldots\\}$ of $S_{1}$, for which the corresponding
set of paths has a limiting path of product operators, $\Pi_{2}(s)$, and by
the same arguments used above for $S_{1}$, the endpoint of this limiting path
is again represented by a rank-$1$ matrix $\hat{C}_{2}$. Since $S_{2}$ is a
subsequence of $S_{1}$, we have that $\lim_{\nu\to\infty,\nu\in
S_{2}}\Pi_{1}^{\nu}(s)=\Pi_{1}(s)$, and $\lim_{\nu\to\infty,\nu\in
S_{2}}\hat{C}_{1}^{\nu}=\hat{C}_{1}$; these limits remain the same. Continuing
this process for each $j$, we find a subsequence $S_{j}$ of $S_{j-1}$ along
with a limiting path $\Pi_{j}(s)$ and rank-$1$ matrix $\hat{C}_{j}$.
Furthermore, the limiting paths and matrices obtained with subsequence
$S_{j^{\prime}}$ continue to be valid limits when restricted to $S_{j}$, for
all $j^{\prime}<j$. Since for each $\nu$,
$\sum_{j}\hat{C}_{j}^{\nu}=I_{\kappa^{\nu}}$, one might naïvely expect that
the set of limiting matrices $\hat{C}_{j}$ satisfies a similar condition,
$\sum_{j}\hat{C}_{j}=I_{\kappa}$. This conclusion is (essentially) correct,
but let us make the argument with care.

As discussed above, rank-$1$ matrices $\hat{C}_{j}^{\nu}$ obey the condition,
$\lim_{\nu\to\infty}\hat{C}_{j}^{\nu}=\hat{C}_{j}$, and their sum behaves as
$\lim_{\nu\to\infty}\sum_{j}\hat{C}_{j}^{\nu}=\lim_{\nu\to\infty}I_{\kappa^{\nu}}$.
And what is the limit of a sequence of identity operators? Note that
$I_{\kappa}-I_{\kappa^{\nu}}$ can never be small unless $\kappa^{\nu}=\kappa$.
From the proof of Lemma 4, we know that $\kappa^{\nu}>\kappa$ is possible, so
this equality may not hold. Nonetheless, we must recognize that this is not
the relevant limit for discussing convergence of a sequence of quantum
channels implemented by LOCC protocols. Rather, as we have shown above, what
is relevant are the operators,
$E_{j}^{\nu}=\sum_{mm^{\prime}}(\hat{C}_{j}^{\nu})_{mm^{\prime}}\hat{K}_{m}^{\nu{\dagger}}\hat{K}_{m^{\prime}}^{\nu}$.
Now,
$\sum_{j}E_{j}^{\nu}=\sum_{mm^{\prime}}(I_{\kappa^{\nu}})_{mm^{\prime}}\hat{K}_{m}^{\nu{\dagger}}\hat{K}_{m^{\prime}}^{\nu}=\sum_{m}\hat{K}_{m}^{\nu{\dagger}}\hat{K}_{m}^{\nu}=I_{\cal
H}$, as required. It may happen that as $\nu$ becomes large, the norm of
$\hat{K}_{m}^{\nu}$ may become negligible, tending asymptotically to zero. In
that case, even though $\sum_{j}\hat{C}_{j}^{\nu}=I_{\kappa^{\nu}}$, the only
non-negligible parts of the sum for $E_{j}^{\nu}$ (see a few sentences
previous to this one) correspond to matrix elements
$(\hat{C}_{j}^{\nu})_{mm^{\prime}}$ for which
$\left\|\hat{K}_{m}^{\nu}\right\|,\left\|\hat{K}_{m^{\prime}}^{\nu}\right\|$
do _not_ tend to zero. Under these circumstances, while
$\sum_{j}\hat{C}_{j}^{\nu}=I_{\kappa^{\nu}}$, it is only the part of
$I_{\kappa^{\nu}}$ lying in that subspace corresponding to the non-negligible
$\hat{K}_{m}^{\nu}$ that is relevant. Since by assumption the limit of this
sequence of channels exists and is equal to ${\cal E}$ of Kraus rank $\kappa$,
then according to these arguments that relevant subspace must be of size
$\kappa$. By Lemma 4, we have $\kappa^{\nu}\geq\kappa$ for large enough $\nu$,
which guarantees the existence of a subspace of size $\kappa$ within the
$\kappa^{\nu}$-dimensional space upon which $I_{\kappa^{\nu}}$ acts, for large
enough $\nu$. Therefore, $I_{\kappa^{\nu}}$ may (essentially) be replaced by
the projector onto this subspace, which is (essentially) equivalent to
$I_{\kappa}$. With this, the proof of Theorem 1 is complete.$\blacksquare$

##  V Detailed example of a quantum channel

In this section, we consider the example [3] of a two-qubit state
transformation which can be arbitrarily closely approximated, but cannot be
implemented exactly by LOCC. In Section VIII, we consider a generalization of
this example to any number of qubits. Consider a _quantum instrument_ [3],
which consists of the following three CP maps, the sum of which form the
quantum channel ${\cal E}$ we consider here, but are not individually trace
preserving,

$\displaystyle{\cal J}_{1}(\rho)$ $\displaystyle=[11]\rho[11]$
$\displaystyle{\cal J}_{2}(\rho)$
$\displaystyle=\sum_{i=1}^{2}(T_{i}\otimes[0])\rho(T_{i}^{\dagger}\otimes[0])$
$\displaystyle{\cal J}_{3}(\rho)$ $\displaystyle=\sum_{i=1}^{2}([0]\otimes
T_{i})\rho([0]\otimes T_{i}^{\dagger}),$ (13)

where $T_{1}=I/\sqrt{3}$ and $T_{2}=\left([0]+2[1]\right)/\sqrt{6}$. This
suggests describing channel ${\cal E}$ by the five product Kraus operators,

$\displaystyle K_{1}^{\prime}$ $\displaystyle=[11]$ $\displaystyle
K_{2}^{\prime}$ $\displaystyle=\sqrt{\frac{1}{3}}\left([00]+[10]\right)$
$\displaystyle K_{3}^{\prime}$
$\displaystyle=\sqrt{\frac{1}{6}}\left([00]+2[10]\right)$ $\displaystyle
K_{4}^{\prime}$ $\displaystyle=\sqrt{\frac{1}{3}}\left([00]+[01]\right)$
$\displaystyle K_{5}^{\prime}$
$\displaystyle=\sqrt{\frac{1}{6}}\left([00]+2[01]\right).$ (14)

It turns out that there is a smaller set of (product) Kraus operators that
represent the same channel. For the sake of simplicity, we will use this
smaller set, which is

$\displaystyle\hat{K}_{1}$ $\displaystyle=[11]$ $\displaystyle\hat{K}_{2}$
$\displaystyle=\frac{2}{3}[00]+[10]$ $\displaystyle\hat{K}_{3}$
$\displaystyle=\frac{2}{3}[00]+[01]$ $\displaystyle\hat{K}_{4}$
$\displaystyle=\frac{1}{3}[00].$ (15)

It is easily checked that $K_{j}^{\prime}=\sum_{j}W_{jl}\hat{K}_{l}$, with $W$
the isometry,

$\displaystyle W=\begin{pmatrix}1&0&0&0\\\
0&\sqrt{\frac{1}{3}}&0&\sqrt{\frac{1}{3}}\\\
0&\sqrt{\frac{2}{3}}&0&-\sqrt{\frac{1}{6}}\\\
0&0&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{3}}\\\
0&0&\sqrt{\frac{2}{3}}&-\sqrt{\frac{1}{6}}\end{pmatrix}.$ (16)

The following LOCC protocol [20, 3], ${\cal P}^{\nu}$, implements a quantum
channel, ${\cal E}^{\nu}$, which approximates the channel ${\cal E}$ of Eq.
(V). Given single qubit (Hermitian) operators,
$A_{0}=\sqrt{1-\epsilon}[0]_{A}+[1]_{A}$ and $A_{1}=\sqrt{\epsilon}[0]_{A}$,
Alice begins by measuring $\\{A_{0},A_{1}\\}$ on her system $A$, with
corresponding POVM elements ${\cal
A}_{j}=A_{j}^{\dagger}A_{j}=A_{j}^{2},~{}j=0,1$. She shares her outcome with
Bob, who follows with the same measurement—having POVM elements ${\cal
B}_{0}=(1-\epsilon)[0]_{B}+[1]_{B}$ and ${\cal B}_{1}=\epsilon[1]_{B}$—on his
system $B$, sharing his outcome in turn with Alice. They halt at this point
unless they both obtained outcome $0$, in which case they repeat their local
measurements as just described. Then, after a maximum of $\nu$ iterations of
this procedure, they stop. Noting that
$A_{0}^{n}=(1-\epsilon)^{n/2}[0]_{A}+[1]_{A}$ and
$A_{1}A_{0}^{n}=\epsilon^{1/2}(1-\epsilon)^{n/2}[0]_{A}$, and similarly for
Bob’s measurement operators, then the first few rounds of this LOCC
protocol—represented as a tree graph with nodes labeled by the corresponding
POVM elements implemented to that point in the protocol—is depicted in Figure
1 (note that the notation is condensed slightly in this figure by defining
$\eta=1-\epsilon$). In the figure, there is a single main branch, stretching
down along the upper right side, which continues (note ellipses at bottom
right) for $2\nu$ segments, while all other branches extend either one or two
segments off of that main branch and then terminate. As discussed in [3], the
limiting channel is ${\cal E}$, as desired, as long as we choose
$\epsilon=\nu^{-c}$ with $0<c<1$, which we will do in what follows.

Figure 1: Tree graph for the first several local measurements performed by
Alice and Bob in any one of the sequence of LOCC protocols described in the
text. Note that we have defined $\eta=1-\epsilon$.

Consider that main branch in each protocol and the sequence of these main
branches formed as we increment $\nu$. Each of these main branches describes a
path, $\Pi^{\nu}(s)$, of product operators in operator space, as described in
Sections III and IV. The sequence of such paths of product operators has a
limiting path, which we will now identify. Defining $s=\textrm{Tr}\left({\cal
A}_{0}^{n}\otimes{\cal B}_{0}^{n}\right)=[(1-\epsilon)^{n}+1]^{2}$, we have
$\sqrt{s}-1=(1-\epsilon)^{n}$, at least at those particular nodes along this
main branch. This suggests the quantity $\sqrt{s}-1$ may play a role. Noting
that we will be considering the limit as $\epsilon\to 0$, and that in this
limit, we will also have the length of the branch, $\nu\to\infty$, care must
be taken in considering how the quantity $(1-\epsilon)^{n}$ behaves in these
limits (since $n$ ranges up to $\nu$). Nonetheless, the quantity $\sqrt{s}-1$
is perfectly well-behaved in these limits, and in fact, we can prove the
following Lemma.

######  Lemma 5.

The limiting path of product operators obtained from the sequence of paths,
$\Pi^{\nu}(s)$, is $\Pi_{M}(s)=\lim_{\nu\to\infty}\Pi^{\nu}(s)=M(s)\otimes
M(s)$, $4\geq s\geq 1$, with $M(s)=\left(\sqrt{s}-1\right)[0]+[1]$.

Note that $\textrm{Tr}\left(\Pi_{M}(s)\right)=s$, so the path is parametrized
by the trace, as has been done in [1].

###### Proof.

Define $t_{n}=(1-\epsilon)^{n}+1$, $s_{2n}=\textrm{Tr}\left({\cal
A}_{0}^{n}\otimes{\cal B}_{0}^{n}\right)=t_{n}^{2}$ and
$s_{2n+1}=\textrm{Tr}\left({\cal A}_{0}^{n+1}\otimes{\cal
B}_{0}^{n}\right)=t_{n}t_{n+1}$. For each $\nu,\epsilon$, and $0\leq
n\leq\nu$, consider a piecewise local segment from ${\cal
A}_{0}^{n}\otimes{\cal B}_{0}^{n}$ to ${\cal A}_{0}^{n+1}\otimes{\cal
B}_{0}^{n}$ along the main branch of the $\nu$th protocol, with $s_{2n}\geq
s\geq s_{2n+1}$ or $t_{n}^{2}\geq s\geq t_{n}t_{n+1}$. This segment along the
path may be parametrized as

$\displaystyle\Pi^{\nu}(s)$
$\displaystyle=\frac{s-s_{2n+1}}{s_{2n}-s_{2n+1}}{\cal A}_{0}^{n}\otimes{\cal
B}_{0}^{n}+\frac{s_{2n}-s}{s_{2n}-s_{2n+1}}{\cal A}_{0}^{n+1}\otimes{\cal
B}_{0}^{n}$
$\displaystyle=\frac{1}{t_{n}(t_{n}-t_{n+1})}\left[(s-t_{n}t_{n+1})\begin{pmatrix}t_{n}-1&0\\\
0&1\end{pmatrix}+(t_{n}^{2}-s)\begin{pmatrix}t_{n+1}-1&0\\\
0&1\end{pmatrix}\right]\otimes\begin{pmatrix}t_{n}-1&0\\\ 0&1\end{pmatrix}$
$\displaystyle=\frac{1}{t_{n}}\begin{pmatrix}s-t_{n}&0\\\
0&t_{n}\end{pmatrix}\otimes\begin{pmatrix}t_{n}-1&0\\\ 0&1\end{pmatrix}.$ (17)

We need to show that for each $s$ and for all $\delta>0$, there exists $\mu$
such that for all $\nu>\mu$,
$\delta>\left\|\Pi^{\nu}(s)-\Pi_{M}(s)\right\|_{1}$. Consider

$\displaystyle t_{n}\langle 00|\left[\Pi^{\nu}(s)-\Pi_{M}(s)\right]|00\rangle$
$\displaystyle=(s-t_{n})(t_{n}-1)-t_{n}(\sqrt{s}-1)^{2}$
$\displaystyle=2t_{n}\sqrt{s}-t_{n}^{2}-s$ $\displaystyle\leq
2t_{n}\sqrt{s_{2n}}-t_{n}^{2}-s_{2n+1}=t_{n}^{2}-t_{n}t_{n+1}=t_{n}(t_{n}-t_{n+1}).$
(18)

Since $t_{n}-t_{n+1}=\epsilon(1-\epsilon)^{n}$, this is small. Similarly,

$\displaystyle t_{n}\langle 01|\left[\Pi_{M}(s)-\Pi^{\nu}(s)\right]|01\rangle$
$\displaystyle=t_{n}(\sqrt{s}-1)-(s-t_{n})$ $\displaystyle=t_{n}\sqrt{s}-s$
$\displaystyle\leq t_{n}(t_{n}-t_{n+1}),$ (19)

and it is easy to see that $\langle
10|\left[\Pi^{\nu}(s)-\Pi_{M}(s)\right]|10\rangle=t_{n}-\sqrt{s}\leq
t_{n}-t_{n+1}$, as well. Since the $11$ diagonal element vanishes identically,
we have that

$\displaystyle\left\|\Pi^{\nu}(s)-\Pi_{M}(s)\right\|_{1}\leq\sqrt{3}(t_{n}-t_{n+1})=\sqrt{3}\epsilon(1-\epsilon)^{n}\leq\sqrt{3}\epsilon=\sqrt{3}\nu^{-c}.$
(20)

Considering the segments from ${\cal A}_{0}^{n+1}\otimes{\cal B}_{0}^{n}$ to
${\cal A}_{0}^{n+1}\otimes{\cal B}_{0}^{n+1}$, corresponding to $s_{2n+1}\geq
s\geq s_{2n+2}$, one may follow the same argument as above—the only difference
being that $t_{n+1}\to t_{n+2}$ and $t_{n}\to t_{n+1}$—yielding the exact same
result. Since this encompasses all cases, then for the entire path along that
main branch, we may choose $\mu=\left(\delta/\sqrt{3}\right)^{-1/c}$ yielding
$\left\|\Pi^{\nu}(s)-\Pi_{M}(s)\right\|_{1}<\delta$ for all $\nu>\mu$ and for
all $s$, and this ends the proof.$\blacksquare$

It is instructive to understand what happens with the other outcomes of our
sequence of protocols. Recalling that $s=[(1-\epsilon)^{n}+1]^{2}$ and that
$\epsilon$ decreases as $\nu$ increases, the value of $s$ at which the $n$th
side-branch emerges from the main branch increases with $\nu$. One may
envision this as these side-branches sliding up the main branch toward the
root node in Fig. 1 as $\nu$ increases, as more and more of these side-
branches are added further down the main branch. In the limit $\nu\to\infty$,
the gap between each side-branch and its nearest neighbors vanishes, and we
have a continuous set of complete separable (generally, not LOCC222Note that
since the limiting main branch is not piecewise local—that is, each point
along this branch differs from the next in both party’s local operators, not
one party’s at a time—these measurements at each $s$ are not LOCC.)
measurements being made, one at each $s$, with one outcome being
$\Pi_{M}(s-\Delta s)$ and the other outcomes we will now determine. By taking
the limit as $\Delta s\to 0$, we see that it is simply neccessary to
differentiate the path along the main branch. This gives

$\displaystyle d\Pi_{M}(s)=\frac{ds}{2\sqrt{s}}\left([0]\otimes
M(s)+M(s)\otimes[0]\right),$ (21)

which is not a product operator, as it must be for the limiting channel to be
in $\overline{\textrm{LOCC}}$. However, we can simply split it into two parts,
each of which is clearly a product operator, providing two additional outcomes
for each of the continuous set of measurements appearing at the various values
of $s$. Notice the path from $\Pi_{M}(s)$ to either one of these parts is, in
fact, piecewise local, extending from $M(s)\otimes M(s)$ to either
$d\sqrt{s}[0]\otimes M(s)$ or $d\sqrt{s}M(s)\otimes[0]$. One of these outcomes
clearly corresponds to those of the form
$\epsilon(1-\epsilon)^{n}[0]\otimes{\cal B}_{0}^{n}$ in each of our sequence
of LOCC protocols, and the other to $\epsilon(1-\epsilon)^{n}{\cal
A}_{0}^{n}\otimes[0]$. One may conjecture that the finite-round parts of
(convergent) sequences of LOCC protocols always converge to piecewise local
paths, as we have now seen happens for this specific example, which also
demonstrates how the parts whose number of rounds tends to infinity can
converge to paths that are not piecewise local. Note that in the limit
$\epsilon\to 0$, there are no paths to outcomes proportional to $[00]$,
consistent with the observation (see Fig. 1) that in each of the sequence of
protocols, these outcomes are proportional to $\epsilon^{2}$ and so become
negligible. On the other hand, outcomes proportional to $\epsilon$ are not
negligible, since their number diverges as $\nu=\epsilon^{-1/c}$, and taken
together, they provide a significant contribution. To summarize then, the POVM
implemented by the protocol illustrated in Fig. 1 in the limit as
$\nu\to\infty$ ($\epsilon\to 0$) consists of the following operators,

$\displaystyle E_{1}$ $\displaystyle=[11]$ $\displaystyle E_{2}(s)$
$\displaystyle=d\sqrt{s}M(s)\otimes[0],~{}4\geq s\geq 1$ $\displaystyle
E_{3}(s)$ $\displaystyle=d\sqrt{s}[0]\otimes M(s),~{}4\geq s\geq 1.$ (22)

One immediately obtains from this the limiting generalized measurement
operators, for example $K_{2}(s)=\sqrt{E_{2}(s)}$, when viewing the sequence
of protocols as implementing generalized measurements. That is,

$\displaystyle K_{1}$ $\displaystyle=[11]$ $\displaystyle K_{2}(s)$
$\displaystyle=\sqrt{d\sqrt{s}M(s)\otimes[0]},~{}4\geq s\geq 1$ $\displaystyle
K_{3}(s)$ $\displaystyle=\sqrt{d\sqrt{s}[0]\otimes M(s)},~{}4\geq s\geq 1.$
(23)

We now calculate the associated Choi-Jamiolkowski operator to demonstrate that
this infinite set of limiting Kraus operators represents the desired channel
${\cal E}$, see Eq. (V). There is one limiting Kraus operator obtained at the
end of the main branch in the limit, which is $\sqrt{M(1)\otimes
M(1)}=[11]=\hat{K}_{1}$. In addition, within the range $4\geq s\geq 1$, we
have two infinite sets of Kraus operators, $K_{2}(s)$ and $K_{3}(s)$.

Define $\sigma=\sqrt{s}-1$ so that $d\sqrt{s}=d\sigma$, and let the Choi-
Jamiolkowski operator be written as $\Omega=\Omega_{1}+\Omega_{2}+\Omega_{3}$,
with

$\displaystyle\Omega_{1}=\sum_{w,x,y,z=0}^{1}|wx\rangle\langle
yz|^{A^{\prime}B^{\prime}}\otimes\hat{K}_{1}|wx\rangle\langle
yz|^{AB}\hat{K}_{1}=[1111],$ (24) $\displaystyle\Omega_{2}$
$\displaystyle=\sum_{w,x,y,z=0}^{1}|wx\rangle\langle
yz|^{A^{\prime}B^{\prime}}\otimes\int_{1}^{2}d\sqrt{s}\left(\sqrt{M(s)}\otimes[0]\right)|wx\rangle\langle
yz|^{AB}\left(\sqrt{M(s)}\otimes[0]\right)$
$\displaystyle=\int_{1}^{2}d\sigma\left(\sqrt{\sigma}|0000\rangle+|1010\rangle\right)\left(\sqrt{\sigma}\langle
0000|+\langle 1010|\right)$
$\displaystyle=\frac{1}{2}[0000]+\frac{2}{3}\left(|0000\rangle\langle
1010|+|1010\rangle\langle 0000|\right)+[1010],$ (25)

and in entirely similar fashion, we find that

$\displaystyle\Omega_{3}$
$\displaystyle=\frac{1}{2}[0000]+\frac{2}{3}\left(|0000\rangle\langle
0101|+|0101\rangle\langle 0000|\right)+[0101].$ (26)

Adding these together yields

$\displaystyle\Omega$ $\displaystyle=I_{{\cal H}\otimes{\cal
H}}+\frac{2}{3}\left(|0000\rangle\langle 0101|+|0101\rangle\langle
0000|+|0000\rangle\langle 1010|+|1010\rangle\langle 0000|\right)`,$ (27)

in agreement with the results of [3] for the Choi-Jamiolkowski operator of
${\cal E}$.333There seem to be typographical errors in [3] in their
calculation of this Choi-Jamiolkowski operator—we believe their $A^{\prime}A$
matrices should be $4\times 4$ instead of $2\times 2$—but their intended
meaning is clear.

Another way to demonstrate that these limiting Kraus operators represent the
desired channel ${\cal E}$ is by showing that they are related to the set in
Eq. (V) by an isometry. In this case, the isometry will have an infinite
number of rows. We have that one of these Kraus operators is $\hat{K}_{1}$, so
that the first row of our isometry is $[1~{}0~{}0~{}0]$. The other rows are
obtained from $K_{2}(s)$ and $K_{3}(s)$, written as

$\displaystyle K_{2}(s)$ $\displaystyle=\sqrt{d\sigma
M(s)\otimes[0]}=\sqrt{d\sigma}\left(\hat{K}_{2}+(3\sqrt{\sigma}-2)\hat{K}_{4}\right),$
$\displaystyle K_{3}(s)$ $\displaystyle=\sqrt{d\sigma[0]\otimes
M(s)}=\sqrt{d\sigma}\left(\hat{K}_{3}+(3\sqrt{\sigma}-2)\hat{K}_{4}\right).$
(28)

That is, we obtain a continuous set of additional rows from each of these:
$\sqrt{d\sigma}[0~{}1~{}0~{}(3\sqrt{\sigma}-2)]$ and
$\sqrt{d\sigma}[0~{}0~{}1~{}(3\sqrt{\sigma}-2)]$. As an isometry, all columns
must be orthonormal. The first column is $[1~{}0~{}0~{}\ldots~{}0]^{T}$, so
this column is normalized as well as orthogonal to the other columns, since
each of their first entries is zero. Normalization of the second and third
columns follows from the fact that $\int_{1}^{2}d\sigma=1$. Orthogonality of
this pair of columns is clear, since the non-zero entries in one are always
zero in the other. Normalization of the last column is

$\displaystyle 2\int_{1}^{2}d\sigma(3\sqrt{\sigma}-2)^{2}$
$\displaystyle=2\int_{0}^{1}d\sigma(9\sigma-12\sqrt{\sigma}+4)=1,$ (29)

where the factor of $2$ is included because there are two entries in this
column for each value of $s$, one from $K_{2}(s)$ and the other from
$K_{3}(s)$, see Eq. (V). Thus, we see that all columns are normalized.
Orthogonality of the last column to either of the middle two is

$\displaystyle\int_{0}^{1}d\sigma(3\sqrt{\sigma}-2)=0$ (30)

Thus, we do indeed see that this continuous set of Kraus operators is related
to those in Eq. (V) by an isometry.

As noted in Item 2 of Theorem 2, the endpoints of our limiting paths should be
described by positive semidefinite matrices which have rank equal to unity.
The current example illustrates this aspect, as well. We have that $d\sigma
M(s)\otimes[0]=\sum_{mn}C_{2}(s)_{mn}\hat{K}_{m}^{\dagger}\hat{K}_{n}$, with
the $\hat{K}_{m}$ given in Eq. (V) and

$\displaystyle C_{2}(s)=d\sigma\begin{pmatrix}0&0&0&0\\\
0&1&0&3\sqrt{\sigma}-2\\\ 0&0&0&0\\\
0&3\sqrt{\sigma}-2&0&\left(3\sqrt{\sigma}-2\right)^{2}\end{pmatrix}.$ (31)

Similarly, $d\sigma[0]\otimes
M(s)=\sum_{mn}C_{3}(s)_{mn}\hat{K}_{m}^{\dagger}\hat{K}_{n}$, with

$\displaystyle C_{3}(s)=d\sigma\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&1&3\sqrt{\sigma}-2\\\
0&0&3\sqrt{\sigma}-2&\left(3\sqrt{\sigma}-2\right)^{2}\end{pmatrix}.$ (32)

Both $C_{2}(s)$ and $C_{3}(s)$ are rank-$1$ matrices for every $s$. They also
provide an easy way to find a matrix $C_{1}(s)$ from which
$\Pi_{M}(s)=\sum_{mn}C_{1}(s)_{mn}\hat{K}_{m}^{\dagger}\hat{K}_{n}$, the path
along the (limit of the) main branch in Fig. 1, may be obtained. That is,
since the collection of these matrices must sum to $I_{\kappa}$, we have

$\displaystyle C_{1}(s)$
$\displaystyle=I_{4}-\int_{\sqrt{s}}^{2}\left(C_{2}(s)+C_{3}(s)\right)$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}+\sigma\begin{pmatrix}0&0&0&0\\\
0&1&0&2\left(\sqrt{\sigma}-1\right)\\\ 0&0&1&2\left(\sqrt{\sigma}-1\right)\\\
0&2\left(\sqrt{\sigma}-1\right)&2\left(\sqrt{\sigma}-1\right)&9\sqrt{s}-1-16\sqrt{\sigma}\end{pmatrix}.$
(33)

This is equal to the identity matrix at $\sigma=1$, of course, and since the
second term in the second line of Eq. (V) vanishes at $\sigma=0$, it leads to
the endpoint of the limiting main branch being described by
$\Pi_{M}(1)=\hat{K}_{1}^{\dagger}\hat{K}_{1}$, with
$\textrm{rank}\left(C_{1}(1)\right)=1$, as required. Furthermore, one
calculates using the set of operators in Eq. (V) that
$\sum_{mn}(C_{1}(s))_{mn}\hat{K}_{m}^{\dagger}\hat{K}_{n}=M(s)\otimes M(s)$
for all $s$, as it must.

Note that $C_{2}(s)$ and $C_{3}(s)$ correspond to the endpoints of side
branches that begin at position $s$ of the main branch. It is easily seen that
the entire length of these side branches may be represented by matrices
$C_{2}(s,x)=(1-x)C_{1}(s)+xC_{2}(s)$ and $C_{3}(s,x)=(1-x)C_{1}(s)+xC_{3}(x)$,
$0\leq x\leq 1$. Since these are convex combinations of positive semidefinite
matrices, they are themselves positive semidefinite for all $s$ and $x$.

According to the definition of $Z_{\cal E}$ in Item 3 of Theorem 1, the fact
that $C_{1}(s)$ and $C_{j}(s,x),~{}j=2,3,$ are all positive semidefinite
matrices shows that the paths associated with all these limiting measurement
outcomes lie entirely within $Z_{\cal E}$. It is not difficult to see that
most of the length of the paths associated with the main branch of the
individual LOCC protocols in our approximating sequence do not lie in $Z_{\cal
E}$, but only approach it in the limit. While the endpoints of those
approximating side branches do lie within $Z_{\cal E}$, they begin at the main
branch so are at least partly outside of $Z_{\cal E}$, as well.

##  VI Quantum instruments

Generalized measurements and quantum channels share similar characteristics,
both involving transformations of state $\rho$ to $K_{j}\rho K_{j}^{\dagger}$,
with the channel involving an unrestricted sum over the latter quantities,
there being no accessible classical information about the outcome $j$. Both
are special cases of the more general notion of a a quantum instrument, which
covers circumstances where some classical information is available but
possibly less than is required for a full generalized measurement. The proof
of Theorem 2 for channels in Section IV is also capable of encompassing all
quantum instruments as long as we make allowances for this required classical
information, which can be embedded in the form of the positive semidefinite
$C$-matrices introduced in Theorem 2, as we now discuss.

In the definition we have given here for zonoid $Z_{\cal E}$, sums over terms
of the form $\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$
appear. On the other hand, in our analysis of quantum measurements [1, 2], the
zonotopes used there involved simple, positive linear combinations of
$K_{m}^{\dagger}K_{m}$. In other words, for measurements, the $C$-matrices are
diagonal, $\hat{C}_{mm^{\prime}}=c_{m}\delta_{mm^{\prime}}$. We will now argue
that a simple generalization to block diagonal matrices will suffice for all
quantum instruments.

Recall that to account for all possible Kraus representations of ${\cal E}$,
we started with a positive linear combination,
$z=\sum_{j}c_{j}K_{j}^{\dagger}K_{j}$ with $\\{K_{j}\\}$ constituting an
arbitrary such representation, and then utilized isometric freedom,
$K_{j}=\sum_{m}W_{jm}\hat{K}_{m}$, to obtain
$z=\sum_{mm^{\prime}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$.
Suppose we have a quantum instrument ${\cal E}$ consisting of CP maps, ${\cal
J}_{r}$, for some set of indices, $r$. To understand what happens to isometric
freedom in this case, consider as illustration instrument ${\cal E}=\\{{\cal
J}_{1},{\cal J}_{2}\\}$, with ${\cal J}_{1}(\rho)=K_{1}\rho
K_{1}^{\dagger}+K_{2}\rho K_{2}^{\dagger}$ and ${\cal J}_{2}(\rho)=K_{3}\rho
K_{3}^{\dagger}+K_{4}\rho K_{4}^{\dagger}$. If an LOCC protocol has a final
outcome $K=\mu_{1}K_{1}+\mu_{3}K_{3}$, this will mix the two CP maps, in
general. This is so unless $K_{1}=\alpha K_{3}+\beta K_{4}$ (or
$K_{3}=\alpha^{\prime}K_{1}+\beta^{\prime}K_{2}$), in which case $K$ can be
written as a linear combination of only those Kraus operators involved in
expressing ${\cal J}_{2}$ (or ${\cal J}_{1}$), and then the parties can obtain
enough classical information to identify $K$ as contributing to one of the CP
maps and not the other. If such a separation between the CP maps is impossible
for one or more outcomes of the protocol, then there is no way to coarse-grain
the collection of outcomes such that each subset exactly represents one of
those CP maps. Thus for a successful implementation, it must be possible to
express each of the final outcomes in a way that does not mix the different CP
maps of the desired instrument ${\cal E}$. As the above example illustrates,
any linear dependence of the full set of Kraus operators allows for multiple
ways of expressing any given outcome, but nonetheless, there must be a choice
of these expressions which keeps the different CP maps of ${\cal E}$ separate.

Thus, we see that the full isometric freedom of quantum channels is no longer
in effect in this case, but it is still allowed within each ${\cal J}_{r}$.
That is, for example, if ${\cal E}=\\{{\cal J}_{r}\\}$ and ${\cal
J}_{r}(\rho)=\sum_{m\in S_{r}}\hat{K}_{m}\rho\hat{K}_{m}$, with
$\\{\hat{K}_{m}\\}_{m\in S_{r}}$ a minimal set of Kraus operators for ${\cal
J}_{r}$, we must be able to write any outcome $K_{j}$ in terms of one, and
only one, of these sets. That is for some fixed $r$, $K_{j}=\sum_{m\in
S_{r}}(W_{r})_{jm}\hat{K}_{m}$ with $W_{r}^{\dagger}W_{r}=I_{\kappa_{r}}$ the
$\kappa_{r}\times\kappa_{r}$ identity matrix and $\kappa_{r}$ is the Kraus
rank of ${\cal J}_{r}$. Note that this leads to
$K_{j}^{\dagger}K_{j}=\sum_{m,m^{\prime}\in
S_{r}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$ with
$\hat{C}_{mm^{\prime}}=(W_{r})_{jm}^{\ast}(W_{r})_{jm^{\prime}}$, so that $C$
is a rank-$1$ matrix, as required by Theorem 2. In addition, since these
outcomes are to be coarse-grained (collected together) to represent ${\cal
J}_{r}$, it must be that

$\displaystyle\sum_{j\in S_{r}^{\prime}}K_{j}\rho K_{j}=\sum_{m\in
S_{r}}\hat{K}_{m}\rho\hat{K}_{m}={\cal J}_{r}(\rho)$ (34)

for some fixed $r$ and index set $S_{r}^{\prime}$. This equality follows
directly from our identification of $W_{r}$ as an isometry acting on a
$\kappa_{r}$-dimensional space.

Defining the direct sum, $W=\oplus_{r}W_{r}$, we have that $W$ is a block-
diagonal isometry mapping the collection of _minimal_ (within each ${\cal
J}_{r}$) Kraus sets to a complete set of Kraus operators $\\{K_{j}\\}$ for the
full protocol implementing instrument ${\cal E}$. Since by Lemma 1 we have
that each node in a finite-round LOCC protocol is a positive linear
combination of the $K_{j}^{\dagger}K_{j}$, then each of these nodes—and
therefore, every point along our piecewise local paths identified from these
protocols—can be written as a positive semidefinite combination of the
$\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$ with the corresponding $C$-matrix
a block diagonal matrix, being itself a sum of block diagonal (rank-$1$)
matrices. Thus, the proof of Theorem 2 holds for the case of quantum
instruments, except that there is this additional constraint that the
$C$-matrices be block diagonal. These matrices keep the different CP maps
separate from each other, precluding any mixing between the Kraus operators
for ${\cal J}_{r}$ and those for ${\cal J}_{r^{\prime}}$ when $r\neq
r^{\prime}$.

These intuitive ideas, which are formalized in Appendix B, lead directly to a
generalization of Theorem 2, encompassing the most general notion of quantum
instruments. Note that to generalize the proof of Lemma 3 for this case, see
Appendix B for details, we simply add this block diagonal constraint on
isometries $W,W^{\nu}$ introduced just before—and matrices $C,C^{\nu}$
appearing in—Eq. (A), and the rest of the proof of this lemma goes through as
before.

######  Theorem 3.

Given quantum instrument ${\cal E}=\\{{\cal J}_{r}\\}$ acting on input space
${\cal B}({\cal H})$, with each ${\cal J}_{r}$ represented by the minimal set
of $\kappa_{r}$ Kraus operators, $\\{\hat{K}_{m}\\}_{m\in S_{r}}$, then if
${\cal E}$ can be implemented by LOCC with vanishingly small error, the
following conditions must hold:

  1. 1.

There exists a set, $\\{\Pi_{j}(s)\\}$, of continuous, monotonic paths of
positive semidefinite product operators, each of which begins at $I_{\cal H}$,
the identity operator on ${\cal H}$, and ends at a positive semidefinite
operator of the form,
$E_{j}=\sum_{mm^{\prime}}(\hat{C}_{j})_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$.

  2. 2.

Each $\hat{C}_{j}$ is a positive semidefinite matrix, $\hat{C}_{j}\geq 0$, and
may be chosen to have rank equal to unity, with the collection satisfying
$\sum_{j}\hat{C}_{j}=I_{\kappa}$, where $I_{\kappa}$ is the
$\kappa\times\kappa$ identity matrix and $\kappa=\sum_{r}\kappa_{r}$.

  3. 3.

Each of these paths of product operators lies entirely within the zonoid,
$Z_{{\cal E}}$, consisting of all positive semidefinite linear combinations of
the operators, $\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}}$: $Z_{{\cal
E}}=\left\\{z\left|z=\sum_{mm^{\prime}}\hat{C}_{mm^{\prime}}\hat{K}_{m}^{\dagger}\hat{K}_{m^{\prime}},I_{\kappa}\geq
C\geq 0\right.\right\\}$, with $1\leq\textrm{rank}(C)\leq\kappa$ and $C$ is
block diagonal, the $r$th diagonal block having size
$\kappa_{r}\times\kappa_{r}$.

  4. 4.

There exists a partition of the index set $j$ into subsets $S_{r}^{\prime}$
such that $\sum_{j\in S_{r}^{\prime}}\hat{C}_{j}=P_{\kappa_{r}}$, with
$P_{\kappa_{r}}$ a $\kappa\times\kappa$ matrix having the
$\kappa_{r}\times\kappa_{r}$ identity matrix embedded within its $r$th block
and everything else being zero. That is, every matrix element of
$P_{\kappa_{r}}$ vanishes except that the diagonal elements within the $r$th
block are equal to unity.

To illustrate this theorem, we return to the example of the preceding section.
The Kraus operators $K_{1},K_{2}(s),K_{3}(s)$ of Eq. (V) can be written in
terms of those of Eq. (V) as

$\displaystyle K_{1}$ $\displaystyle=K_{1}^{\prime}$ $\displaystyle K_{2}(s)$
$\displaystyle=\sqrt{d\sigma
M(s)\otimes[0]}=\sqrt{3d\sigma}\left[\left(2\sqrt{\sigma}-1\right)K_{2}^{\prime}+\sqrt{2}\left(1-\sqrt{\sigma}\right)K_{3}^{\prime}\right],$
$\displaystyle K_{3}(s)$ $\displaystyle=\sqrt{d\sigma[0]\otimes
M(s)}=\sqrt{3d\sigma}\left[\left(2\sqrt{\sigma}-1\right)K_{4}^{\prime}+\sqrt{2}\left(1-\sqrt{\sigma}\right)K_{5}^{\prime}\right].$
(35)

We see that these Kraus operators, obtained from the limit of our sequence of
protocols, are indeed related to those of the quantum instrument, see Eq. (V),
by a block diagonal isometry. The first diagonal block has size $1\times 1$,
whereas the next two blocks are each infinite in length and two columns wide,
and the five columns form an orthonormal set. This implies that the
$C$-matrices describing our paths are also block diagonal, as required by
Theorem 3. From another point of view, we may coarse-grain all outcomes
$K_{2}(s)$ together on the one hand, and all outcomes $K_{3}(s)$ on the other
to find

$\displaystyle\int K_{2}(s)\rho K_{2}(s)$
$\displaystyle=\int_{1}^{2}d\sigma\sqrt{M(s)}\otimes[0]\rho\sqrt{M(s)}\otimes[0]=K_{2}^{\prime}\rho
K_{2}^{\prime}+K_{3}^{\prime}\rho K_{3}^{\prime}={\cal J}_{2}(\rho)$
$\displaystyle\int K_{2}(s)\rho K_{3}(s)$
$\displaystyle=\int_{1}^{2}d\sigma[0]\otimes\sqrt{M(s)}\rho[0]\otimes\sqrt{M(s)}=K_{4}^{\prime}\rho
K_{4}^{\prime}+K_{5}^{\prime}\rho K_{5}^{\prime}={\cal J}_{3}(\rho),$ (36)

where we have used Eq. (VI). Thus, recalling that ${\cal
J}_{1}(\rho)=K_{1}^{\prime}\rho K_{1}^{\prime}$, then by retaining the
classical information about which outcome is obtained and when, one is able to
coarse-grain the outcomes to separately obtain each of the three CP maps of
the full quantum instrument.

##  VII LOCC$\neq\overline{\textrm{LOCC}}$

In this section, we consider the difference between LOCC and
$\overline{\textrm{LOCC}}$ for quantum channels and quantum measurements, the
extremes of the range of quantum instruments. For quantum channels, we are
unable to shed much light on this question. Certainly as we have seen in
Section V, while the side-branches of the limit of the LOCC protocols
implementing channels ${\cal E}^{\nu}$ are all piecewise local, the limiting
main branch, $\Pi_{M}(s)$, is not. At first blush, one may wonder if this
demonstrates that channel ${\cal E}$ cannot be exactly implemented by LOCC,
which requires a piecewise local path to each outcome, even if one or more
paths may contain an infinite number of pieces. However, it is possible that
there is a different set of paths, possibly to the same Kraus representation
of ${\cal E}$ or possibly to a different one, all of which are piecewise
local. We suspect that this is not the case but have not found a proof.

While we are presently unable to answer this question for quantum channels, we
are able to show, we believe for the first time, that LOCC is not closed when
considering implementation of quantum measurements, whether the measurements
are viewed as POVMs or as generalized measurements. That is, we have the
following theorem.

######  Theorem 4.

The set of LOCC measurements is not closed.

###### Proof.

We continue with the example of [23, 3], for which we have obtained the
limiting POVM, see Eq. (V), of the sequence of protocols illustrated in Fig.
1, and the associated limiting generalized measurement, see Eq. (V). Our
measurement operators for the limiting generalized measurement are $[11]$,
$\sqrt{d\sigma}[0]\otimes\sqrt{M(s)}$, and
$\sqrt{d\sigma}\sqrt{M(s)}\otimes[0]$, and recall that $0\leq\sigma\leq 1$ and
$\sigma=\sqrt{s}-1$. We wish to show that this measurement cannot be
implemented by LOCC, even though it is the limit of a sequence of LOCC
measurements. To show this, we use an argument very similar to the one used in
[3] for the same sequence of LOCC protocols, but there coarse-grained to the
quantum instrument of Eq. (V). That is, we will consider the action of this
generalized measurement—each element of which is tensored with the identity
operator on a third party, $C$—on the $W$ state,
$|W\rangle=\left(|001\rangle+|010\rangle+|100\rangle\right)/\sqrt{3}$. Then,
it is not difficult to see that

$\displaystyle(K_{1}\otimes I_{C})|W\rangle$ $\displaystyle=\left([11]\otimes
I_{C}\right)|W\rangle=0$ $\displaystyle(K_{2}(s)\otimes I_{C})|W\rangle$
$\displaystyle=\sqrt{d\sigma}\left([0]_{A}\otimes\sqrt{M(s)_{B}}\otimes
I_{C}\right)|W\rangle=\sqrt{\frac{d\sigma}{3}}[0]_{A}\otimes\left(\sqrt{\sigma}|01\rangle+|10\rangle\right)_{BC}$
$\displaystyle(K_{3}(s)\otimes I_{C})|W\rangle$
$\displaystyle=\sqrt{d\sigma}\left([0]_{B}\otimes\sqrt{M(s)_{A}}\otimes
I_{C}\right)|W\rangle=\sqrt{\frac{d\sigma}{3}}[0]_{B}\otimes\left(\sqrt{\sigma}|01\rangle+|10\rangle\right)_{AC}.$
(37)

The latter pair are a continuum of elements, and what we’re interested in are
outcomes in the form $\left(K_{2}(s)\otimes
I_{C}\right)[W]\left(K_{2}(s)\otimes I_{C}\right)^{\dagger}$, which we may sum
up by integration over $\sigma$. This yields

$\displaystyle\int_{0}^{1}\left(K_{2}(s)\otimes
I_{C}\right)[W]\left(K_{2}(s)\otimes
I_{C}\right)^{\dagger}=[0]_{A}\otimes\left([01]/3+4|01\rangle\langle
10|/9+4|10\rangle\langle 01|/9+2[10]/3\right)_{BC}$ (38)

with probability equal to one-half, and the exact same thing with the same
probability but with parties $A$ and $B$ interchanged for the term involving
$K_{3}(s)$. Given that Eq. (38) is equal to $\left({\cal J}_{2}\otimes{\cal
I}_{C}\right)([W])$, compare Eq. (VI) (and similarly for $\left({\cal
J}_{3}\otimes{\cal I}_{C}\right)([W])$; ${\cal I}_{C}$ is the identity channel
on the Hilbert space, ${\cal H}_{C}$, representing system $C$), this is in
agreement with the result obtained in Section $4.3$ of [3], from which they
showed by a concurrence argument [30] that this could not be achieved by LOCC.
That argument applies here, as well, proving that this is an example of a
generalized measurement that is in $\overline{\textrm{LOCC}}$ but not in LOCC.
To put it in simple terms, if this measurement could be implemented by LOCC,
then by coarse-graining, so could the instrument of Eq. (V), but this is known
to be impossible [3], so the result follows directly.

Note that if the corresponding POVM could be implemented within LOCC, then so
could the generalized measurement. Since we have just argued that the latter
is impossible, we have also now provided an example of a POVM that is in
$\overline{\textrm{LOCC}}$ but not in LOCC, and this ends the
proof.$\blacksquare$

It is perhaps of interest that this POVM involves an infinite number of
outcomes, as does the corresponding generalized measurement. Therefore, one
may ask whether or not LOCC is closed for measurements that are restricted to
have a finite number of outcomes. If the latter question can be answered in
the affirmative, it would show an interesting relationship between the closure
properties of LOCC and the number of outcomes allowed in a measurement. The
same relationship would then apparently also hold when considering the size of
Kraus representation needed to implement a quantum channel or quantum
instrument.

Note that these arguments require that one knows which outcomes are in the
set, $K_{2}(s)$, and which are in $K_{3}(s)$, treating the two sets separately
while coarse-graining over the range of $s$. Thus, this example also requires
classical information about the outcomes, so it does not apply to the full
quantum channel, ${\cal E}$. Therefore, the question of whether or not LOCC is
closed for implementing quantum channels remains open.

##  VIII Generalization of ${\cal E}$ to any number of qubits

Consider a $P$-qubit system with the parties performing the same local
measurements as described in Section V. That is, each of their local POVM
elements are $R=(1-\epsilon)[0]+[1]$ and $I-R$. The parties measure one after
the other, and if after any cycle of their $P$ local measurements, any one of
them has gotten the second outcome, they stop. Only if they all got $R$ do
they continue to the next cycle, and we shall refer to the continued sequence
of these cycles as _the main branch_ of the protocol and denote it as
$\Pi^{\nu}(s)$, as has been done in Section V for the two-qubit case. The
parties always stop after $\nu$ rounds, with $\epsilon=\nu^{-c}$ for some
$0<c<1$, even if they’ve always gotten the first outcome, having thus
implemented channel ${\cal E}^{\nu}$. The limiting paths have a form that are
a simple generalization of those for the two-qubit case.

######  Lemma 6.

In the limit $\nu\to\infty$ (equivalently, $\epsilon\to 0$), the main branch
of this protocol has a limit given by
$\Pi_{M}(s)=\lim_{\nu\to\infty}\Pi^{\nu}(s)=M(s)^{\otimes P}$, with
$M(s)=(\sqrt[P]{s}-1)[0]+[1]$.

The proof, which requires one to show that
$\lim_{\nu\to\infty}\left\|\Pi_{M}(s)-\Pi^{\nu}(s)\right\|_{1}=0$, is
straightforward and is left as an exercise for the reader. We share a few
hints. Note that after $n$ complete cycles followed by $l$ of the parties
having completed their part of the $(n+1)$st cycle, the segment of the
(piecewise local) path from this point to the point after the $(l+1)$st party
performs the next measurement may be written as

$\displaystyle\Pi^{\nu}(s)=\frac{s-t_{n+1}^{l+1}t_{n}^{P-l-1}}{t_{n+1}^{l}t_{n}^{P-l-1}(t_{n}-t_{n+1})}\left(R^{n+1}\right)^{\otimes
l}\left(R^{n}\right)^{\otimes(P-l)}+\frac{t_{n+1}^{l}t_{n}^{P-l}-s}{t_{n+1}^{l}t_{n}^{P-l-1}(t_{n}-t_{n+1})}\left(R^{n+1}\right)^{\otimes
l+1}\left(R^{n}\right)^{\otimes(P-l-1)},$ (39)

with $t_{n+1}^{l}t_{n}^{P-l}\geq s\geq t_{n+1}^{l+1}t_{n}^{P-l-1}$, and
$t_{n}=\textrm{Tr}\left(R^{n}\right)=\left(1-\epsilon\right)^{n}+1$. Since all
operators are diagonal, the proof of this lemma simply requires one to show
that $\left|\langle
N|\left(\Pi_{M}(s)-\Pi^{\nu}(s)\right)|N\rangle\right|\leq\epsilon P$ for
every $0\leq N\leq 2^{P}-1$, where using the binary expansion,
$N=b_{P-1}\ldots b_{1}b_{0}$ with $b_{j}=0$ or $1$, we define

$\displaystyle|N\rangle=|b_{0}\rangle|b_{1}\rangle\ldots|b_{P-1}\rangle.$ (40)

Recall that all non-zero local matrix elements are equal to unity unless that
party’s part of the binary expansion of $N$ is equal to $0$, in which case one
obtains a factor of a power of $\left(1-\epsilon\right)$; for example,
$\langle 0|R^{n}|0\rangle=(1-\epsilon)^{n}=t_{n}-1$.

As done in the case of $P=2$ above, we may differentiate $\Pi_{M}(s)$ to
obtain other possible Kraus operators for our limiting channel, ${\cal
E}_{P}$. This gives

$\displaystyle d\Pi_{M}(s)=d\\!\\!\sqrt[P]{s}\left([0]\otimes
M(s)^{\otimes(P-1)}+M(s)\otimes[0]\otimes
M(s)^{\otimes(P-2)}+\ldots+M(s)^{\otimes(P-1)}\otimes[0]\right),$ (41)

suggesting a separable Kraus representation as consisting of $[11\ldots 1]$
and each of the individual terms appearing in the preceding equation. In fact,
we can show this is correct by calculating the Choi-Jamiolkowski operator for
this conjectured channel, as well as the limit of those for the approximating
sequence of channels described above in terms of local operators $R$. In the
following, $|i\rangle,|j\rangle$ are to be understood as a binary expansion,
in similar fashion to that for $|N\rangle$ in Eq. (40), $l_{i}$ is the number
of $0$’s appearing in that binary expansion of $i$, and $m_{ij}$ counts the
number of positions where both $i$ and $j$ each have a zero. In other words,
$l_{i}=h(\neg i)$, the Hamming weight of $\neg{i}$ with $\neg$ denoting the
bitwise NOT function, and $m_{ij}=h\left(\neg(i|j)\right)$ with $|$ the
bitwise OR function. Then, we have

######  Lemma 7.

The Choi-Jamiolkowski operator for channel ${\cal E}_{P}$ and that for the
$\nu\to\infty$ limit of the above-described approximating channels, ${\cal
E}^{\nu}$, are both given by

$\displaystyle\Phi_{P}=2^{-P}[2^{P}-1]\otimes[2^{P}-1]+2^{-P}\sum_{i,j=0}^{2^{P}-2}|i\rangle\langle
j|\otimes\frac{2m_{ij}}{l_{i}+l_{j}}|i\rangle\langle j|,$ (42)

and since this is equal to $2^{-P}\sum_{i,j=0}^{2^{P}-1}|i\rangle\langle
j|\otimes{\cal E}_{P}(|i\rangle\langle j|)$, the corresponding channels are
given by

$\displaystyle{\cal E}_{P}(\rho)=\langle
2^{P}-1|\rho|2^{P}-1\rangle[2^{P}-1]+\sum_{i,j=0}^{2^{P}-2}\frac{2m_{ij}\rho_{ij}}{l_{i}+l_{j}}|i\rangle\langle
j|,$ (43)

with $\rho_{ij}=\langle i|\rho|j\rangle$.

The proof is given in Appendix C.

One may consider the action of ${\cal E}_{P}$ on the state
$[W]\otimes[11\ldots 1]$ where the first tensor factor is a projector onto the
$W$-state of three qubits and the second one is the tensor product of $P-3$
local (qubit) projectors onto the $|1\rangle$ state. Since the action of
$M(s)$ on $|1\rangle$ is identical to the action of the identity operator, as
is the action of $[1]$, whereas $[0]$ annihilates $|1\rangle$, the last $P-3$
parties’ actions are irrelevant to the evolution of the state of the other
three parties. It has been shown that the transformation ${\cal E}_{3}([W])$
cannot be achieved exactly by LOCC [23]. Therefore, it is also the case that
${\cal E}_{P}([W])$ cannot be implemented exactly by LOCC—since if it could,
then so could ${\cal E}_{3}([W])$—even though, as we’ve demonstrated above,
LOCC can be used to approximate ${\cal E}_{P}$ arbitrarily closely. Given the
results of [3, 23], these last points are perhaps unsurprising, but the
limiting measurement operators in Eq. (41) may be of some interest.

##  IX Conclusions

In this paper, we have extended our earlier result providing a necessary
condition that a quantum measurement can be closely approximated by LOCC [1,
2], to a similar condition which now applies to the full range of quantum
instruments from the most refined up to the fully coarse-grained case of
quantum channels. The proof of this new result includes a demonstration that
if a sequence of quantum instruments converges to some limiting instrument,
then there also exists a sequence of geometric objects known as zonoids, which
converges to a limiting zonoid. Each such zonoid is uniquely determined by the
instrument from which it is derived, and the limiting zonoid plays a critical
role in determining whether or not the limiting instrument can be closely
approximated by LOCC. When this approximation is possible, there must exist
certain paths of product operators lying within the limiting zonoid, see
Theorems 2 and 3.

We have studied in detail a particular quantum instrument on two-qubits,
${\cal E}$, which has been shown [3] to be the limit of a sequence of
instruments, each of which can be implemented by a finite-round LOCC protocol,
but that ${\cal E}$ itself cannot be exactly achieved using LOCC. Our analysis
shows why this is so. For each of the approximating instruments in the given
sequence, there exists a set of piecewise local paths to each of a set of
outcomes, where that set of outcomes can be coarse-grained to represent the
given instrument. When one considers the limit of these paths, however, one
finds that while they correspond to a set of outcomes representing ${\cal E}$,
one of the limiting paths fails to remain piecewise local in the limit. Since
for LOCC implementation, a crucial characteristic of these paths is that they
must be piecewise local, see the discussion above Theorem 1, this failure is
consistent with the finding that ${\cal E}$ cannot be exactly implemented by
LOCC [3].

We initially treated this example as a quantum channel, see Section V, and
found the Kraus representation of ${\cal E}$ corresponding to the limit of the
Kraus representations for the approximating channels ${\cal E}^{\nu}$. This
allowed us to identify the isometry which transforms the minimal
representation of ${\cal E}$ to the representation implemented in the limit,
which contains an infinite number of Kraus operators. Then after generalizing
Theorem 2 to the case of general quantum instruments, Theorem 3, we treated
this same example as a (less than fully coarse-grained) quantum instrument,
see Section VI. For the latter case, it was shown [3] that this instrument
cannot be exactly implemented by LOCC, but when considered as a channel, a
proof of this statement is still lacking. Thus, it remains an open question
whether LOCC$=\overline{\textrm{LOCC}}$ in the case of quantum channels. On
the other hand, we have succeeded in answering this same question for the case
of quantum measurements, which as far as we are aware, had not been previously
answered. That is, we have given an example of a quantum measurement taken
directly from the limiting Kraus representation for the above-mentioned ${\cal
E}$, which can be arbitrarily closely approximated, but not implemented
exactly, by LOCC. This measurement consists of an infinite number of outcomes.

We conjecture that LOCC is closed for measurements that are restricted to have
a finite number of outcomes. It is certainly possible this is false, but our
reasoning is as follows: That LOCC is not closed would mean that within
$Z_{\cal M}$, the zonotope associated with measurement ${\cal M}$, any
sequence of protocols whose limiting measurement is ${\cal M}$ leads to a
limiting path that is not piecewise local, since otherwise it would be LOCC.
But as shown in our example of Section V, obtaining a path that isn’t
piecewise local appears to require that the sequence of approximating LOCC
protocols involves paths that branch off at closer and closer intervals,
leaving a continuum of branches in the limit. This continuum of branches
implies a continuum of leaf nodes at the ends of those branches, and this
suggests an infinite number of outcomes. However, it is possible to have
multiple leaf nodes that are equal to each other, including possibly an
infinite number of them. Therefore, it is not entirely clear whether or not
the conjecture is correct. Of course, in our example we find that each leaf
node is unique, and it is at least conceivable that this must be true for any
case demonstrating this property of sequences of piecewise local paths having
a limiting path that is not piecewise local.

Finally, we would like to encourage further study of multipartite quantum
channels, for which there appear to be relatively few examples of physical
interest that have been studied in the literature. Hopefully, the results
presented here will stimulate new interest in exploring such avenues of
research.

Acknowledgments — We would like to thank Vlad Gheorghiu and Fei Shi for
comments on an early version of this manuscript.

##  Appendix A Convergence of channels implies convergence of the associated
zonoids

Suppose we have a sequence of LOCC protocols ${\cal P}^{\nu}$, each
implementing a quantum channel ${\cal E}^{\nu}$, and $\lim_{\nu\to\infty}{\cal
E}^{\nu}={\cal E}$, where ${\cal E}:{\cal B}({\cal H})\to{\cal B}({\cal
H}_{o})$ and ${\cal E}^{\nu}:{\cal B}({\cal H})\to{\cal B}({\cal
H}_{o}^{\nu})$, having recognized that the input Hilbert space ${\cal B}({\cal
H})$ must be the same for all these channels. (Recall that superscript $\nu$
denotes the position in the sequence of quantum channels and is not to be read
as an exponent.) That is, for every $\delta>0$, there exists natural number
$\mu$ such that for every $\nu>\mu$

$\displaystyle\delta/16>\left\|{\cal E}-{\cal E}^{\nu}\right\|_{\diamond},$
(44)

where $\left\|\cdot\right\|_{\diamond}$ is the diamond norm, see Eq. (47)
below. From the results of Ref. [31, 32], we have that

$\displaystyle\sqrt{\delta}/4>\sqrt{\left\|{\cal E}-{\cal
E}^{\nu}\right\|_{\diamond}}\geq\inf_{V,V^{\nu}}\left\|V-V^{\nu}\right\|_{\infty}\geq\frac{\left\|{\cal
E}-{\cal E}^{\nu}\right\|_{\diamond}}{\sqrt{\left\|{\cal
E}\right\|_{\diamond}}+\sqrt{\left\|{\cal E}^{\nu}\right\|_{\diamond}}}.$ (45)

where the operator norm, $\left\|X\right\|_{\infty}$, is equal to the largest
singular value of $X$, and the infimum appearing here is over all Stinespring
dilations [33], $V:{\cal H}\to{\cal H}_{o}\otimes{\cal H}_{e}$ and
$V^{\nu}:{\cal H}\to{\cal H}_{o}^{\nu}\otimes{\cal H}_{e}^{\nu}$, of channels
${\cal E}$ and ${\cal E}^{\nu}$, respectively. Below, we will make use of
channels, ${\cal F}:{\cal H}\to{\cal H}_{e}$ complementary to ${\cal E}$, and
${\cal F}^{\nu}:{\cal H}\to{\cal H}_{e}^{\nu}$ complementary to ${\cal
E}^{\nu}$, and without loss of generality, we will consider minimal
representations of ${\cal E}$ $({\cal E}^{\nu})$, so that the dimension of
${\cal H}_{e}$ $({\cal H}_{e}^{\nu})$ is equal to the Kraus rank $\kappa$ of
${\cal E}$ ($\kappa^{\nu}$ of ${\cal E}^{\nu}$). The dimensions of ${\cal
H},{\cal H}_{o}$ and ${\cal H}_{o}^{\nu}$ are $d,d_{o}$ and $d_{o}^{\nu}$,
respectively.

The trace norm of a quantum channel $T$ is defined as

$\displaystyle\left\|T\right\|_{1}=\max_{\left\|X\right\|_{1}\leq
1}\left\|T(X)\right\|_{1},$ (46)

where $\left\|X\right\|_{1}=\textrm{Tr}\left(\sqrt{X^{\dagger}X}\right)$ is
the trace norm of operator $X$. For $X\geq 0$, we have that
$\left\|X\right\|_{1}=\textrm{Tr}\left(X\right)$. With ${\cal I}_{\cal H}$ the
identity channel on input Hilbert space ${\cal H}$, recall the definition of
the diamond norm as

$\displaystyle\left\|T\right\|_{\diamond}=\left\|T\otimes{\cal I}_{\cal
H}\right\|_{1}=\max_{\left\|X\right\|_{1}\leq 1}\left\|(T\otimes{\cal I}_{\cal
H})X\right\|_{1},$ (47)

We can find a lower bound of this diamond norm over all quantum channels by
choosing $X=\rho\otimes\sigma$ with $\rho,\sigma$ each positive semidefinite
and having trace equal to unity, and then omitting the maximum in Eq. (47).
This tells us that for all quantum channels, $T$,

$\displaystyle\left\|T\right\|_{\diamond}\geq\left\|T(\rho)\otimes\sigma\right\|_{1}=\textrm{Tr}\left(T(\rho)\right)\textrm{Tr}\left(\sigma\right)=1,$
(48)

and we have used the fact that a quantum channel is trace-preserving. Using
this result, we may rewrite Eq. (45) as [32]

$\displaystyle\sqrt{\delta}/4>\sqrt{\left\|{\cal E}-{\cal
E}^{\nu}\right\|_{\diamond}}\geq\inf_{V,V^{\nu}}\left\|V-V^{\nu}\right\|_{\infty}\geq\frac{1}{2}\left\|{\cal
E}-{\cal E}^{\nu}\right\|_{\diamond}.$ (49)

From Eq. (44) we have an (arbitrarily small) upper bound proportional to
$\delta$ on $\left\|{\cal E}-{\cal E}^{\nu}\right\|_{\diamond}$, but we are
here seeking to find a similar bound on the Hausdorff distance between
corresponding zonotopes, $Z_{\cal E},Z_{{\cal E}^{\nu}}$. The following
arguments achieve that goal.

While the infimum in Eq. (49) may not be achieved, it is always arbitrarily
close to at least one point in the set over which it is taken. Therefore, for
every $\delta>0$ there exists Stinespring dilations
$\tilde{V},\tilde{V}^{\nu}$ of ${\cal E},{\cal E}^{\nu}$, respectively, such
that
$\sqrt{\delta}/2>\inf_{V,V^{\nu}}\left\|V-V^{\nu}\right\|_{\infty}+\sqrt{\delta}/4\geq\left\|\tilde{V}-\tilde{V}^{\nu}\right\|_{\infty}$.
Therefore,

$\displaystyle\sqrt{\delta}>2\left\|\tilde{V}-\tilde{V}^{\nu}\right\|_{\infty}\geq\left\|{\cal
E}-{\cal E}^{\nu}\right\|_{\diamond}.$ (50)

Since each of these isometries $\tilde{V}$ $(\tilde{V}^{\nu})$ is a dilation
not only of channel ${\cal E}$ $({\cal E}^{\nu})$ but also of a corresponding
complementary channel ${\cal F}$ $({\cal F}^{\nu})$, it also holds that [32]

$\displaystyle\sqrt{\delta}>2\left\|\tilde{V}-\tilde{V}^{\nu}\right\|_{\infty}\geq\left\|{\cal
F}-{\cal F}^{\nu}\right\|_{\diamond}.$ (51)

Since the diamond norm is a maximum over inputs $X$, see Eq. (47), we get a
lower bound on this norm by evaluating ${\cal F}-{\cal F}^{\nu}$ acting on any
density operator $X$, so choose $X=|\Omega\rangle\langle\Omega|$, where
$|\Omega\rangle=\sum_{i}|i\rangle\otimes|i\rangle/\sqrt{d}$ is a maximally
entangled state on input Hilbert space ${\cal H}\otimes{\cal H}$. Then, we
obtain

$\displaystyle\sqrt{\delta}>2\left\|\tilde{V}-\tilde{V}^{\nu}\right\|_{\infty}\geq\left\|\Phi-\Phi^{\nu}\right\|_{1},$
(52)

where $\Phi=(I_{\cal H}\otimes{\cal F})(|\Omega\rangle\langle\Omega|)$ is the
Choi-Jamiolkowski operator [34, 35] for channel ${\cal F}$, and similarly for
$\Phi^{\nu}$ and ${\cal F}^{\nu}$.

Consider the two Kraus representations $\\{K_{m}\\}$ of ${\cal E}$ with
$d_{e}$ members, and $\\{K_{m}^{\nu}\\}$ of ${\cal E}^{\nu}$ with
$d_{e}^{\nu}$ members. Define $\tilde{d}_{e}=\max(d_{e},d_{e}^{\nu})$, expand
the smaller of ${\cal H}_{e},{\cal H}_{e}^{\nu}$ to dimension $\tilde{d}_{e}$,
and add extra zero operators to the smaller of the two representations, so
that these each have $\tilde{d}_{e}$ operators. For the smaller of these two
Hilbert spaces, ${\cal F}$ or ${\cal F}^{\nu}$ still maps into the same
unexpanded space, ${\cal H}_{e}$ or ${\cal H}_{e}^{\nu}$, which is now a
subspace of the new, expanded space. This can be seen, in the case that
$d_{e}<d_{e}^{\nu}$ for example, from the relation noted below Eq. (2) of the
main text, $\langle l|K_{m}=\langle m|Q_{l}$, implying that for those $m$ such
that $K_{m}$ is the null operator, the added kets correspond to a subspace of
${\cal H}_{e}$ which ${\cal F}$ does _not_ map into. With these
considerations, we have

$\displaystyle\Phi$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{l=1}^{d_{o}}|i\rangle\langle
j|\otimes Q_{l}|i\rangle\langle j|Q_{l}^{\dagger}$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\sum_{l=1}^{d_{o}}|i\rangle\langle
j|\otimes|m^{\prime}\rangle\langle m^{\prime}|Q_{l}|i\rangle\langle
j|Q_{l}^{\dagger}|m\rangle\langle m|$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\sum_{l=1}^{d_{o}}|i\rangle\langle
j|\otimes\langle l|K_{m^{\prime}}|i\rangle\langle
j|K_{m}^{\dagger}|l\rangle|m^{\prime}\rangle\langle m|$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\sum_{l=1}^{d_{o}}|i\rangle\langle
j|\otimes\langle j|K_{m}^{\dagger}|l\rangle\langle
l|K_{m^{\prime}}|i\rangle|m^{\prime}\rangle\langle m|$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\langle
j|K_{m}^{\dagger}K_{m^{\prime}}|i\rangle|i\rangle\langle
j|\otimes|m^{\prime}\rangle\langle m|,$ (53)

and similarly for $\Phi^{\nu}$, the only difference being the replacement of
$K_{m}$ by $K_{m}^{\nu}$. In going from the second to the third line of Eq.
(A), we have used the relationship, $\langle m|Q_{l}=\langle l|K_{m}$, see
just below Eq. (2) of the main text. Note that matrices $\Phi$ and
$\Phi^{\nu}$ are of the same size, and we have

$\displaystyle\left\|\Phi-\Phi^{\nu}\right\|_{1}=\frac{1}{d}\left\|\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\langle
j|\left(K_{m}^{\dagger}K_{m^{\prime}}-K_{m}^{\nu{\dagger}}K_{m^{\prime}}^{\nu}\right)|i\rangle|i\rangle\langle
j|\otimes|m^{\prime}\rangle\langle m|\right\|_{1}.$ (54)

Let $\hat{K}_{n}$ ($\hat{K}_{n}^{\nu})$ be one of $\kappa$ $(\kappa^{\nu})$
members of a minimal Kraus representation for ${\cal E}$ $({\cal E}^{\nu})$.
Then, $K_{m}=\sum_{n}W_{mn}\hat{K}_{n}$ with $W$ a $\tilde{d}_{e}\times\kappa$
isometry so that $W^{\dagger}W=I_{\kappa}$, and
$K_{m}^{\nu}=\sum_{n}W_{mn}^{\nu}\hat{K}_{n}^{\nu}$ with $W^{\nu}$ a
$\tilde{d}_{e}\times\kappa^{\nu}$ isometry so that
$W^{\nu{\dagger}}W^{\nu}=I_{\kappa^{\nu}}$. Recalling from Lemma 4 that
$\kappa^{\nu}\geq\kappa$, add zero operators if needed to $\\{\hat{K}_{n}\\}$,
so that this set also has $\kappa^{\nu}$ members, along with zero columns to
$W$ so that $W^{\dagger}W=P_{\kappa}$, a $\kappa^{\nu}\times\kappa^{\nu}$
projector onto a $\kappa$-dimensional subspace. Then, from Eqs. (52) and (54),
we can write

$\displaystyle\sqrt{\delta}$
$\displaystyle>\frac{1}{d}\left\|\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\langle
j|\left(K_{m}^{\dagger}K_{m^{\prime}}-K_{m}^{\nu{\dagger}}K_{m^{\prime}}^{\nu}\right)|i\rangle|i\rangle\langle
j|\otimes|m^{\prime}\rangle\langle m|\right\|_{1}$
$\displaystyle=\frac{1}{d}\left(\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\left|\langle
j|\left(K_{m}^{\dagger}K_{m^{\prime}}-K_{m}^{\nu{\dagger}}K_{m^{\prime}}^{\nu}\right)|i\rangle\right|^{2}\right)^{1/2}$
$\displaystyle\geq\frac{1}{d}\sup_{0\leq C\leq
I_{\tilde{d}_{e}}}\left(\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\left|C_{mm^{\prime}}\langle
j|\left(K_{m}^{\dagger}K_{m^{\prime}}-K_{m}^{\nu{\dagger}}K_{m^{\prime}}^{\nu}\right)|i\rangle\right|^{2}\right)^{1/2}$
$\displaystyle\geq\frac{1}{d}\sup_{0\leq C\leq
I_{\tilde{d}_{e}}}\left(\sum_{i,j=1}^{d}\left|\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}C_{mm^{\prime}}\langle
j|\left(K_{m}^{\dagger}K_{m^{\prime}}-K_{m}^{\nu{\dagger}}K_{m^{\prime}}^{\nu}\right)|i\rangle\right|^{2}\right)^{1/2}$
$\displaystyle=\frac{1}{d}\sup_{0\leq C\leq I_{\tilde{d}_{e}}}\inf_{0\leq
C^{\nu}\leq
I_{\tilde{d}_{e}}}\left(\sum_{i,j=1}^{d}\left|\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\sum_{n,n^{\prime}=1}^{\kappa^{\nu}}\langle
j|\left(W_{mn}^{\ast}C_{mm^{\prime}}W_{m^{\prime}n^{\prime}}\hat{K}_{n}^{\dagger}\hat{K}_{n^{\prime}}-W_{mn}^{\nu\ast}C_{mm^{\prime}}^{\nu}W_{m^{\prime}n^{\prime}}^{\nu}\hat{K}_{n}^{\nu{\dagger}}\hat{K}_{n^{\prime}}^{\nu}\right)|i\rangle\right|^{2}\right)^{1/2}$
$\displaystyle=\frac{1}{d}\sup_{0\leq\hat{C}\leq
P_{\kappa}}\inf_{0\leq\hat{C}^{\nu}\leq
I_{\kappa^{\nu}}}\left(\sum_{i,j=1}^{d}\left|\sum_{n,n^{\prime}=1}^{\kappa^{\nu}}\langle
j|\left(\hat{C}_{nn^{\prime}}\hat{K}_{n}^{\dagger}\hat{K}_{n^{\prime}}-\hat{C}_{nn^{\prime}}^{\nu}\hat{K}_{n}^{\nu{\dagger}}\hat{K}_{n^{\prime}}^{\nu}\right)|i\rangle\right|^{2}\right)^{1/2}$
$\displaystyle=\frac{1}{d}\sup_{0\leq\hat{C}\leq
P_{\kappa}}\inf_{0\leq\hat{C}^{\nu}\leq
I_{\kappa^{\nu}}}\left\|\sum_{n,n^{\prime}=1}^{\kappa^{\nu}}\left(\hat{C}_{nn^{\prime}}\hat{K}_{n}^{\dagger}\hat{K}_{n^{\prime}}-\hat{C}_{nn^{\prime}}^{\nu}\hat{K}_{n}^{\nu{\dagger}}\hat{K}_{n^{\prime}}^{\nu}\right)\right\|_{1},$
(55)

with $\hat{C}=W^{\dagger}CW$ and
$\hat{C}^{\nu}=W^{\nu{\dagger}}C^{\nu}W^{\nu}$. To obtain the third line, we
inserted $\left|C_{mm^{\prime}}\right|\leq 1$, which will hold for all
$m,m^{\prime}$ when $0\leq C\leq I_{\tilde{d}_{e}}$, and then we used the
triangle inequality in the line after that. In entirely analogous fashion, one
finds that

$\displaystyle\sqrt{\delta}>\frac{1}{d}\sup_{0\leq\hat{C}^{\nu}\leq
I_{\kappa^{\nu}}}\inf_{0\leq\hat{C}\leq
P_{\kappa}}\left\|\sum_{n,n^{\prime}=1}^{\kappa^{\nu}}\left(\hat{C}_{nn^{\prime}}\hat{K}_{n}^{\dagger}\hat{K}_{n^{\prime}}-\hat{C}_{nn^{\prime}}^{\nu}\hat{K}_{n}^{\nu{\dagger}}\hat{K}_{n^{\prime}}^{\nu}\right)\right\|_{1}.$
(56)

Setting $\delta=(\epsilon/d)^{2}$, it immediately follows from Eqs. 44, A and
56 that for every $\epsilon>0$ there exists $\mu$ such that for every
$\nu>\mu$

$\displaystyle\epsilon>d_{H}\left(Z_{\cal E},Z_{{\cal E}^{\nu}}\right),$ (57)

where $d_{H}(X,Y)=\max\\{\sup_{x\in X}\inf_{y\in Y}\left|x-y\right|,\sup_{y\in
Y}\inf_{x\in X}\left|x-y\right|\\}$ is the Hausdorff distance between sets
$X,Y$, with $\left|x-y\right|$ the distance between elements $x,y$, which for
present purposes is the operator trace norm, $\left\|\cdot\right\|_{1}$.

We therefore have that the sequence of zonoids, $Z_{{\cal E}^{\nu}}$,
associated with quantum channels ${\cal E}^{\nu}$, converges to the zonoid
$Z_{\cal E}$, associated with the limiting quantum channel ${\cal E}$. This
completes the proof of Lemma 3.

##  Appendix B Generalization of $Z_{\cal E}$ for the case of quantum
instruments

Assume that the sequence of instruments, ${\cal E}^{\nu}$, has as its limit
instrument ${\cal E}=\\{{\cal J}_{r}\\}$. To formalize the intuitive arguments
leading up to the statement of Theorem 3 in Section VI, let us consider ${\cal
E}_{qc}$ the _quantum-classical_ channel, $\rho\to\sum_{r}{\cal
J}_{r}(\rho)\otimes[r]_{c}$, uniquely associated with instrument ${\cal E}$
[3], and ${\cal E}_{qc}^{\nu}$ associated with each ${\cal E}^{\nu}$, as well.
Then, $\lim_{\nu\to\infty}{\cal E}_{qc}^{\nu}={\cal E}_{qc}$. Here, subscript
$c$ on $[r]_{c}$ denotes an ancillary output containing the classical
information as to which CP map, ${\cal J}_{r}$, was obtained. As these are
quantum channels, the analysis in Appendix A holds, but let us take care in
identifying the Kraus operators representing these channels. Recognizing that
a set of Kraus operators for ${\cal E}_{qc}$ is
$K_{m}\otimes|f(m)\rangle_{c}$, where $f(m)=r$ for $K_{m}$ a Kraus operator
representing ${\cal J}_{r}$, then returning to Eq. (A) and noting that the
first two lines there remain unchanged, we may follow through the same steps
(but with minor re-arrangement of factors to make clear how the classical
register is to be handled) to derive the Choi-Jamiolkowski operator for the
channel complementary to ${\cal E}_{qc}$. We find

$\displaystyle\Phi$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}\sum_{l=1}^{d_{o}}|i\rangle\langle
j|\otimes|m^{\prime}\rangle\langle m|\langle
j|\left(K_{m}^{\dagger}\otimes\langle f(m)|_{c}\right)\left(|l\rangle\langle
l|\otimes
I_{c}\right)\left(K_{m^{\prime}}\otimes|f(m^{\prime})\rangle_{c}\right)|i\rangle$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}|i\rangle\langle
j|\otimes|m\rangle\langle m^{\prime}|\langle
j|K_{m}^{\dagger}K_{m^{\prime}}|i\rangle\langle f(m)|f(m^{\prime})\rangle_{c}$
$\displaystyle=\frac{1}{d}\sum_{i,j=1}^{d}\sum_{m,m^{\prime}=1}^{\tilde{d}_{e}}|i\rangle\langle
j|\otimes|m\rangle\langle m^{\prime}|\langle
j|K_{m}^{\dagger}K_{m^{\prime}}|i\rangle\delta_{f(m),f(m^{\prime})},$ (58)

and similarly for $\Phi^{\nu}$ associated with each ${\cal E}_{qc}^{\nu}$ but
with a different coarse-graining map, $f^{\nu}(m)$. Due to the appearance of
the Kronecker delta in the last line, this is a block-diagonal operator, each
block associated with one of the CP maps ${\cal J}_{r}$. Furthermore, each CP
map has its own set of minimal Kraus operators, the collection of which (over
all the CP maps in instrument ${\cal E}$) we denote as $\\{\hat{K}_{n}\\}$.
For the associated quantum-classical channel, these Kraus operators become
$\hat{K}_{n}\otimes|g(n)\rangle_{c}$, where $g(n)$ is another coarse-graining
map, possibly having a different domain than, but the same range as, $f(n)$.
As before, $g(n)=r$ for those operators representing ${\cal J}_{r}$. Then,
isometric freedom of this channel implies

$\displaystyle K_{m}\otimes|f(m)\rangle_{c}$
$\displaystyle=\sum_{n}W_{mn}\hat{K}_{n}\otimes|g(n)\rangle_{c}$
$\displaystyle K_{m}^{\nu}\otimes|f^{\nu}(m)\rangle_{c}$
$\displaystyle=\sum_{n}W_{mn}^{\nu}\hat{K}_{n}^{\nu}\otimes|g^{\nu}(n)\rangle_{c}.$
(59)

Clearly, since the kets appearing here represent classical states and are
therefore mutually orthogonal (and we can always take them to be normalized by
absorbing a factor into $K_{m}$ or $\hat{K}_{n}$, for example), there should
be no contributions on the right-hand sides of these expressions except for
those $n$ satisfying $g(n)=f(m)$ or $g^{\nu}(n)=f^{\nu}(m)$. That is to say,
isometries $W~{}(W^{\nu})$, introduced on the fifth line of Eq. (A), are here
block diagonal with blocks matching those appearing in $C~{}(C^{\nu})$, also
introduced in Eq. (A). By arguments completely analogous to those in the proof
of Lemma 4, the blocks defined by $f^{\nu}(m)$ must _contain_ those blocks
defined by $f(m)$, at least for large enough $\nu$. This means that a CP map
of ${\cal E}^{\nu}$ cannot have smaller Kraus rank than the corresponding CP
map of ${\cal E}$ (or possibly CP maps, plural, since the blocks defined by
$f^{\nu}(m)$ may contain more than one of those defined by $f(m)$).

Thus, by continuing with the arguments following Eq. (A) in Appendix A, one
obtains the exact same result as previously obtained in Eqs. (56) and (57),
but with the additional condition that matrices $\hat{C}~{}(\hat{C}^{\nu})$
appearing in the definition of $Z_{\cal E}~{}(Z_{{\cal E}^{\nu}})$ must be
block diagonal. The remainder of the proof of Theorem 3 follows precisely the
arguments in the proof of Theorem 2.

##  Appendix C Proof of Lemma 7

Define

$\displaystyle
K_{\alpha}(s)=\sqrt{d\\!\\!\sqrt[P]{s}M(s)^{\otimes(\alpha-1)}\otimes[0]\otimes
M(s)^{\otimes(P-\alpha)}},$ (60)

where the $[0]$ appearing here is a local operator for party $\alpha$. Then,
defining $\sigma=\sqrt[P]{s}-1$, we have that

$\displaystyle\int_{0}^{1}d\sigma K_{\alpha}(s)|i\rangle\langle
j|K_{\alpha}^{\dagger}=\int_{0}^{1}d\sigma\sqrt{\sigma}^{(l_{i}+l_{j}-2)}|i\rangle\langle
j|=\frac{2}{l_{i}+l_{j}}|i\rangle\langle j|,$ (61)

when the binary representation of $i$ and $j$ each have a $0$ in the
$\alpha$th position, whereas this integral vanishes otherwise. Summing over
all $\alpha$ gives an additional factor of $m_{ij}$. This accounts for all the
Kraus operators except for the one equal to $[2^{P}-1]$, which gives
$[2^{P}-1]$ when acting on itself, and zero otherwise. Therefore, adding all
the terms together, we see that Eq. (42) and Eq. (43) in the main text are
correct for channel ${\cal E}_{P}$.

For the approximating channel ${\cal E}^{\nu}$, the Kraus operators are
$K_{M}^{\nu}=\sqrt{R}^{\otimes\nu}$ at the end of the main branch, which
asymptotically approaches $[2^{P}-1]$ in the limit $\nu\to\infty$
(equivalently, $\epsilon\to 0$). In addition, we have $K_{\alpha
n}^{\nu}=\sqrt{\epsilon(1-\epsilon)^{(n-1)}}\sqrt{R^{n}}^{\otimes(\alpha-1)}\otimes[0]\otimes\sqrt{R^{n}}^{\otimes(P-\alpha-1)}$,
while all other Kraus operators are $\cal{O}(\epsilon)$ and will yield
vanishing contributions in the limit. Now, $K_{\alpha n}^{\nu}|i\rangle\langle
j|K_{\alpha
n}^{\nu{\dagger}}=\epsilon(1-\epsilon)^{(n-1)}\left(1-\epsilon\right)^{n(l_{i}+l_{j}-2)/2}|i\rangle\langle
j|$ when $i,j$ each have a $0$ appearing at position $\alpha$ in their binary
expansions. Summing over $\alpha$ gives a factor of $m_{ij}$, and then the sum
over $n$ from $1$ to $\nu$ yields factor $2/(l_{i}+l_{j})+\cal{O}(\epsilon)$.
Therefore, the limiting channel also agrees with Eq. (42) and Eq. (43), and
the proof is complete.$\blacksquare$

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