# Quantum Origins of the Density Operator

Mark G. Kuzyk  Department of Physics and Astronomy, Washington State
University, Pullman, Washington 99164-2814

###### Abstract

Students in quantum mechanics class are taught that the wave function contains
all knowable information about an isolated system. Later in the course, this
view seems to be contradicted by the mysterious density matrix, which
introduces a new set of probabilities in addition to those that are built into
the wave function. This paper brings attention to the fact that the density
matrix can be reconciled with the underlying quantum-mechanical description
using a two-particle entangled state with a one-particle subsystem as the
simplest illustration of the basic principle. The extra-quantum probabilities
are traced to the coefficients of superposition of the quantum state vector
and the seemingly irreversible exponential population decay is shown to be
compatible with the unitary time evolution of a pure state when the two
particles interact. The two-particle universe thus provides the student with a
tool for understanding how the density operator, with all its richness,
emerges from quantum mechanics.

###### Contents

  1. I Introduction
  2. II Universe as an Object in an Environment
    1. II.1 Particle as System
    2. II.2 Spatial boundary between Object and Environment
  3. III Interacting Indistinguishable Particles
    1. III.1 Two-Particle State Vector Basis
      1. III.1.1 A Particle as the Object
  4. IV Discussion
    1. IV.1 Spin Mixtures
    2. IV.2 Time Evolution
    3. IV.3 Many Particles
    4. IV.4 Multiple-Particle Operators
  5. V Conclusion
  6. A Summation Gymnastics
    1. A.1 Sums of Differences
    2. A.2 Continuum Limit of Finite Sums
  7. B Finding the Coefficients

##  I Introduction

The textbook developmentsakur10.01 ; sakur94.01 ; huang63.01 ; pathr01.11 ;
gottf01.74  and pedagogical treatmentpatte62.01 ; sabba07.01  of the density
operatorneuma27.01 ; neuma18.01  typically calls upon a hybrid
classical/quantum argument as follows. Consider a collection of particles,
such as electrons. If $w_{i}$ represents the fraction of electrons in state
$i$, the expectation value of the observable represented by the operator $A$
is given by

$\Braket{A}=\sum_{i}w_{i}\Braket{i}{A}{i}.$ (1)

Equation 1 makes intuitive sense; $\Braket{i}{A}{i}$ is the quantum
expectation value of observable $A$ in state $i$ and $w_{i}$ provides the
statistical weighting to the quantum expectations.

The index $i$ can span a number of states that exceeds the dimensionality of
the Hilbert space of the quantum system and the individual states in the sum
need not be orthogonal, so it is often the case that
$\Braket{i}{i^{\prime}}\neq\delta_{i,i^{\prime}}$. For example, the spin
(living in a two-dimensional space) can have a fifth of the electrons oriented
along $\hat{z}$, a tenth along $\hat{x}$, a quarter along
$(\hat{x}+\hat{y})/\sqrt{2}$, and so on. As such, closure need not apply to
$\Ket{i}$, that is $\sum_{i}\Ket{i}\Bra{i}\neq\mathds{1}$.

For a complete orthonormal basis $\\{\Ket{n}\\}$, closure holds and can be
inserted into Equation 1, yielding

$\Braket{A}=\sum_{n}\sum_{i}w_{i}\Braket{i}{(\Ket{n}\Bra{n})A}{i}.$ (2)

Interchanging the scalars $\Braket{n}{A}{i}$ and $\Braket{i}{n}$, Equation 2
takes the form

$\Braket{A}=\sum_{n}\sum_{i}w_{i}\Braket{n}{A}{i}\Braket{i}{n}.$ (3)

Defining the operator

$\rho=\sum_{i}w_{i}\Ket{i}\Bra{i},$ (4)

Equation 3 can be expressed as

$\Braket{A}=\sum_{n}\Braket{n}{A\rho}{n}=\text{Tr}(\rho A).$ (5)

$\rho$ is the density operator, which contains probabilities $w_{i}$ of
finding the population in state $\Ket{i}$.

We call the set $\\{w_{i}\\}$ classical probabilities because they represent
the fraction of systems within an ensemble that are in state $\Ket{i}$, a
classical concept. The coefficients $a_{i}$ in the state
$\Ket{\psi}=\sum_{i}a_{i}\Ket{i}$ give what we call the quantum probabilities
$\left|a_{i}\right|^{2}$ because they arise from superposition, a quantum
mechanical property.

The density operator in this form was first introduced by von
Neumann,neuma27.01  who explains the need for the classical probabilities,
“But such statistical considerations acquire a new aspect when we do not even
know what state is actually present…”neuma18.01

The above pedagogical derivation can be found at least as far back as in
Richard Tolman’s 1938 book, where he suggests the need for “…taking the mean
first over the range of possibilities presented by each member of the ensemble
and then over the members of the ensemble,” where the “possibilities” refers
to quantum probabilities and the “members of the ensemble” the classical
average from the distribution of populations.tolma38.01

In 1957, a review article by Ugo Fano reinforces this sentiment by
explainingfano57.01  “States that are not ‘pure’ have been called ‘mixed’
states because they can be described by the incoherent superposition of pure
states. Incoherent superposition means, by definition, that to calculate the
probability of finding a certain experimental result with a system in the
mixed state one must first calculate the probability for each of the pure
states and then take an average, attributing to each of the pure states an
assigned ‘weight.’ ”

This development again mixes classical and quantum concepts. The state vectors
are quantum mechanical in nature but the separation into populations is a
classical construct. Landau warns us that the universe is governed by quantum
mechanics and viewing this separation classically is wrong.landa69.01  In his
own words, “…it would be quite incorrect to suppose that the description by
means of the density matrix signifies that the subsystem can be in various
$\Psi$ states with various probabilities and that averaging is over these
probabilities. Such a treatment would be in conflict with the basic principles
of quantum mechanics.” He then continues by emphasizing that, “…the
statistical averaging [is] necessitated by the incompleteness of our
information concerning the objects considered.” Subsequently, he stresses that
the two probabilities cannot be separated and that “the whole averaging
procedure is carried out as a single operation, and cannot be represented as
the result of successive averaging, one purely quantum-mechanical and the
other purely statistical.” Many standard textbooks present the student with an
explanation that does not heed Landau’s warning. Band and Park disagree with
this all-quantum view, arguing that measurements are often repeated many
times, so the statistics resides in the variability of preparing the
system.band79.01

Landau and Lifshitz (L&L) present what appears to be a more precise derivation
of the density operator along the following lineslanda77.01  by defining
$\psi(x,q)$ to be the wavefunction of the universe, which can be formally
calculated from the state vector $\ket{\psi}$ of the universe using the inner
product $\psi(x,q)=\braket{xq}{\psi}$. $\left|\psi(x,q)\right|^{2}$ is
understood to be the probability density of finding the object in a state
defined by the parameter $x$ and a particle in the environment described by
parameter $q$. Since $x$ and $q$ can each describe many degrees of freedom,
this wavefunction depends on $x=(x_{1},x_{2},\dots)$ in the object and
$q=(q_{1},q_{2},\dots)$ in the environment. The wavefunction is usually not
separable, so that $\psi(x,q)\neq\psi(x)\psi(q)$.

The expectation of the operator $A$ is given by

$\displaystyle\Braket{A}=\int dx\int dq\,\psi^{*}(x,q)A(x,q)\psi(x,q).$ (6)

Since only the system is being measured, L&L argue that the operator $A$ is
restricted to parameters $x$. Then, $A(x,q)\rightarrow A(x)$ – a somewhat
confusing assumption, which we clarify later. With A(x) depending only on $x$,
Equation 6 after some regrouping of terms becomes

$\displaystyle\Braket{A}_{x}=\int dx\,A(x)\int
dq\,\psi^{*}(x,q)\psi(x,q)\equiv\int dx\,A(x)\rho(x),$ (7)

where $\rho(x)$ is the diagonal part of the density matrix, which is defined
by

$\displaystyle\rho(x,x^{\prime})=\int dq\,\psi^{*}(x,q)\psi(x^{\prime},q).$
(8)

Clearly, $\rho^{*}(x,x^{\prime})=\rho(x^{\prime},x)$ so the density matrix
represents a hermitian operator.

The diagonal elements of the density matrix are the probability densities of
finding the system with parameter $x$ and are of quantum origin. It is
impossible to know the state of the environment, which is entangled with the
system, so the density matrix is modeled with phenomenological parameters
$w_{i}$ in Equation 4, which have the feel of classical probabilities. Our
simple two-particle model will provide a simple example of how this works.

The L&L derivation does not provide a clear way of getting the density matrix
since $\psi(x,q)$ includes the coordinates of the environment, whose effects
are too complex to take into account. As we show later, defining a system by
its spatial region leads to additional confusion for identical particles,
which must be entangled. At the other extreme, one can use Equation 4 to
interpret the coefficients $w_{i}$ as populations, which as we have seen is
justifiably objectionable to Landau. Feynman’s approach follows Landau’s
development,feynm72.01  but connects the quantum calculation to the classical
interpretation more clearly. However, Feynman’s book lacks simple examples to
explain how the universe is parsed and contains the seemingly erroneous
statement that $\Ket{i}$ must generally form a compete orthonormal set of
state vectors.

Our goal is to use simple examples that show the student how the density
matrix emerges from an entangled state and to clear up confusion that might
arise from textbooks and the literature. In a nutshell, this paper can be
summarized by the following example of how a density matrix representing a
mixed state emerges from two non-interacting electrons with state vector

$\displaystyle\Ket{\psi}=\frac{1}{\sqrt{2}}\Big{(}\Ket{\uparrow}\ket{\downarrow}-\ket{\downarrow}\Ket{\uparrow}\Big{)}.$
(9)

Landau first noticed the emergence of mixed states in this way.landau27.01
For an operator $A_{1}$ that acts only on the first particle (leftmost ket in
each term), its expectation value is given by

$\displaystyle\Braket{\psi}{A_{1}}{\psi}$
$\displaystyle=\frac{1}{2}\Big{(}\Braket{\uparrow}{A_{1}}{\uparrow}\Braket{\downarrow}{\downarrow}+\Braket{\downarrow}{A_{1}}{\downarrow}\Braket{\uparrow}{\uparrow}$
$\displaystyle-\Braket{\uparrow}{A_{1}}{\downarrow}\Braket{\downarrow}{\uparrow}-\Braket{\downarrow}{A_{1}}{\uparrow}\Braket{\uparrow}{\downarrow}\Big{)}$
$\displaystyle=\frac{1}{2}\Big{(}\Braket{\uparrow}{A_{1}}{\uparrow}+\Braket{\downarrow}{A_{1}}{\downarrow}\Big{)}.$
(10)

Equation I is equivalent to Equation 5 with $A_{1}=A$ and a density matrix
given by

$\displaystyle\rho=\left(\begin{array}[]{cc}\frac{1}{2}&0\\\ 0&\frac{1}{2}\\\
\end{array}\right)$ (13)

– a mixed stated.

This simple example shows that:

  1. 1.

The progenitor of a one-particle mixed state can be a pure state (in this
case, two entangled particles).

  2. 2.

The coefficients $w_{i}$ originate from the coefficients of superposition in
Equation 9, which represent quantum probability amplitudes, i.e. the factors
$1/\sqrt{2}$.

  3. 3.

Mixed states do not require classical populations.

  4. 4.

A mixed state emerges when measuring an observable whose associated operator
acts on a one-particle subset of a pure and entangled two-particle state.

These observations can be generalized to many particles, but the important
points can be made with two particles alone. Lerner showed in a similar way
how decoherence can be understood without a density matrixlerne17.01  and
Press avoided the density matrix to glean insights that are lost about
entangled spins when using the density matrix. Asher, on the other hand, uses
arguments similar to ours using two spins but calls upon the time evolution of
the density matrix.peres74.01  von Neumann’s excellent book, which has been
typeset anew and published in 2018 provides many insights on projections into
subsystems,neuma18.01  but at a level more advanced than the intended audience
of this paper.

This view is not new and the quantum information theory literature commonly
accepts the fact that the mixed state is part of a larger purely quantum
system. In fact, it is a simple matter to find a many-particle pure state that
gives a particular density matrix through a process called “purification,” and
there are practical ways to implement purification.klein06.01 ; Nielsen01.10
However, the more specialized textbooks do not make such connections clear at
an elementary level, and are not used in core physics classes. For example,
Nielsen and Chuang start with “suppose that a quantum system is in one of a
number of states $\Ket{\psi_{i}}$, where $i$ is an index, with respective
probabilities $p_{i}$” then they define the density matrix by
$\sum_{i}p_{i}\Ket{\psi_{i}}\Bra{\psi_{i}}$ with no explanation, taking for
granted that the reader already has an understanding of the topic.Nielsen01.10
A significant effort would be required of a beginning quantum mechanics
student to use such texts to come to the understanding embodied in the simple
example leading to Eq. 13.

One motivation for this paper is to point out that this connection between
pure quantum states and density matrices should be made when the topic is
first introduced to students, and to urge instructors to consider doing so
using simple two-particle models. A central theme of this paper is to
emphasize that the richness of phenomena described by density operators, such
as mixed states and the time evolution of populations, can be understood using
an entangled pair of two particles.

In the sections that follow, we start by defining what is meant by an object
and environment, then consider such universes made of distinct then identical
particles. Interactions between particles are shown to be required to get
density matrices with diagonal components that decay exponentially over time
to their equilibrium values, as is observed in experiments and
phenomenologically modeled with the time evolution of density matrices. Most
importantly, we show how the coefficients used in quantum superposition are
related to the density matrix elements, thus providing insights into the
connection between the two. We use these observations to highlight common
misconceptions and the confusion that might result.

##  II Universe as an Object in an Environment

Figure 1: (a) A position coordinate within the universe is labelled $x$ when
inside the system under study and $q$ when in the environment. $x$ and $q$ can
represent many degrees of freedom. (b) Alternatively, the system can be
defined to be Particle #1 (or a collection of particles) with coordinate(s)
$x$ and the environment as Particle #2 (or a collection of particles) with
coordinate(s) $q$. In this case $x$ and $q$ can occupy the same spatial
region.

There are two common ways to divide the universe into a system/object and
environment. Figure 1a shows a spatially-isolated system. Then, the
environment is the part of the universe that falls outside the system. In this
case, $x$ and $q$ label the coordinates inside the object and in the
environment, respectively. Alternatively, as shown in Figure 1b, the system is
not defined by the region it occupies, but by the type of particle – for
example a large blue one or a small red one. This makes it possible to ask
“What is the probability of finding the red particle at position $x$?” In
contrast, for indistinguishable particles, one can only ask “What is the
probability of finding a particle at position $x$?” without specifying which
one.

The following subsection starts with the simplest example of two
distinguishable particles to illustrate how the density matrix is constructed
for one of the particles as the system, and the other particle as the
environment. The concluding subsection derives the density matrix when the
system and environment are two spatially separated regions. The two regions
can be viewed as two wells that each contain a particle, thus forming a two-
particle pure state. In the presence of infinite wells, either method of
identification is equivalent and will yield the same result, though splitting
the universe into two particles by type is more intuitive than spatially
separating them when including spin statistics.kuzyk19.01

###  II.1 Particle as System

The most general two-particle wave function is of the form

$\displaystyle\Ket{\psi}=\sum_{n_{1},n_{2}}c_{n_{1},n_{2}}\Ket{n_{1}}\Ket{n_{2}},$
(14)

where the subscripts $1$ and $2$ are the particle labels. Then, the
expectation value of the observable $A_{1}$ of Particle #1 is given by

$\displaystyle\Braket{\psi}{A_{1}}{\psi}=\sum_{n_{1},n_{2}}\sum_{n_{1}^{\prime},n_{2}^{\prime}}c_{n_{1},n_{2}}^{*}c_{n_{1}^{\prime},n_{2}^{\prime}}\Braket{n_{1}}{A_{1}}{n_{1}^{\prime}}\delta_{n_{2},n_{2}^{\prime}}.$
(15)

Defining

$\displaystyle\rho_{n_{1},n_{1}^{\prime}}=\sum_{n_{2}}c_{n_{1},n_{2}}^{*}c_{n_{1}^{\prime},n_{2}},$
(16)

Equation 15 becomes

$\displaystyle\Braket{\psi}{A_{1}}{\psi}=\sum_{n_{1},n_{1}^{\prime}}\rho_{n_{1},n_{1}^{\prime}}\Braket{n_{1}}{A_{1}}{n_{1}^{\prime}}=\text{Tr}(\rho
A_{1}).$ (17)

Equation 17 is equivalent to Equation 5, so the expectation value of the
“system” originates from a two-particle pure state in a quantum superposition,
but can be expressed in terms of a density matrix, which depends on the
coefficients $c_{n_{1},n_{2}}$.

The operator form of Equation 16 is

$\displaystyle\rho=\sum_{n_{2}}\Braket{n_{2}}{\psi}\Braket{\psi}{n_{2}},$ (18)

where the density operator given by Equation 18 of the object is the part that
is left over after taking the trace over the environment of the pure state
operator of the universe $\Ket{\psi}\Bra{\psi}$. Convince yourselves that this
is true by calculating $\Braket{n_{1}^{\prime}}{\rho}{n_{1}}$ of Equation 18
with the help of Equation 14 to get Equation 16.

I leave it as an exercise for the student to use Equation 16 to show that the
density matrix is hermitian and has unit trace. Hermiticity and unit trace
allows the density matrix to be diagonalized to yield an expression in the
form of Equation 4 with what appear to be classical population fractions
$w_{i}$ that are related to the $c_{n_{1},n_{2}}$ coefficients of Equation 16.
While this diagonal form has the appearance of classical probabilities given
by $w_{i}$, their origin here is purely quantum in nature.

Viewing a system as being made of an ensemble, whose properties are determined
with statistical weighting, has been a highly successful paradigm for building
models of a wide variety of systems and phenomena that result. However, one
should not lose sight of the fact that our universe need not naturally divide
into little quantum compartments that form a larger collection of classical
populations.

###  II.2 Spatial boundary between Object and Environment

To treat spatial regions, we label the coordinates of the distinguishable
particles $r_{1}$ and $r_{2}$, where $r_{i}$ is given by $x_{i}$ inside the
object and $q_{i}$ outside. We restrict the calculation that follows to the
case where each particle is fully localized in its own well.

For a universe described fully by the state vector $\Ket{\psi}$, the
probability amplitude of finding Particle #1 in the system at position $x_{1}$
and Particle #2 in the environment at position $q_{2}$ is given by

$\displaystyle\psi(x_{1},q_{2})=\Bra{x_{1}}\Bra{q_{2}}\cdot\Ket{\psi}.$ (19)

Therefore, in the position representation, the two-particle state vector is

$\displaystyle\Ket{\psi}=\int dq_{2}\,\int
dx_{1}\,\psi(x_{1},q_{2})\Ket{x_{1}}\Ket{q_{2}}.$ (20)

The student should apply $\Bra{x_{1}^{\prime}}\Bra{q_{2}^{\prime}}$ to both
sides of Equation 20 to verify that Equation 19 results. Equation 20 is the
continuous form of Equation 14. The expectation of observable $A$ of the
system is then

$\displaystyle\Braket{\psi}{A}{\psi}=\int dq_{2}^{\prime}\,\int
dx_{1}^{\prime}\,\int dq_{2}\,\int
dx_{1}\,\psi^{*}(x_{1}^{\prime},q_{2}^{\prime})$
$\displaystyle\times\psi(x_{1},q_{2})\Braket{x_{1}^{\prime}}{A}{x_{1}}\braket{q_{2}^{\prime}}{q_{2}}$
(21) $\displaystyle=\int dx_{1}^{\prime}\,\int dq_{2}\,\int
dx_{1}\,\psi^{*}(x_{1}^{\prime},q_{2})\psi(x_{1},q_{2})\Braket{x_{1}^{\prime}}{A}{x_{1}},$

which is a more general result than that of Landau and Lifshitz given by
Equation 7 but reduces to their result when
$\Braket{x_{1}^{\prime}}{A}{x_{1}}=\delta(x_{1}-x_{1}^{\prime})A(x_{1})$. We
stress that the derivation leading to Equation II.2 assumes that the single-
particle wavefunctions are each confined to one well or the other one, a way
in which the universe is typically separated, but an idealization of
reality.kuzyk19.01

Equation II.2 can be written as

$\displaystyle\Braket{\psi}{A}{\psi}=\int dx^{\prime}\int
dx\,\rho(x,x^{\prime})\Braket{x^{\prime}}{A}{x},$ (22)

where $\rho(x^{\prime},x)$ is given by Equation 8. Here we have dropped the
subscript to label the particle because we know that Particle #1 is in Well #1
with spatial coordinates $x$.

Equation II.2 makes clear that the double integral over the environment
collapses into a single integral due to the inner product
$\Braket{q_{2}^{\prime}}{q_{2}}$. In contrast, Landau states Eq. 6 without
proof, which is an obvious fact to experts but not to students learning the
material for the first time. Furthermore, Equation II.2 is a more general
result that allows for off-diagonal elements of the operator $A$.

To summarize, a universe with two distinguishable particles in two separate
infinite wells can be referenced in two ways; if Particle #$i$ is placed in
Well #$i$, then the particle label is the same as the well label. Then, the
projector onto Particle #i is the same as the projector into Well #i. Then

$\displaystyle\sum_{n_{1}}\Ket{n_{1}}\Bra{n_{1}}=\int_{\mbox{Well}\\#1}dx\,\Ket{x}\Bra{x}$
(23)

and

$\displaystyle\sum_{n_{2}}\Ket{n_{2}}\Bra{n_{2}}=\int_{\mbox{Well}\\#2}dq\,\Ket{q}\Bra{q}.$
(24)

In this example, each well is in a different physical subspace – one here and
one there. Particles, on the other hand, always live in different Hilbert
spaces even when they occupy the same physical space. In the special case
where each particle is confined to its own infinite well, thus defining two
distinct regions of physical space, there is a direct correspondence between
the Hilbert space and the physical space, so they can be used interchangeably.
This correspondence does not hold in the absence of such localized
wavefunctions, where the Hilbert space is unrelated to the physical space.
Pedagogically, it is preferable to parse the universe by particle type,
because it clearly shows that Hilbert space is not the same as physical
space.kuzyk19.01

To summarize, this section has shown that when the system and the environment
is defined by two regions of space or by two distinguishable particles, the
expectation values of an operator for one of the particles takes the form
given by Equation 5, where the density matrix is a function of the
coefficients from the superposition of states that defines the universe.

##  III Interacting Indistinguishable Particles

This section shows how the density matrix of one particle or one confined
region of space emerges from the pure state of two indistinguishable fermions
or bosons, which must obey the spin statistics theorem so therefore must be
entangled.duck98.01  Spatial entanglement is an interesting topic, but
applying it here adds no new physics to the arguments that follow.schro17.01

###  III.1 Two-Particle State Vector Basis

If two particles are indistinguishable, we cannot label them individually so
the wave functions must be either symmetric or antisymmetric upon exchanging
them. Thus, we can only compute the probability amplitude of finding one
particle in one state and one particle in another one without referencing
which one is which. In Dirac notation, the state vector of two noninteracting
particles that occupy any two of the single-particle eigenstates $\Ket{n}$ and
$\Ket{m}$ is of the form

$\displaystyle\Ket{n,m}=\frac{1}{\sqrt{2}}\Big{[}\Ket{n}\Ket{m}\pm\Ket{m}\Ket{n}\Big{]}.$
(25)

Note that two kets together such as $\Ket{n}\Ket{m}$ represent a two-particle
state, with Particle #1 in a one-particle eigenstate $\Ket{n}$ with eigenvalue
$n$ and Particle #2 in a one-particle eigenstate $\ket{m}$ with eigenvalue
$m$. The eignevalue associated with $n$ could, for example, be the energy
$E_{n}$, but consider it to be general.

The first ket in a pair refers to Particle #1 and the second ket to Particle
#2. Each of the two terms in Equation 25 distinguish between the two
particles, yet when added together, the state $\Ket{n,m}$ does not; that is,
state $\Ket{n,m}$ corresponds to each particle being in both states. This
result applies to multiple particles in one well or particles in two different
wells.

Equation 25 in position space is given by

$\displaystyle\psi(x,q)$
$\displaystyle=\Braket{x,q}{n,m}=\frac{1}{2}\Big{[}\psi_{n}(x)\psi_{m}(q)\pm\psi_{n}(q)\psi_{m}(x)$
$\displaystyle+\psi_{m}(x)\psi_{n}(q)\pm\psi_{m}(q)\psi_{n}(x)\Big{]},$ (26)

where $\Ket{x,q}$ is in the same (anti)symmetrized form as Equation 25, so
each particle lives in both the object and the environment. Note that we have
kept the wavefunctions ordered to keep track of which particle is in which
state and in which region. For example, $\psi_{m}(x)\psi_{n}(q)$ means that
Particle #1 is in state $m$ and inside the object (position $x$) while
particle #2 is in state $n$ and in the environment (position $q$).

Let’s pick two wells that have an irrational ratio of widths so that the
single-particle energy eigenfunction vanishes in one well when it is nonzero
in the other one. Since one particle occupies each well, let’s arbitrarily
pick state $n$ to be found in the object and state $m$ in the environment.
Then, $\psi_{n}(q)=\psi_{m}(x)=0$ because state $n$ vanishes in the
environment and state $m$ vanishes in the object. Equation III.1 then becomes

$\displaystyle\psi(x,q)$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{[}\psi_{n}^{(1)}(x)\psi_{m}^{(2)}(q)\pm\psi_{m}^{(1)}(q)\psi_{n}^{(2)}(x)\Big{]},$
(27)

where we have labeled each particle’s identity as a superscript to remind us
which wavefunction is associated with which particle.

We note that the wavefunction given by equation 27 associates an energy
eigenstate index ($n$ or $m$) with a particular well ($x$ or $q$). In contrast
to the previous case where particle type is associated with a spatial region,
here the particle types are indistinguishable. We arbitrarily associate the
energy index $n$ with the spatial region defined by $x$ and $m$ with $q$.
Thus, all of the previous discussions for distinguishable particles caries
forward here by replacing “particle type” with “energy index.” We need not
rehash these discussions for indistinguishable particles. Instead, we will
focus on the particle as the object for a system of two indistinguishable
particles, but will add interactions between them to show that this will lead
to all possible density matrices.

####  III.1.1 A Particle as the Object

When two particles interact with each other, the energy eigenstate vectors
$\Ket{\psi^{(k)}}$ of the full Hamiltonian

$H=H_{1}+H_{2}+V({\bf r}_{1},{\bf r}_{2})$ (28)

can be expressed as the superposition

$\Ket{\psi^{(k)}}=\frac{1}{\sqrt{2}}\sum_{n,m}c_{n,m}^{(k)}\Ket{nm},$ (29)

where we have used the fact that the set of state vectors
$\left\\{\Ket{nm}\right\\}$ form an orthonormal basis. For bosons/fermions,
since $\ket{nm}=\pm\ket{mn}$, we choose the complex coefficients to obey
$c_{n,m}^{(k)}=\pm c_{m,n}^{(k)}$. For each state $k$, normalization requires

$\sum_{n,m}\left|c_{n,m}^{(k)}\right|^{2}=1.$ (30)

Alternatively, we are free to restrict the sums in Equation 29 to terms with
$m>n$ for fermions (since $\Ket{nm}=0$) and $m\geq n$ for bosons. We can
choose either form based on convenience.

The $k^{\text{th}}$ eigen energy $E^{(k)}$ is given by

$H\Ket{\psi^{(k)}}=E^{(k)}\Ket{\psi^{(k)}}.$ (31)

Note that $H_{i}$ in Equation 28 is the one-particle Hamiltonian of Particle
#i so $H_{1}\Ket{n}\Ket{m}=E_{n}\Ket{n}\Ket{m}$ and yields
$\left(H_{1}+H_{2}\right)\Ket{nm}=\left(E_{n}+E_{m}\right)\Ket{nm}$. The
student should verify this relationship.

The most general state vector, then, will be a superposition of the two-
particle state vectors given by Equation 29, yielding

$\Ket{\psi}=\sum_{k}a_{k}\Ket{\psi^{(k)}}=\frac{1}{\sqrt{2}}\sum_{k}a_{k}\sum_{n,m}c_{n,m}^{(k)}\Ket{nm},$
(32)

where $\sum_{k}\left|a_{k}\right|^{2}=1$ to maintain normalization. The
expectation value of operator $A_{1}$, which acts only on particle #1, yields

$\displaystyle\Braket{\psi}{A_{1}}{\psi}$
$\displaystyle=\frac{1}{2}\sum_{k,k^{\prime}}a_{k^{\prime}}^{*}a_{k}\sum_{n^{\prime},m^{\prime}}\sum_{n,m}c_{n^{\prime},m^{\prime}}^{(k^{\prime})*}c_{n,m}^{(k)}$
$\displaystyle\times\Braket{n^{\prime},m^{\prime}}{A_{1}}{n,m}$ (33)

Re-expressing Eq. III.1.1 in terms of the single-particle states using Eq. 25,
and recognizing that only the first bras and kets in the pair act on $A_{1}$
and the second ones pass through – as illustrated in Eq. I, we get

$\displaystyle\Braket{\psi}{A_{1}}{\psi}$
$\displaystyle=\sum_{k,k^{\prime}}a_{k^{\prime}}^{*}a_{k}\sum_{n^{\prime},m^{\prime}}\sum_{n,m}\frac{c_{n^{\prime},m^{\prime}}^{(k^{\prime})*}c_{n,m}^{(k)}}{4}$
$\displaystyle\Big{(}\Braket{n^{\prime}}{A_{1}}{n}\delta_{m,m^{\prime}}+\Braket{m^{\prime}}{A_{1}}{m}\delta_{n,n^{\prime}}$
$\displaystyle\pm\Braket{n^{\prime}}{A_{1}}{m}\delta_{n,m^{\prime}}\pm\Braket{m^{\prime}}{A_{1}}{n}\delta_{m,n^{\prime}}\Big{)},$
(34)

where the $\pm$ signs are for bosons and fermions.

Next we combine terms in Equation III.1.1 by changing the dummy indices in the
sums and using the fact that $c_{nm}^{(k)}=\pm c_{mn}^{(k)}$, which is a
consequence of $\Ket{n,m}=\pm\Ket{m,n}$. For example, the second term in
parentheses in Equation III.1.1 with its pre-factor
$c_{n^{\prime},m^{\prime}}^{(k^{\prime})*}c_{n,m}^{(k)}$ can be converted into
the first term by interchanging the dummy index $m^{\prime}\rightleftharpoons
n^{\prime}$ then interchanging the indices $m\rightleftharpoons n$. It is left
as an exercise for the student to manipulate the other terms in this way,
which yields

$\displaystyle\Braket{\psi}{A_{1}}{\psi}$
$\displaystyle=\sum_{k,k^{\prime}}a_{k^{\prime}}^{*}a_{k}\sum_{n^{\prime}}\sum_{n,m}c_{n^{\prime},m^{\prime}}^{(k^{\prime})*}c_{n,m}^{(k)}\Braket{n^{\prime}}{A_{1}}{n}$
$\displaystyle\equiv\sum_{n,n^{\prime}}\rho_{n,n^{\prime}}\Braket{n^{\prime}}{A_{1}}{n}=\mbox{TR[$\rho
A_{1}$]},$ (35)

where the density matrix is given by

$\displaystyle\rho_{n^{\prime},n}=\sum_{k^{\prime},k}a_{k^{\prime}}^{*}a_{k}\sum_{m}c_{n^{\prime},m}^{(k^{\prime})*}c_{n,m}^{(k)}.$
(36)

Equation 36 is the most general form resulting from two interacting particles
that obey the spin statistics theorem.duck98.01  This two-interacting-particle
state vector yields the general form of the density matrix, where its matrix
elements originate in the coefficients of the superposition expansion given by
Equation 32.

As an example of imprecise language that implies the density matrix’ ability
to account for phenomena beyond the quantum realm, consider Blum’s statement
on page 7 in his book Density Matrix Theory and Applications when referring to
a beam of spin-1/2 particles,blum12.01  “Clearly it is not possible to
characterize the state of the beam in terms of a single state vector
$\Ket{\chi}$ since associated with any of these states there is necessarily a
direction in which all spins are pointing; the direction of the polarization
vector. If the Stern-Gerlach filter were placed in this orientation the whole
beam would have to be transmitted. Since no such orientation exists it is not
possible to describe a mixture by a single state vector.” Articles in this
journal repeat such arguments with even stronger statements such as “…hence it
cannot be represented by any state vector,”peres74.01  my emphasis.

We derived Equation 36 from a single state vector made of just two entangled
particles, and it is the same as Blum’s Equation 2.11. Hence, we have found a
pure state of two entangled particles in superposition with properties that
Blum and others imply are impossible for a pure state. Thus, the density
matrix is not more general than a two-particle pure state. Statements of the
sort quoted in the previous paragraph are not true and confusing to a student
who notices the inconsistency while trying to learn about density matrices.
Most often, such contradictions reinforce the notion that density matrices are
in some way more general than state vectors and mask the simple fact that all
these properties emerge form a pure state that is represented by a state
vector.

Blum’s narrative is commonly found in textbooks, lectures and the literature –
which leaves the impression that only a density matrix will do. In reality, an
experimental observation is most conveniently characterized with a density
matrix because an infinite number of progenitor pure states lead to the same
observation, making it impossible to determine the unique one for that
particular system. Even within a given basis, there are an infinite number of
ways to construct a particular density matrix by virtue of the fact that
mixtures can include non-orthogonal states. However, using Equation 36, a two-
particle state vector can be identified that in the one-particle subspace
fully describes any experimental observation of that system, so can be used in
lieu of the density matrix.

The density operator for indistinguishable particles cannot be determined by
inspection as it was in obtaining Equation 18 for distinguishable particles
because indistinguishable-particle wave functions are entangled. However, the
matrix elements $\rho_{n,n^{\prime}}$ given by Equation 36 can be used to
construct the density operator using

$\displaystyle\rho=\sum_{n,n^{\prime}}\Ket{n}\rho_{n,n^{\prime}}\Bra{n^{\prime}},$
(37)

which can be diagonalized to get Equation 4 by virtue of its hermiticity and
unit trace.

##  IV Discussion

This section applies the general results presented in prior sections to two
important examples that are often used to argue for the need of density
matrices representing mixed states in lieu of state vectors. These include (1)
the Stern Gerlach experiment, which in certain preparations finds the signal
to be independent of magnetic field direction and (2) the fact that many
processes – such as population decay – are irreversible and thus not
describable by unitary time evolution. This section illustrates that a simple
two-particle pure state can have these properties. Our aim is to show that
such behavior can result, and not to provide a many-particle calculation for a
real system, countering the suggestion of many texts that a density matrix is
required.

###  IV.1 Spin Mixtures

Fano argues that density matrices are needed when information about a system
is lacking. He uses as an example a particular preparation of an atom beam
measured by a Stern-Gerlach apparatus that yields a vanishing expectation
value of the spin along any arbitrary axis. He then asserts that a pure state
cannot explain this result. Specifically, he states that , “…lack of
information about a variable can often be expressed as a statement that
certain operators have expectation value zero. For example if the spin
orientation of a particle is wholly unknown, the expectation value of each
component of its angular momentum vanishes.” We offer Equation 9 as a
counterexample. It is a well-defined pure state that yields vanishing
expectation values of the spin of each of the two particles.

Using the state vector defined in Equation 9 the student can use Equation I
with $A_{1}$ being the one-particle spin operator to verify that the
expectation value of the spin of the $n^{\text{th}}$ electron is

$\displaystyle{\bf
S}^{(n)}=\Braket{\psi}{\sum_{p}S_{p}^{(n)}\hat{p}}{\psi}=0,$ (38)

where $S_{p}^{(n)}$ is the $p^{\text{th}}$ cartesian component of the spin
vector of the $n^{\text{th}}$ electron. The same result will be obtained
independent of the coordinate system. ${\bf S}^{(n)}=0$ satisfies Fano’s
requirement, which he claims implies that the state of the system is unknown,
yet we have shown that his requirement is met with a well-defined two-particle
state vector.

This example presents a teachable moment for how the density matrix can be
presented without giving the impression that the density matrix is an
extension of quantum mechanics. One possible set of exercises follows.

  1. 1.

Have the student calculate the expectation value of the spin, let’s say
$S_{z}$, for the general single-particle state
$\psi(\alpha)=\cos\alpha\Ket{\uparrow}+e^{i\phi}\sin\alpha\Ket{\downarrow}$ as
a function of $\alpha$ to illustrate that the expectation value depends on
$\alpha$. Alternatively, the state vector can be fixed and the spin operator
rotated to show that the expectation values agree with the behavior of
projections of classical spins.

  2. 2.

Have the student calculate the expectation value of the spin of one particle
in an entangled pair using the state vector given by Equation 9 to show that
the spin vanishes for all orientations of the single-particle spin operator.
This illustrates that the expectation value of one particle that is part of a
collection of many particles behaves quite differently than a particle in
isolation.

  3. 3.

Starting from the density matrix given by Equation 13, show that the
expectation value of the one-particle spin is the same as that given by Item
2.

  4. 4.

Explain to the student that the density matrix can be interpreted as
describing a population of spins, where half of them are spin up and half of
them are spin down. Stress that this property emerges from a pure state vector
of entangled particles, so the system is fully describable by quantum
mechanics, but that it is often convenient to use a density matrix rather than
a state vector for a many particle system.

After dong the above exercises, the student will appreciate that an
experimentalist measuring an expectation value of ${\bf S}=0$ would conclude
that the density matrix is given by Equation 13 but would have no knowledge of
the progenitor pure state. A more accurate statement than Fano’s is that a
particular density matrix determined from an experiment can correspond to many
possible pure states, but that does not imply that no such pure state exists;
a pure state is unique but a density matrix can be the offspring of an
infinite number of pure states.

This example dispels the notion that the density matrix embeds more
information than the state vector and that quantum mechanics is in some ways
lacking. Weinberg suggests that the density matrix might be the fundamental
object of nature and that physics can be described without state
vectors.weinb14.01  Since any density matrix of a one-particle subsystem can
be constructed from a superposition of two-particle states, the state vector
and density matrix for such a system are mathematically equivalent as long as
we keep in mind that the system is part of a larger universe. Thus, one is no
more fundamental than the other. This view is often referred to as “Church of
the Larger Hilbert Space,” following John Smolin of IBM.gottes00.01

###  IV.2 Time Evolution

This section shows that a pure two-particle state in an entangled stated and
in superposition can yield what appears to be non-unitary time evolution. In
particular, we answer the question of how an exponential population decay is
compatible with the unitary time evolution of a pure state, where probability
amplitudes normally oscillate in time. The research literature is filled with
examples where “Breakdown in unitarity can lead to exponential
behavior…”Beau17.01  Our narrative shows that a breakdown of unitarity is not
required.

Physicists model exponential population decay by adding a phenomenological
parameter to the time evolution equations of the density matrix. At first
glance, it would appear that quantum mechanics must be augmented or
generalized beyond the traditional theory to get the observed behavior.

In the density matrix formalism, exponential population decay originates in
the diagonal components of the density matrix. In contrast, the diagonal
components of a pure state are time independent. When the particles interact,
the diagonal elements become time dependent. In a nutshell, we use the one
particle subsystem of two interacting particles to find the specific
superposition that gives exponential decay. Why a system would be in that
particular superposition is not a question we address here; rather, our aim is
to show that an exponential decay is not at odds with the quantum-mechanical
description of nature.

The energy eigenvalue equation for interacting particles is given by Equation
31. Applying the time-evolution operator to Equation 36, which we emphasize
comes from a pure two-particle state, yields the time-dependent density matrix

$\displaystyle\rho_{n^{\prime},n}(t)$
$\displaystyle=\sum_{k^{\prime},k}a_{k^{\prime}}^{*}a_{k}\exp(-i(E_{k}-E_{k^{\prime}})t/\hbar)$
$\displaystyle\times\sum_{m}c_{n^{\prime},m}^{(k^{\prime})*}c_{n,m}^{(k)}.$
(39)

The diagonal component $\rho_{n,n}(t)$ is here time dependent and given by

$\displaystyle\rho_{n,n}(t)$
$\displaystyle=\sum_{k^{\prime},k}A_{k^{\prime},k}^{(n)}\exp(i(\omega_{k^{\prime}}-\omega_{k})t)\stackrel{{\scriptstyle?}}{{=}}b_{n}\exp(-\alpha
t),$ (40)

where $\omega_{k}=E_{k}/\hbar$ and $A_{k^{\prime},k}^{(n)}$ is obtained by
comparison of Equations 40 and IV.2, yielding

$\displaystyle
A_{k^{\prime},k}^{(n)}=a_{k^{\prime}}^{*}a_{k}\sum_{m}c_{n,m}^{(k^{\prime})*}c_{n,m}^{(k)},$
(41)

with

$\displaystyle A_{k^{\prime},k}^{(n)^{*}}=A_{k,k^{\prime}}^{(n)}.$ (42)

The question then is if the coefficients $A_{k,k^{\prime}}^{(n)}$ can be
chosen such that the time dependence is an exponential function. The answer is
“yes.” Appendix B shows the derivation for the special case where the energy
levels are equally spaced, as one finds in a harmonic oscillator or a particle
in free space. Equation 61 explicitly gives the coefficients. Thus, one of two
interacting particles that are in a superposition of one-particle states can
have diagonal density matrix elements that represent exponential population
decay. Asher makes a similar point, arguing that the apparent irreversibility
of a measurement resides in the fact that only part of the full state is being
observed, but calls upon populations using density matrices.peres74.01  Such
an exponential decay is usually modelled only with density matrices, but here
we see that the full quantum picture suffices, as should be the case for
measurements that obey unitary time evolution.maxwe72.01  This example closes
the loop, showing the student that properties that are beleived to require a
density matrix naturally emerge form a pure state.

###  IV.3 Many Particles

The most general many-particle state vector can be expanded as a superposition
of direct products of single-particle state vectors of the form

$\Ket{\psi^{(k)}}=\sum_{n,m,\dots,\ell}c_{n,m,\dots,\ell}^{(k)}\Ket{n,m,\dots,\ell},$
(43)

where normalization demands that

$\sum_{n,m,\dots,\ell}c_{n,m,\dots,\ell}^{(k)^{*}}c_{n,m,\dots,\ell}=1.$ (44)

It is not difficult to see that the generalization of Equation 36 to $N$
particles yields

$\displaystyle\rho_{n^{\prime},n}=\sum_{k^{\prime},k}a_{k^{\prime}}^{*}a_{k}\sum_{m,\dots,\ell}c_{n^{\prime},m,\dots,\ell}^{(k^{\prime})*}c_{n,m,\dots,\ell}^{(k)}.$
(45)

There are many ways that a two-particle universe and a many-particle one can
give the same density matrix. We stress again that full knowledge of the
density matrix does not allow one to determine a unique progenitor pure state.

###  IV.4 Multiple-Particle Operators

All calculations in this paper are limited to the class of one-particle
operators. If we were interested instead in a quantity such as the interaction
energy between a pair of particles, we would need to generalize the results to
two-particle operators $V_{12}$, which act on the first two kets. Then, we
would need to add at least one more particle to act as the environment. This
type of generalization is messy but straightforward and adds little insight
into density matrices.

##  V Conclusion

We have shown how the density matrix emerges from a purely quantum treatment
of a system and environment made of two particles. What appear to be classical
population fractions originate in the superposition coefficients of the purely
quantum state vector, illustrating that the density matrix does not imply that
the universe is made of a classical collection of populations of tiny quantum
systems but that the whole system can be quantum in nature. Rather, the
density matrix is a useful tool for studying systems that are part of a larger
universe whose state vector is unknown, but this does not necessarily imply
that new physics is required – as may be interpreted by some students in the
way this material is often presented.

We showed how the system and environment can be parsed by particle or region
of space, and find that particle parsing leads to less confusion than the more
intuitive separation by regions of space. The two-particle state vector model
of the universe as a system of particle interacting with an environment of one
particle yields every possible density matrix, showing that density matrices
can be a direct consequence of quantum mechanics without need for
augmentation.

Finally, the time evolution of the populations is shown to be a consequence of
particle interactions. While this calculation does not prove that the dynamics
of a particular many-body quantum system will obey exponential decay to
equilibrium, it illustrates how it is possible to get this behavior even in
the two-particle case.

Feynman once mused “It is unknown whether or not the unverse is in a pure
state.”feynm72.01  We may never know the state of the full universe; but, this
paper brings attention to the fact that if mixed states are not required to
explain physical phenomena, it is within the realm of possibilities that our
universe is in a pure state. Popescu and coworkers take this view a step
further, showing that thermodynamics may be a consequence of measuring a small
entangled subspace within a universe that is in a pure state.Popescu06.01  The
simple two-particle picture, which can be easily generalized to a larger
system, gives the student insights into such musings and could serve as an
inspiration to learn more.

Acknowledgements We thank the National Science Foundation (EFRI-
ODISSEI:1332271 and ECCS-1128076) for generously supporting work that
meandered into this unexpected direction. I am indebted to Mark C. Kuzyk for
illuminating discussions, Fred Gittes for insightful suggestions, Ethan
Crowell for helpful comments and suggestions, and Daniel Steck for useful
references.

##  Appendix A Summation Gymnastics

Here we describe useful manipulations of sums.

###  A.1 Sums of Differences

Consider the sum

$\displaystyle\sum_{i=0}^{N}\sum_{j=0}^{N}f(i-j),$ (46)

which is a function of the differences of the indices. Making the variable
change

$\displaystyle\delta=i-j$ (47)

Sum 46 can be re-expressed as

$\displaystyle\sum_{i=0}^{N}\sum_{j=0}^{N}f(i-j)=\sum_{\delta=-N}^{N}g(\delta)f(\delta),$
(48)

where $g(\delta)$ corresponds to the number of times $\delta=i-j$ appears in
the sum. For example, $\delta=0$ appears $N+1$ times. The terms with
$\delta=0$ are $i-j=0-0,1-1,\dots,N-N$. Convince yourselves that

$\displaystyle g(\delta)=N+1-|\delta|$ (49)

by listing some examples.

Next let’s add a coefficient $a_{i,j}$ to the sum, making it of the form

$\displaystyle\sum_{i=0}^{N}\sum_{j=0}^{N}a_{i,j}f(i-j).$ (50)

Expressing Sum 50 in terms of $\delta$ along the same lines as we did in
arriving at Equation 48 yields

$\displaystyle\sum_{i=0}^{N}\,$ $\displaystyle\,\sum_{j=0}^{N}a_{i,j}f(i-j)$
$\displaystyle=\sum_{k=-N}^{N}a_{k,k}f(0)$
$\displaystyle+\sum_{\delta=1}^{N}\sum_{k=0}^{N-\delta}\Big{[}a_{k+\delta,k}f(\delta)+a_{k,k+\delta}f(-\delta)\Big{]}.$
(51)

The details of the math are not important but the student should verify that
when $a_{i,j}=1$ for all $i$ and $j$, the $k$ sum gives Equation 49 yielding
the full sum given by Equation 48, as expected.

###  A.2 Continuum Limit of Finite Sums

We seek to evaluate the sum

$\displaystyle\sum_{\ell=j}^{k}f(\ell\omega_{0})=\sum_{\ell=j}^{k}f(\ell\omega_{0})\Delta\ell,$
(52)

where we use the fact that $\Delta\ell=(\ell+1)-\ell=1$, so the continuum
limit is given by

$\displaystyle\sum_{\ell=j}^{k}f(\ell\omega_{0})\Delta\ell\rightarrow\lim_{N\rightarrow\infty}\sum_{\ell=jN}^{kN}f(\frac{\ell}{N}\omega_{0})\frac{\Delta\ell}{N}.$
(53)

Defining $x=\ell/N$, Equation 53 becomes

$\displaystyle\lim_{N\rightarrow\infty}\sum_{\ell=jN}^{kN}f(\frac{\ell}{N}\omega_{0})\frac{\Delta\ell}{N}\rightarrow\int_{j}^{k}f(\omega_{0}x)\,dx.$
(54)

For convenience, we can make the variable change $\omega=\omega_{0}x$. Then,
the continuum limit of 54 becomes

$\displaystyle\sum_{\ell=j}^{k}f(\ell\omega_{0})\xrightarrow{\mbox{continuum}}\int_{j\omega_{0}}^{k\omega_{0}}d\omega\,\frac{f(\omega)}{\omega_{0}}=\int_{\omega_{j}}^{\omega_{k}}d\omega\,\frac{f(\omega)}{\omega_{0}}.$
(55)

##  Appendix B Finding the Coefficients

We evaluate Equation 40 in the special case where the frequencies are equally
spaced, that is, $\omega_{k^{\prime}}-\omega_{k}=\omega_{0}(k^{\prime}-k)$ –
true for free particles and the harmonic oscillator in one dimension. Such a
spectrum is amenable to re-indexing the sums in terms of the difference
$\delta=k^{\prime}-k$ using the procedure described in Appendix A.1. This
yields for the first equality in Equation 40

$\displaystyle\rho_{n,n}(t)=\lim_{N\rightarrow\infty}\sum_{k=0}^{N}\,\sum_{k^{\prime}=0}^{N}A_{k,k^{\prime}}^{(n)}e^{i\omega_{0}\delta
t}=\lim_{N\rightarrow\infty}\Big{(}\sum_{k=0}^{N}A_{k,k}^{(n)}$
$\displaystyle+\sum_{\delta=1}^{N}\sum_{k=0}^{N-\delta}\Big{[}A_{k+\delta,k}^{(n)}e^{+i\omega_{0}\delta
t}+A_{k,k+\delta}^{(n)}e^{-i\omega_{0}\delta t}\Big{]}\Big{)}.$ (56)

Defining

$\displaystyle B_{\delta}^{(n)}=\sum_{k=0}^{N-\delta}A_{k+\delta,k}^{(n)}$
(57)

and using Equation 42, Equation B takes the simplified form

$\displaystyle\rho_{n,n}(t)$
$\displaystyle=\lim_{N\rightarrow\infty}\Big{(}B_{0}^{(n)}$
$\displaystyle+\sum_{\delta=1}^{N}\Big{[}B_{\delta}^{(n)}e^{+i\omega_{0}\delta
t}+B_{\delta}^{(n)^{*}}e^{-i\omega_{0}\delta t}\Big{]}\Big{)}.$ (58)

Next, we express Equation B in the more compact form

$\displaystyle\rho_{n,n}(t)$
$\displaystyle=\lim_{N\rightarrow\infty}\sum_{\delta=-N}^{+N}B_{\delta}^{(n)}e^{+i\omega_{0}\delta
t},$ (59)

where evidently $B_{-\delta}^{(n)}=B_{\delta}^{(n)^{*}}$. Equation 59 is
transformed into the continuous limit as described in Appendix A.2 with
Equation 55, resulting in the continuum form of Equation 40:

$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega}{\omega_{0}}B^{(n)}(\omega/\omega_{0})\exp(i\omega
t),\stackrel{{\scriptstyle?}}{{=}}\theta(t)b_{n}\exp(-\alpha t),$ (60)

where we have added the step function $\theta(t)$ to set $t=0$ at the start of
decay.

Multiplying both sides of Equation 60 by $\exp(i\omega^{\prime}t)$,
integrating over time and solving for $B^{(n)}$ yields

$\displaystyle
B^{(n)}(\omega^{\prime}/\omega_{0})=\frac{\omega_{0}}{2\pi}\cdot\frac{b_{n}}{i\omega^{\prime}+\alpha}.$
(61)

These are the coefficients of the two-particle superposition that yields a
pure state that decays exponentially, showing that exponentials are compatible
with unitary time evolution.

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# A combinatorial view on derivations in bimodules

. . rutyunov

###### Abstract

This paper is devoted to derivations in bimodules over group rings using
previously proposed methods which are related to character spaces over
groupoids. The theorem describing the arising spaces of derivations is proved.
We consider some examples, in particular the case of
$(\sigma,\tau)$-derivations.

##  1 Introduction

Derivations in various associative algebras have been actively studied since
the middle of the last century. In particular, the following question, known
as the Johnson Problem (or ’’Derivation Problem’’), is widely known:

######  Question ([6], question 5.6.B).

Is it true that all derivations in $L_{1}(G)$ are inner?

Here and hereafter $G$ is a finitely generated generally noncommutative group.

A partial answer to this question was given by B. Johnson himself in [7], and
the most complete answer was found by V. Losert’s in [12]. A more detailed
description of the history of this problem is given in [4, 6].

In purely algebraic form, consider the group ring ${\mathbb{C}}[G]$, that is,
the space of all linear combinations of the form $\sum_{g\in G}x(g)g$, where
$x(\cdot)$ is a finite function – a function with a finite support. The
derivation in this case is a linear operator
$d:{\mathbb{C}}[G]\to{\mathbb{C}}[G]$ satisfying the Leibniz rule

$d(uv)=d(u)v+ud(v),\quad\forall u,v\in{\mathbb{C}}[G].$

The Johnson’s problem is whether there are derivation other than inner, i.e.,
having the form

$d_{a}:x\to[a,x],a\in{\mathbb{C}}[G].$

In this formulation of the problem, the algebra of outer derivations will be
nontrivial (see central derivations from [3] as well as [4]).

In the present paper we focus on the study of Banach spaces equipped with a
bimodule structure over a group ring ${\mathbb{C}}[G]$ of the following form.
Let us fix a norm $\||\cdot\|$ in ${\mathbb{C}}[G]$ and let ${\mathcal{A}}$ be
the closure of the group ring by this norm. ${\mathcal{A}}$ is not an algebra,
but it is naturally understood as a bimodule over the ring ${\mathbb{C}}[G]$.

The ring ${\mathbb{C}}[G]$ will be defined as space with the supremum norm
$\|\cdot\||_{s}$, i.e. for $\omega=\sum\limits_{g\in G}x(g)g$ give

$\|\omega\|_{s}:=\sup\limits_{g\in G}|x(g)|.$

The boundedness of operators will be defined as follows. Let us define the
norm of the operator $d$ as

$\|d\|=\sup\limits_{\omega\neq
0\in{\mathbb{C}}[G]}\frac{\|d(\omega)\|}{\|\omega\|_{s}}.$ (1.1)

######  Definition 1.1.

By the derivation over ${\mathbb{C}}[G]$ with values in bimodule
${\mathcal{A}}$ we will call a linear bounded operator
$d:{\mathbb{C}}[G]\to{\mathcal{A}}$ such that

$d(uv)=d(u)v+ud(v),\quad\forall u,v\in{\mathbb{C}}[G].$ (1.2)

The space of such operators we will denote as $Der({\mathcal{A}})$.

For a wide class of norms (including natural $\ell_{p}$) the following
statement will be proved.

######  Theorem 1.

If the norm $\|\cdot\|$ is subordinate to the supremum norm, then all
derivations in the bimodule ${\mathcal{A}}$ are quasi-inner.

Let us explain about which norms we are talking about.

######  Definition 1.2.

We will say that the norm $\|\cdot\|$ is subordinated to the norm
$\|\cdot\|_{s}$ iff

$\|\omega\|<\infty\implies\|\omega\|_{s}<\infty$

A rigorous definition of quasi-inner derivations is given below (see
definition 2.1). Informally speaking, quasi-inner derivations are such
operators that can be represented as a formal infinite sum of inner
derivations, i.e. the commutator $x\to[a,x]$, for the element $a$ not lying in
${\mathcal{A}}$ in general.

##  2 Preliminaries

The following definitions and results are based on [4, 5].

Let $\Gamma$ be a groupoid of connected action in which objects coincide with
elements of the group ${\hbox{\bf Obj}}(\Gamma)=G$, and morphisms are pairs of
elements of the group, i.e. ${\hbox{\bf Hom}\;}(\Gamma)=G\times G$. Moreover,
the morphism $\phi:=(u,v)$ has a beginning $s(\phi)=v^{-1}u$ and an end
$t(\phi)=uv^{-1}$. We will define endomorphisms (i.e. morphisms which source
and target coincide) $\phi$ as loops for clarity.

Then we define a character as a complex-valued function $\chi:{\hbox{\bf
Hom}\;}(\Gamma)\to{\mathbb{C}}$ such that

$\chi(\psi\circ\phi)=\chi(\psi)+\chi(\phi),$ (2.1)

for all pairs of linkable morphisms $\phi,\psi$.

The groupoid $\Gamma$ will be represented as an uncoupled union of
subgroupoids $\Gamma_{[u]}$, where $[u]$ is a class of conjugate elements.

$[u]=\\{tut^{-1}|t\in G\\}.$

Objects of the subgroupoid $\Gamma_{u}$ coincide with the class $u$.

######  Lemma 2.1.

For every derivation $d\in Der({\mathcal{A}}_{s})$ there exists a character
$\chi\in X(\Gamma)$ such that the following formula holds

$d(g)=g\left(\sum\limits_{t\in G}\chi(gt,g)t\right),\forall g\in G.$ (2.2)

###### Proof.

The proof Literally repeats the proof of Theorem 1 of [3] except for the
property of local finiteness, which was a consequence of the finiteness of
nonzero terms in elements of group algebras. ∎

Let us make it clear that character values are coefficients of the expansion
of a linear operator by standard basis of the ring ${\mathbb{C}}[G]$. Due to
the implementation of the Leibniz’s rule, the calculation shows that the
property (2.1) is fulfilled.

The space of quasi-inner derivations was introduced earlier in [5]. It will
play an important role below.

######  Definition 2.1.

We will define the derivation $d\in Der({\mathcal{A}}(G))$ as quasi-inner if
the character $\chi$ in the formula (2.2) is zero on all loops.

It is easy to see that the inner derivations are quasi-inner. Indeed, let
$a\in G$ be a basis element, then the character $\chi^{a}$ corresponding to
the inner derivation $d_{a}:x\to[x,a]$ has the form (see Section 2.2 in [3]
for details)

$\chi^{a}(\phi)=\begin{cases}1,&\phi\in{\hbox{\bf Hom}\;}(a,b),b\neq a,\\\
-1&\phi\in{\hbox{\bf Hom}\;}(b,a),b\neq a,\\\ 0&\textit{else}.\end{cases}$

The character $\chi^{a}$ is zero on all loops. The inner derivation is a sum
of (possibly infinite) characters $\chi^{a}$, and hence is itself trivial on
loops.

The role of quasi-inner derivations for group rings is that they form an ideal
containing the ideal of inner derivations (Theorem 4.1, [5]), with examples of
quasi-inner derivations that would not be inner in the group ring (Section 3.3
of [3] – the case of Heisenberg group).

The question of coincidence spaces of inner and quasi-inner derivations in a
bimodule ${\mathcal{A}}$ is equivalent to the fact that inner derivations are
dense in the space of quasi-inner derivations.

##  3 Derivations in bimodules

Recall that in the group ring ${\mathbb{C}}[G]$ we have a ’’supremum norm’’.
For $x=\sum\limits_{g\in G}x(g)g\in{\mathbb{C}}[G]$, where $x(\cdot)$ is a
finite function

$\|x\|_{s}:=\sup\limits_{g\in G}|x(g)|.$ (3.1)

The Banach bimodule which is the closure of ${\mathbb{C}}[G]$ by the supremum
norm is denoted by ${\mathcal{A}}_{s}(G)$. The bimodule ${\mathcal{A}}_{s}(G)$
can be understood as the space of elements of the form $\sum\limits_{g\in
G}x(g)g$, where the function $x(g)$ is bounded.

######  Lemma 3.1.

All derivations with values in the Banach bimodule ${\mathcal{A}}_{s}$ are
quasi-inner.

###### Proof.

Let $\chi$ be the character corresponding in the sense of the lemma 2.1 to a
nontrivial derivation $d$. Let us show that if the character $\chi$ takes a
nonzero value on some loop, then the operator $d$ is unbounded.

Consider the loop $\phi\in Hom(g,g)$. As can be seen from the definition, for
some $t\in Z(g)$, the notation $\phi=(gt,t)$ is valid.

Let $\chi(\phi)\neq 0$. Without loss of generality, we may assume that
$\chi(\phi)=1$.

Note that the element $t$ cannot be of finite order in the group $G$,
otherwise we would get that $\chi((g,e))=ord(t)$, which is impossible since
$(g,e)$ is a neutral morphism.

Given that $t$ is of infinite order, using the property (2.1), we have

$\chi((gt^{n},t^{n}))=\chi(\phi^{n})=n.$

with the formula (2.2) we get that $\|d(t^{n})\|_{s}\geq n$. So
$\|d(t^{n})\||_{s}\to\infty$ at $n\to\infty$. At the same time
$\|gt^{n}\||_{s}=1$. In this case the operator $d$ – converts a bounded
sequence into an unbounded one, and hence is not bounded itself. ∎

######  Proposition 3.1.

If the character $\chi$ gives the derivation $d\in Der({\mathcal{A}}_{s})$,
then for any two morphisms $\phi,\psi\in{\hbox{\bf Hom}\;}(a,b)$ we have that
$\chi(\phi)=\chi(\psi)$.

###### Proof.

By the lemma 3.1 for each derivation $d\in Der({\mathcal{A}}_{s})$, the
corresponding character $\chi$, in the sense of the lemma 2.1, is zero on
loops. For two different objects $a\neq b$, and two morphisms
$\phi,\psi\in{\hbox{\bf Hom}\;}(a,b)$ (if they exist, of course) there exists
a loop $\zeta\in{\hbox{\bf Hom}\;}(a,a)$ such that $\phi\circ\zeta=\psi$. So,
since $\chi(\zeta)=0$, by the formula (2.1) we have that
$\chi(\phi)=\chi(\psi)$. ∎

Let us proceed to the proof of the 1 theorem, i.e. we show that if norm
$\|\cdot\|$ is subject to the supremum norm $\|\cdot\|_{s}$, then all
derivations with values in the bimodule ${\mathcal{A}}$ are quasi-inner.

###### Proof of theorem 1.

Consider a bounded operator $d$ over ${\mathcal{A}}$. From boundedness we
obtain that for some $A$: $\|d(g)\|<A$ for all basis elements $g\in G$.

From the norm subordination we have that $\|d(g)\|_{s}<CA$. So the character
$\chi$ corresponding to the derivation $d$ is trivial on loops by the lemma
3.1, and so $d$ is a quasi-inner derivation. ∎

##  4 Examples and applications

Consider the space $\ell_{p}(G)$, i.e., all elements of the form
$\omega=\sum_{g\in G}x(g)g$, bounded by the $\ell_{p}$-norm

$\|\omega\|_{p}:=\sqrt[p]{\sum\limits_{g\in G}|x(g)|^{p}}.$ (4.1)

It is clear that such norm is subordinate to the supremum if $p\geq 1$:

$\|\omega\|_{p}\leq\sup_{g\in G}|x(g)|.$

######  Example 4.1.

All derivations in $Der(\ell_{p}(G)),p\geq 1$ are quasi-inner.

In [8] was shown that for $p=1$ all derivations are inner. The case $p>1$ is
new.

### The case of $(\sigma,\tau)$-derivations

It is clear that in the proof of Theorem 1 the ability to represent
derivations via groupoid characters plays the key role. This construction can
be applied to other structures as well, with a corresponding change in the
groupoid structure. Thus, in [9], $(\sigma,\tau)$-derivations, i.e., operators
satisfying a ’’twisted’’ Leibniz rule, were investigated. Conditions and
applications of $(\sigma,tau)$-derivations can be found in [10, 11].

Let us give a slightly more general definition to fit our case.

######  Definition 4.1.

A linear bounded operator $d:{\mathbb{C}}[G]\to{\mathcal{A}}$ such that for
some endomorphisms $\sigma,\tau:{\mathbb{C}}[G]\to{\mathbb{C}}[G]$

$d(ab)=d(a)\sigma(b)+\tau(a)d(v),\quad a,b\in{\mathbb{C}}[G],$ (4.2)

let’s call $(\sigma,\tau)-$derivation.

As shown in [9] for the ’’twisted’’ groupoid (see [9], Section 3)
$(\sigma,\tau)-$derivation can be presented by its characters ([9], Theorem
1). Applying the same reasoning as in the proof of the theorem 1 to the case
of $(\sigma,\tau)-$derivations we obtain the ’’twisted’’ analog of our
theorem.

######  Example 4.2.

All $(\sigma,\tau)-$derivations with values in bimodule ${\mathcal{A}}$
generated by a norm subordinate supremum – will be ($\sigma,\tau$)-quasi-
inner.

Here ($\sigma,\tau$)-quasi-inner derivations are defined similarly as
($\sigma,\tau$)-derivations given by characters identically equal to zero on
loops.

### Central derivations

The proved theorem and the proposed approach also suggest examples of norms in
which outer derivations appear.

Recall (see [3], Section 2.3) that the central derivation $d^{t}_{z}$ is an
operator which is given by a center element of the group $z\in Z(G)$ and a
non-trivial homomorphism $t:G\to({\mathbb{C}},+)$ into the additive group of
complex numbers on generators $g\in G\subset{\mathbb{C}}[G]$ by the formula

$d^{t}_{z}:g\to\tau(g)gz.$ (4.3)

Central derivations form a subalgebra ([3], Theorem 2). It is easy to see that
the character $\chi^{t}_{z}$ corresponding in the sense of the lemma 2.1 is
non-trivial on loops (see [3], proposition 5). More precisely, its carrier is
a subgroupoid with one object (just a fixed central element). From which we
obtain that the central derivation cannot be quasi-inner.

A simple calculation shows that the central derivations are not bounded in the
norm supremum, which of course follows from the theorem 1. However, it is
possible to choose a norm which is not subordinate to the supremum111The
following example is proposed by A. Nayanzin.

We define the norm $\|\cdot\|^{*}_{\alpha}$ for $\omega=\sum_{g\in G}x(g)g$ as
follows

$\|\omega\|^{*}_{\alpha}:=\sum\limits_{g\in G}|x(g)|e^{-\alpha|g|}$ (4.4)

We denote the closure of ${\mathbb{C}}[G]$ by this norm by
${\mathcal{A}}^{*}_{\alpha}$.

######  Example 4.3.

For all sufficiently large $\alpha>0$, central derivations with values in the
bimodule ${\mathcal{A}}^{*}_{\alpha}$ are bounded.

The proof is achieved by a simple calculation.

In the case of a group ring (i.e. the case of finite linear combinations
without normalized space structure) the ideals of inner and quasi-inner
derivations do not coincide (see [3], the case of Heisenberg group). Moreover,
the difference between inner and quasi-inner derivations is determined by the
combinatorial properties of the group. As the proof of the theorem 1 shows in
the case of normalized bimodules, this dependence vanishes. So the following
assumption seems plausible.

######  Question.

All derivations in $\ell_{p}(G)$ are inner.

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