https://cs.uwaterloo.ca/$\scriptstyle\sim$watrous/

# Time-reversal of rank-one quantum strategy
functions

Yuan Su  Department of Computer Science, Institute for Advanced Computer
Studies, and Joint Center for Quantum Information and Computer Science,
University of Maryland, USA  http://quics.umd.edu/people/yuan-su John Watrous
Institute for Quantum Computing and School of Computer Science, University of
Waterloo, Canada  Canadian Institute for Advanced Research, Toronto, Canada

(October 1, 2018)

###### Abstract

The _quantum strategy_ (or _quantum combs_) framework is a useful tool for
reasoning about interactions among entities that process and exchange quantum
information over the course of multiple turns. We prove a time-reversal
property for a class of linear functions, defined on quantum strategy
representations within this framework, that corresponds to the set of rank-one
positive semidefinite operators on a certain space. This time-reversal
property states that the maximum value obtained by such a function over all
valid quantum strategies is also obtained when the direction of time for the
function is reversed, despite the fact that the strategies themselves are
generally not time reversible. An application of this fact is an alternative
proof of a known relationship between the conditional min- and max-entropy of
bipartite quantum states, along with generalizations of this relationship.

##  1 The quantum strategy framework

The _quantum strategy framework_ [9], which is also known as the _quantum
combs framework_ [2, 4], provides a useful framework for reasoning about
networks of quantum channels. It may be used to model scenarios in which two
or more entities, which we will call _players_ , process and exchange quantum
information over the course of multiple rounds of communication; and it is
particularly useful when one wishes to consider an optimization over all
possible behaviors of one player, for any given specification of the other
player or players. Various developments, applications, and variants of the
quantum strategy framework can be found in [3, 5, 1, 8, 10], for instance, and
in a number of other sources.

In the discussion of the quantum strategy framework that follows, as well as
in the subsequent sections of this paper, we assume that the reader is
familiar with quantum information theory and semidefinite programming.
References on this material include [13, 15, 11, 16] as well as [14], which we
follow closely with respect to notation and terminology. In particular, we
denote quantum registers by capital sans serif letters such as $\mathsf{X}$,
$\mathsf{Y}$, and $\mathsf{Z}$ (sometimes with natural number subscripts),
while the same letters (with matching subscripts) in a scripted font, such as
$\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ denote the complex Euclidean
spaces (i.e., finite-dimensional complex Hilbert spaces) associated with the
corresponding registers. The set $\mathrm{L}(\mathcal{X},\mathcal{Y})$ denotes
the set of all linear operators from $\mathcal{X}$ to $\mathcal{Y}$;
$\mathrm{L}(\mathcal{X})$ is a shorthand for
$\mathrm{L}(\mathcal{X},\mathcal{X})$; $\mathrm{Herm}(\mathcal{X})$,
$\mathrm{Pos}(\mathcal{X})$, $\mathrm{D}(\mathcal{X})$, and
$\mathrm{U}(\mathcal{X})$ denote the sets of all Hermitian operators, positive
semidefinite operators, density operators, and unitary operators acting on
$\mathcal{X}$; $\mathrm{C}(\mathcal{X},\mathcal{Y})$ denotes the set of all
channels (i.e., completely positive and trace-preserving maps) mapping
$\mathrm{L}(\mathcal{X})$ to $\mathrm{L}(\mathcal{Y})$; and
$\mathrm{C}(\mathcal{X})$ is a shorthand for
$\mathrm{C}(\mathcal{X},\mathcal{X})$. The adjoint of an operator $A$ is
denoted $A^{\ast}$, the entry-wise complex conjugate is denoted
$\overline{A}$, and the transpose is denoted
$A^{{\scriptscriptstyle\mathsf{T}}}$. A similar notation is used for the
adjoint and transpose of a channel $\Phi$ (the meaning of which, in the case
of the transpose, will be clarified later). The (Hilbert-Schmidt) inner-
product is defined as $\langle A,B\rangle=\operatorname{Tr}(A^{\ast}B)$ for
all operators $A,B\in\mathrm{L}(\mathcal{X})$. Some additional notation will
be introduced as it is used.

### An example of a six-message interaction

To explain the aspects of the quantum strategy framework that are relevant to
this paper, we will begin by discussing an example of an interaction structure
involving six messages exchanged between two players, Alice and Bob. We have
chosen to describe a six-message interaction because it is simple and
concrete, but nevertheless clearly suggests the underlying structure of an
interaction having any finite number of message exchanges. Our main result
holds in the general case, which will be considered later, where an arbitrary
finite number of message exchanges may take place.

$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$$\Psi_{4}$$\Phi_{1}$$\Phi_{2}$$\Phi_{3}$$\mathsf{W}_{1}$$\mathsf{W}_{2}$$\mathsf{W}_{3}$$\mathsf{W}_{4}$$\mathsf{X}_{1}$$\mathsf{X}_{2}$$\mathsf{X}_{3}$$\mathsf{Y}_{1}$$\mathsf{Y}_{2}$$\mathsf{Y}_{3}$$\mathsf{Z}_{1}$$\mathsf{Z}_{2}$AliceBob
Figure 1: A six message interaction between Alice and Bob, after which Bob
produces a measurement outcome.

Figure 1 illustrates an interaction between Alice and Bob. In this figure,
time proceeds from left to right, and the arrows represent registers either
being sent from one player to the other (as is the case for the registers
$\mathsf{X}_{1}$, $\mathsf{Y}_{1}$, $\mathsf{X}_{2}$, $\mathsf{Y}_{2}$,
$\mathsf{X}_{3}$, and $\mathsf{Y}_{3}$), or momentarily stored by one of the
two players (as is the case for $\mathsf{Z}_{1}$ and $\mathsf{Z}_{2}$, stored
by Alice, and $\mathsf{W}_{1}$, $\mathsf{W}_{2}$, $\mathsf{W}_{3}$,
$\mathsf{W}_{4}$, stored by Bob). Alice’s actions are represented by the
channels $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$, and Bob’s actions are
represented by the channels $\Psi_{1}$, $\Psi_{2}$, $\Psi_{3}$, and
$\Psi_{4}$, as well as a final measurement, which is not given a name in the
figure.

Suppose that Bob’s specification has been fixed, including his choices for the
channels $\Psi_{1}$, $\Psi_{2}$, $\Psi_{3}$, and $\Psi_{4}$, as well as his
final measurement, and suppose further that one of Bob’s possible measurement
outcomes is to be viewed as desirable to Alice. It is then natural to consider
an optimization over Alice’s possible actions, maximizing the probability that
Bob’s measurement produces the outcome Alice desires. The quantum strategy
framework reveals that this optimization problem can be expressed as a
semidefinite program, in the manner that will now be described.

First, a single channel $\Xi_{3}$ that transforms
$(\mathsf{X}_{1},\mathsf{X}_{2},\mathsf{X}_{3})$ to
$(\mathsf{Y}_{1},\mathsf{Y}_{2},\mathsf{Y}_{3})$ is associated with any given
choice for Alice’s actions. That is, the channel $\Xi_{3}$ takes the form

$\Xi_{3}\in\mathrm{C}(\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3},\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3}),$
(1)

and for a particular selection of $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$ may
be expressed as

$\Xi_{3}=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2})}\otimes\Phi_{3}\bigr{)}\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1})}\otimes\Phi_{2}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{3})}\bigr{)}\bigl{(}\Phi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\mathcal{X}_{3})}\bigr{)}.$
(2)

Formally speaking, this composition requires that we view $\Phi_{1}$,
$\Phi_{2}$, and $\Phi_{3}$ as channels of the form
$\Phi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1}\otimes\mathcal{Z}_{1})$,
$\Phi_{2}\in\mathrm{C}(\mathcal{Z}_{1}\otimes\mathcal{X}_{2},\mathcal{Y}_{2}\otimes\mathcal{Z}_{2})$,
and
$\Phi_{3}\in\mathrm{C}(\mathcal{Z}_{2}\otimes\mathcal{X}_{3},\mathcal{Y}_{3})$,
as opposed to the forms
$\Phi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Z}_{1}\otimes\mathcal{Y}_{1})$,
$\Phi_{2}\in\mathrm{C}(\mathcal{Z}_{1}\otimes\mathcal{X}_{2},\mathcal{Z}_{2}\otimes\mathcal{Y}_{2})$,
and
$\Phi_{3}\in\mathrm{C}(\mathcal{Z}_{2}\otimes\mathcal{X}_{3},\mathcal{Y}_{3})$
suggested by Figure 1, so that the ordering of the tensor factors of the
various input and output spaces is consistent with the composition. Similar
re-orderings of tensor factors should be assumed implicitly throughout this
paper as needed. This understanding should not be a source of confusion
because we always assign distinct names to distinct registers (and their
associated spaces). Figure 2 illustrates the action of the channel $\Xi_{3}$,
which in words may be described as the channel obtained if all three of the
registers $(\mathsf{X}_{1},\mathsf{X}_{2},\mathsf{X}_{3})$ are provided
initially, and then Alice’s actions are composed in the natural way to produce
$(\mathsf{Y}_{1},\mathsf{Y}_{2},\mathsf{Y}_{3})$ as output registers.

$\Phi_{1}$$\Phi_{2}$$\Phi_{3}$$\mathsf{X}_{1}$$\mathsf{X}_{2}$$\mathsf{X}_{3}$$\mathsf{Y}_{1}$$\mathsf{Y}_{2}$$\mathsf{Y}_{3}$$\mathsf{Z}_{1}$$\mathsf{Z}_{2}$
Figure 2: The channel $\Xi_{3}$ that describes Alice’s actions in the
interaction illustrated in Figure 1.

It may appear that by considering the channel $\Xi_{3}$, one is ignoring the
possibility that Bob’s actions could, for instance, allow the contents of
$\mathsf{Y}_{1}$ or $\mathsf{Y}_{2}$ to influence what is input into
$\mathsf{X}_{2}$ or $\mathsf{X}_{3}$. Despite this appearance, the influence
that Alice’s actions have from the viewpoint of Bob, including the probability
for each of his measurement outcomes to appear, is uniquely determined by the
channel $\Xi_{3}$.

Naturally, not all channels of the form (1) will arise from a composition of
channels $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$ as in (2); the fact that
$\Phi_{1}$ is effectively performed first, $\Phi_{2}$ is performed second, and
$\Phi_{3}$ is performed third imposes constraints on the channels $\Xi_{3}$
that can be obtained. In particular, consider the channel that results when
$\Xi_{3}$ is performed and then the partial trace is performed on
$\mathcal{Y}_{3}$. As $\Phi_{3}$ is a channel, discarding its output is
equivalent to discarding its inputs, from which it follows that

$\text{Tr}_{\mathcal{Y}_{3}}\circ\Xi_{3}=\Xi_{2}\circ\text{Tr}_{\mathcal{X}_{3}},$
(3)

where the circles represent channel compositions and
$\Xi_{2}\in\mathrm{C}(\mathcal{X}_{1}\otimes\mathcal{X}_{2},\mathcal{Y}_{1}\otimes\mathcal{Y}_{2})$
is the channel defined as

$\Xi_{2}=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1})}\otimes(\textup{Tr}_{\mathcal{Z}_{2}}\circ\Phi_{2})\bigr{)}\bigl{(}\Phi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2})}\bigr{)}.$
(4)

That is, $\Xi_{2}$ is the channel obtained from $\Phi_{1}$ and $\Phi_{2}$,
followed by the partial trace over $\mathcal{Z}_{2}$, by a similar process to
the one used to obtain $\Xi_{3}$. By similar reasoning, one finds that

$\text{Tr}_{\mathcal{Y}_{2}}\circ\Xi_{2}=\Xi_{1}\circ\text{Tr}_{\mathcal{X}_{2}},$
(5)

where $\Xi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1})$ is the channel
given by $\Xi_{1}=\textup{Tr}_{\mathcal{Z}_{1}}\circ\Phi_{1}$.

Somewhat remarkably, this is not only a necessary condition on the channel
$\Xi_{3}$, but also a sufficient one, for it to be obtained from a composition
of channels $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$ as described above. That
is, given any channel

$\Xi_{3}\in\mathrm{C}(\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3},\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3})$
(6)

satisfying (3) and (5), for some choice of channels

$\begin{gathered}\Xi_{2}\in\mathrm{C}(\mathcal{X}_{1}\otimes\mathcal{X}_{2},\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}),\\\
\Xi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1}),\end{gathered}$ (7)

there must exist channels

$\begin{gathered}\Phi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1}\otimes\mathcal{Z}_{1}),\\\
\Phi_{2}\in\mathrm{C}(\mathcal{Z}_{1}\otimes\mathcal{X}_{2},\mathcal{Y}_{2}\otimes\mathcal{Z}_{2}),\\\
\Phi_{3}\in\mathrm{C}(\mathcal{Z}_{2}\otimes\mathcal{X}_{3},\mathcal{Y}_{3}),\end{gathered}$
(8)

for spaces $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ having sufficiently large
dimension, so that (2) holds. This fact is proved in [2, 4, 9], and we note
that a key idea through which this equivalence is proved may be found in [7].

The next step toward an expression of the optimization problem suggested above
as a semidefinite program makes use of the Choi representation of channels.
The Choi representation of the channel $\Xi_{3}$ takes the form

$J(\Xi_{3})\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}),$
(9)

as the complete positivity of $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$ implies
that $\Xi_{3}$ is also completely positive, and therefore $J(\Xi_{3})$ is
positive semidefinite. The constraints on the channel $\Xi_{3}$ described
previously correspond (conveniently) to linear constraints; one has that (3)
and (5) hold, for some choice of channels $\Xi_{2}$ and $\Xi_{1}$, if and only
if the Choi representation $X_{3}=J(\Xi_{3})$ of $\Xi_{3}$ satisfies

$\begin{gathered}\operatorname{Tr}_{\mathcal{Y}_{3}}(X_{3})=X_{2}\otimes\mathds{1}_{\mathcal{X}_{3}},\\\
\operatorname{Tr}_{\mathcal{Y}_{2}}(X_{2})=X_{1}\otimes\mathds{1}_{\mathcal{X}_{2}},\\\
\operatorname{Tr}_{\mathcal{Y}_{1}}(X_{1})=\mathds{1}_{\mathcal{X}_{1}},\end{gathered}$
(10)

for some choice of operators

$\begin{gathered}X_{2}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}),\\\
X_{1}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{X}_{1}).\end{gathered}$
(11)

These operators correspond to the Choi representations $X_{2}=J(\Xi_{2})$ and
$X_{1}=J(\Xi_{1})$.

Finally, the probability that Bob’s measurement produces a given outcome is a
linear function of the channel $\Xi_{3}$, and is therefore a linear function
of the Choi representation $X_{3}=J(\Xi_{3})$. Although this process is not
relevant to the main result of this paper, we note that it is possible to
obtain an explicit description of this linear function given a specification
of Bob’s actions, including his final measurement. In somewhat vague terms,
the linear function describing Bob’s probability to produce a particular
measurement outcome is given by $\langle P,X_{3}\rangle$, where

$P\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3})$
(12)

is an operator that is obtained from $\Psi_{1}$, $\Psi_{2}$, $\Psi_{3}$,
$\Psi_{4}$, and the measurement operator corresponding to the outcome being
considered by a process very similar to the one through which $X_{3}$ is
obtained from $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$. The reader is again
referred to [9, 2, 4] for further details.

More generally, an arbitrary real-valued linear function of the operator
$X_{3}$ may be expressed as $\langle H,X_{3}\rangle$ for some choice of a
Hermitian operator

$H\in\mathrm{Herm}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}),$
(13)

which need not represent the probability with which a particular measurement
outcome is obtained for channels $\Psi_{1},\ldots,\Psi_{4}$ followed by a
measurement. Such a function could, for instance, represent an expected payoff
for Alice’s actions, under the assumption that a real-valued payoff is
associated with each of Bob’s measurement outcomes.

### General semidefinite programming formulation

As mentioned previously, the six-message example just described generalizes to
any finite number of message exchanges. If the number of message exchanges is
equal to $n$, the input registers to Alice (the player whose actions are being
optimized) are $\mathsf{X}_{1},\ldots,\mathsf{X}_{n}$, and the output
registers of Alice are $\mathsf{Y}_{1},\ldots,\mathsf{Y}_{n}$, then the
possible strategies for Alice are represented by channels of the form

$\Xi_{n}\in\mathrm{C}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n},\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})$
(14)

that obey constraints that generalize (3) and (5). Specifically, there must
exist channels

$\begin{gathered}\Xi_{n-1}\in\mathrm{C}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n-1},\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1})\\\
\vdots\\\ \Xi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1})\end{gathered}$
(15)

such that

$\text{Tr}_{\mathcal{Y}_{k}}\circ\Xi_{k}=\Xi_{k-1}\circ\text{Tr}_{\mathcal{X}_{k}}$
(16)

for all $k\in\\{2,\ldots,n\\}$. For the maximization of a real-valued linear
function over all strategies for Alice, represented by a Hermitian operator

$H\in\mathrm{Herm}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}),$
(17)

one obtains the semidefinite program described in Figure 3. The primal problem
corresponds to an optimization over all Choi representations of the channels
$\Xi_{1},\ldots,\Xi_{n}$. This semidefinite programming formulation is
implicit in [9], and first appeared explicitly in [8]. It also appears in [1],
where it was used to define a generalized notion of min-entropy for quantum
networks.

Primal problem

maximize: $\displaystyle\langle H,X_{n}\rangle$ subject to:
$\displaystyle\operatorname{Tr}_{\mathcal{Y}_{n}}(X_{n})=X_{n-1}\otimes\mathds{1}_{\mathcal{X}_{n}},$
$\displaystyle\qquad\vdots$
$\displaystyle\operatorname{Tr}_{\mathcal{Y}_{2}}(X_{2})=X_{1}\otimes\mathds{1}_{\mathcal{X}_{2}},$
$\displaystyle\operatorname{Tr}_{\mathcal{Y}_{1}}(X_{1})=\mathds{1}_{\mathcal{X}_{1}},$
$\displaystyle
X_{n}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}),$
$\displaystyle\qquad\vdots$ $\displaystyle
X_{2}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}),$
$\displaystyle X_{1}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{X}_{1}).$

Dual problem

minimize: $\displaystyle\operatorname{Tr}(Y_{1})$ subject to: $\displaystyle
Y_{n}\otimes\mathds{1}_{\mathcal{Y}_{n}}\geq H,$ $\displaystyle
Y_{n-1}\otimes\mathds{1}_{\mathcal{Y}_{n-1}}\geq\operatorname{Tr}_{\mathcal{X}_{n}}(Y_{n}),$
$\displaystyle\qquad\vdots$ $\displaystyle
Y_{1}\otimes\mathds{1}_{\mathcal{Y}_{1}}\geq\operatorname{Tr}_{\mathcal{X}_{2}}(Y_{2}),$
$\displaystyle
Y_{n}\in\mathrm{Herm}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}),$
$\displaystyle
Y_{n-1}\in\mathrm{Herm}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-2}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n-1}),$
$\displaystyle\qquad\vdots$ $\displaystyle
Y_{1}\in\mathrm{Herm}(\mathcal{X}_{1}).$

Figure 3: The semidefinite program representing a maximization of a linear
function of an $n$-turn strategy.

It may be noted that the general problem just formulated concerns interactions
involving an even number of register exchanges, where Alice (the player whose
actions are being optimized) always receives the first transmission,
represented by $\mathsf{X}_{1}$, and sends the last transmission, represented
by $\mathsf{Y}_{n}$. However, one is free to take either or both of the
registers $\mathsf{X}_{1}$ and $\mathsf{Y}_{n}$ to be trivial registers, so
that correspondingly $\mathcal{X}_{1}=\mathbb{C}$ and/or
$\mathcal{Y}_{n}=\mathbb{C}$. This is tantamount to allowing either an odd
number of register exchanges or an even number in the situation that Alice
sends the first (nontrivial) register and receives the last.

##  2 Statement and proof of the main result

The main result of the current paper concerns the optimization problem
described in the previous section, as represented by the semidefinite program
in Figure 3, in the case that $H=uu^{\ast}$ is a rank one positive
semidefinite operator. The result to be described does not hold in general
when $H$ does not take this form.

In order to explain the main result in precise terms, it will be helpful to
introduce some notation. Suppose that a positive integer $n$ along with spaces
$\mathcal{X}_{1},\ldots,\mathcal{X}_{n}$ and
$\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n}$ have been fixed. For each
$k\in\\{1,\ldots,n\\}$, let

$\mathcal{S}_{k}(\mathcal{X}_{1},\ldots,\mathcal{X}_{k};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{k})\subset\mathrm{Pos}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{k}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{k})$
(18)

denote the primal-feasible choices for the operator $X_{k}$ in the
semidefinite program specified in Figure 3. That is, we define

$\mathcal{S}_{1}(\mathcal{X}_{1};\mathcal{Y}_{1})=\bigl{\\{}X_{1}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\mathcal{X}_{1})\,:\,\operatorname{Tr}_{\mathcal{Y}_{1}}(X_{1})=\mathds{1}_{\mathcal{X}_{1}}\bigr{\\}}$
(19)

(which is the set of all Choi operators of channels of the form
$\Xi_{1}\in\mathrm{C}(\mathcal{X}_{1},\mathcal{Y}_{1})$), and

$\begin{multlined}\mathcal{S}_{k}(\mathcal{X}_{1},\ldots,\mathcal{X}_{k};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{k})\\\\[5.69054pt]
=\bigl{\\{}X_{k}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{k}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{k})\,:\,\operatorname{Tr}_{\mathcal{Y}_{k}}(X_{k})=X_{k-1}\otimes\mathds{1}_{\mathcal{X}_{k}}\\\
\text{for
some}\;X_{k-1}\in\mathcal{S}_{k-1}(\mathcal{X}_{1},\ldots,\mathcal{X}_{k-1};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{k-1})\bigr{\\}}\end{multlined}\mathcal{S}_{k}(\mathcal{X}_{1},\ldots,\mathcal{X}_{k};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{k})\\\\[5.69054pt]
=\bigl{\\{}X_{k}\in\mathrm{Pos}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{k}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{k})\,:\,\operatorname{Tr}_{\mathcal{Y}_{k}}(X_{k})=X_{k-1}\otimes\mathds{1}_{\mathcal{X}_{k}}\\\
\text{for
some}\;X_{k-1}\in\mathcal{S}_{k-1}(\mathcal{X}_{1},\ldots,\mathcal{X}_{k-1};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{k-1})\bigr{\\}}$
(20)

for $k\in\\{2,\ldots,n\\}$. The primal form of the semidefinite program
described in Figure 3 can therefore be expressed succinctly as

maximize: $\displaystyle\langle H,X\rangle$ (21) subject to: $\displaystyle
X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n}).$

We will refer to operators in the sets defined above as _strategy operators_ ,
as they represent $n$-turn strategies with respect to the quantum strategy
framework.

Let us also define an isometry

$W\in\mathrm{U}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n},\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{1}\otimes\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1})$
(22)

by the action

$W(y_{1}\otimes\cdots\otimes y_{n}\otimes x_{1}\otimes\cdots\otimes
x_{n})=x_{n}\otimes\cdots\otimes x_{1}\otimes y_{n}\otimes\cdots\otimes y_{1}$
(23)

for all vectors $x_{1}\in\mathcal{X}_{1},\ldots,x_{n}\in\mathcal{X}_{n}$ and
$y_{1}\in\mathcal{Y}_{1},\ldots,y_{n}\in\mathcal{Y}_{n}$. In words, $W$ simply
reverses the order of the tensor factors of the space
$\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}$,
yielding a vector in
$\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{1}\otimes\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1}$
that, aside from this re-ordering of tensor factors, is the same as its input
vector.

### Statement of the main result

With the notation just introduced in hand, the main theorem may now be stated.

######  Theorem 1.

Let $\mathcal{X}_{1},\ldots,\mathcal{X}_{n}$ and
$\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n}$ be complex Euclidean spaces, for $n$
a positive integer, let

$u\in\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}$
(24)

be a vector, and let

$X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$
(25)

be a strategy operator. There exists a strategy operator

$Y\in\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$
(26)

such that

$\langle Wuu^{\ast}W^{\ast},Y\rangle\geq\langle uu^{\ast},X\rangle.$ (27)

If it is the case that
$\dim(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})\leq\dim(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$,
then the operator $Y$ may be chosen so that equality holds in (27).

######  Corollary 2.

Let $\mathcal{X}_{1},\ldots,\mathcal{X}_{n}$ and
$\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n}$ be complex Euclidean spaces, for $n$
a positive integer, and let

$u\in\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}$
(28)

be a vector. The semidefinite optimization problems

maximize: $\displaystyle\langle uu^{\ast},X\rangle$ (29) subject to:
$\displaystyle
X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$

and

maximize: $\displaystyle\langle Wuu^{\ast}W^{\ast},Y\rangle$ (30) subject to:
$\displaystyle
Y\in\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$

have the same optimum value.

######  Remark.

Using the notation introduced in [1], which defines a quantum network
generalization of conditional min-entropy, the equivalence expressed by
Corollary 2 may alternatively be written

$\begin{multlined}\operatorname{H}_{\text{min}}\bigl{(}\mathsf{Y}_{n}\mid\mathsf{X}_{1},\mathsf{Y}_{1},\ldots,\mathsf{X}_{n-1},\mathsf{Y}_{n-1},\mathsf{X}_{n}\bigr{)}_{uu^{\ast}}\\\\[2.84526pt]
=\operatorname{H}_{\text{min}}\bigl{(}\mathsf{X}_{1}\mid\mathsf{Y}_{n},\mathsf{X}_{n},\ldots,\mathsf{Y}_{2},\mathsf{X}_{2},\mathsf{Y}_{1}\bigr{)}_{uu^{\ast}}\end{multlined}\operatorname{H}_{\text{min}}\bigl{(}\mathsf{Y}_{n}\mid\mathsf{X}_{1},\mathsf{Y}_{1},\ldots,\mathsf{X}_{n-1},\mathsf{Y}_{n-1},\mathsf{X}_{n}\bigr{)}_{uu^{\ast}}\\\\[2.84526pt]
=\operatorname{H}_{\text{min}}\bigl{(}\mathsf{X}_{1}\mid\mathsf{Y}_{n},\mathsf{X}_{n},\ldots,\mathsf{Y}_{2},\mathsf{X}_{2},\mathsf{Y}_{1}\bigr{)}_{uu^{\ast}}$
(31)

for every vector
$u\in\mathcal{X}_{1}\otimes\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{X}_{n}\otimes\mathcal{Y}_{n}$.

### Interpretations of the main theorem

Theorem 1 establishes a _time-reversal property_ of rank-one strategy
functions. Intuitively speaking, the linear function

$Y\mapsto\langle Wuu^{\ast}W^{\ast},Y\rangle$ (32)

defined on
$\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$
represents the _time-reversal_ of the linear function

$X\mapsto\langle uu^{\ast},X\rangle$ (33)

defined on
$\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$,
in the sense that the two functions differ only in the reversal of the
ordering of the register exchanges: $\mathsf{X}_{1}$, $\mathsf{Y}_{1}$,
$\ldots$, $\mathsf{X}_{n}$, $\mathsf{Y}_{n}$ for the function corresponding to
$uu^{\ast}$ and $\mathsf{Y}_{n}$, $\mathsf{X}_{n}$, $\ldots$,
$\mathsf{Y}_{1}$, $\mathsf{X}_{1}$ for the function corresponding to
$Wuu^{\ast}W^{\ast}$.

For a given choice of
$X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$,
it is generally not the case that
$W^{\ast}XW\in\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$.
It may not even be the case that $W^{\ast}XW$ is the Choi representation of a
channel, and in the case that $W^{\ast}XW$ is the Choi representation of a
channel, it will generally not be the case that this channel obeys the
constraints necessary for it to be a valid strategy operator. When combined
with the observation that
$\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$
and
$\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$
are compact and convex sets, this fact implies that the main theorem cannot
possibly hold for all Hermitian operators $H$ by the separating hyperplane
theorem. For small values of $n$ and for spaces having small dimensions,
simple examples of operators $H$ for which the main theorem fails may also
easily be obtained through random selections.

In Section 4 we discuss another interpretation of Theorem 1, which concerns
multiple round entanglement manipulation.

### Proof of Theorem 1

We will now prove Theorem 1. The first step is to express the strategy
represented by $X$ as a sequence of channels corresponding to invertible
isometries (i.e., unitary operators for which the input and output spaces have
different names but necessarily the same dimension), assuming an auxiliary
input space initialized to a pure state is made available.

Through the repeated application of the Stinespring dilation theorem, together
with the result of [9, 2, 4] establishing that $X=J(\Xi_{n})$ is the Choi
representation of a channel $\Xi_{n}$ arising from a valid $n$-turn strategy,
one finds that there must exist complex Euclidean spaces
$\mathcal{Z}_{0},\ldots,\mathcal{Z}_{n}$ satisfying
$\dim(\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k})=\dim(\mathcal{Z}_{k}\otimes\mathcal{Y}_{k})$
for all $k\in\\{1,\ldots,n\\}$, a unit vector $v\in\mathcal{Z}_{0}$, and
invertible isometries $U_{1},\ldots,U_{n}$ of the form

$U_{k}\in\mathrm{U}(\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k},\mathcal{Y}_{k}\otimes\mathcal{Z}_{k})$
(34)

such that

$\Xi_{n}(Z)=\operatorname{Tr}_{\mathcal{Z}_{n}}\bigl{(}U(vv^{\ast}\otimes
Z\bigr{)}U^{\ast}\bigr{)}$ (35)

for all $Z\in\mathrm{L}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$,
where

$\begin{multlined}U=(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1}}\otimes
U_{n})\cdots(U_{1}\otimes\mathds{1}_{\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n}})\\\\[2.84526pt]
\in\mathrm{U}(\mathcal{Z}_{0}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n},\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{Z}_{n}).\end{multlined}U=(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1}}\otimes
U_{n})\cdots(U_{1}\otimes\mathds{1}_{\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n}})\\\\[2.84526pt]
\in\mathrm{U}(\mathcal{Z}_{0}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n},\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{Z}_{n}).$
(36)

In words, the strategy represented by the operator $X$ is implemented by first
initializing a register $\mathsf{Z}_{0}$ to the pure state $v$, then applying
the invertible isometric channels corresponding to $U_{1},\ldots,U_{n}$, and
finally discarding $\mathsf{Z}_{n}$ after the interaction has finished. (The
top picture in Figure 4 illustrates this for the case $n=3$.)

$v$$U_{1}$$U_{2}$$U_{3}$$\mathsf{Z}_{0}$$\mathsf{Z}_{1}$$\mathsf{Z}_{2}$$\mathsf{Z}_{3}$$\mathsf{Y}_{1}$$\mathsf{Y}_{2}$$\mathsf{Y}_{3}$$\mathsf{X}_{1}$$\mathsf{X}_{2}$$\mathsf{X}_{3}$

$w$$U_{3}^{{\scriptscriptstyle\mathsf{T}}}$$U_{2}^{{\scriptscriptstyle\mathsf{T}}}$$U_{1}^{{\scriptscriptstyle\mathsf{T}}}$$\mathsf{Z}_{3}$$\mathsf{Z}_{2}$$\mathsf{Z}_{1}$$\mathsf{Z}_{0}$$\mathsf{X}_{3}$$\mathsf{X}_{2}$$\mathsf{X}_{1}$$\mathsf{Y}_{3}$$\mathsf{Y}_{2}$$\mathsf{Y}_{1}$

Figure 4: An arbitrary strategy may be implemented by initializing a register
$\mathsf{Z}_{0}$ to a pure state $v$, followed by the application of an
invertible isometric channel on each turn, and finally by discarding the last
memory register $\mathsf{Z}_{n}$ (which is $\mathsf{Z}_{3}$ in the picture).
The time-reversed strategy whose existence is implied by the main theorem is
obtained by setting the register $\mathsf{Z}_{n}$ ($\mathsf{Z}_{3}$ in the
picture) to an appropriate choice of a pure state $w$, followed by the
application of invertible isometric channels obtained by transposing the
original isometries, and finally by discarding the memory register
$\mathsf{Z}_{0}$.

The vector $u$ may be expressed as

$u=\sum_{\begin{subarray}{c}a_{1},\ldots,a_{n}\\\
b_{1},\ldots,b_{n}\end{subarray}}u(b_{1},\ldots,b_{n},a_{1},\ldots,a_{n})\lvert
b_{1}\rangle\cdots\lvert b_{n}\rangle\lvert a_{1}\rangle\cdots\lvert
a_{n}\rangle,$ (37)

where the sum is over all standard basis states $\lvert
a_{1}\rangle,\ldots,\lvert a_{n}\rangle$ of
$\mathcal{X}_{1},\ldots,\mathcal{X}_{n}$ and $\lvert
b_{1}\rangle,\ldots,\lvert b_{n}\rangle$ of
$\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n}$, respectively. Based on this
expression, define an operator
$A\in\mathrm{L}(\mathcal{Z}_{0},\mathcal{Z}_{n})$ as

$\begin{multlined}A=\sum_{\begin{subarray}{c}a_{1},\ldots,a_{n}\\\
b_{1},\ldots,b_{n}\end{subarray}}u(b_{1},\ldots,b_{n},a_{1},\ldots,a_{n})(\langle
b_{n}\rvert\otimes\mathds{1}_{\mathcal{Z}_{n}})U_{n}(\mathds{1}_{\mathcal{Z}_{n-1}}\otimes\lvert
a_{n}\rangle)\rule{28.45274pt}{0.0pt}\\\\[-11.38109pt] \cdots(\langle
b_{1}\rvert\otimes\mathds{1}_{\mathcal{Z}_{1}})U_{1}(\mathds{1}_{\mathcal{Z}_{0}}\otimes\lvert
a_{1}\rangle).\end{multlined}A=\sum_{\begin{subarray}{c}a_{1},\ldots,a_{n}\\\
b_{1},\ldots,b_{n}\end{subarray}}u(b_{1},\ldots,b_{n},a_{1},\ldots,a_{n})(\langle
b_{n}\rvert\otimes\mathds{1}_{\mathcal{Z}_{n}})U_{n}(\mathds{1}_{\mathcal{Z}_{n-1}}\otimes\lvert
a_{n}\rangle)\rule{28.45274pt}{0.0pt}\\\\[-11.38109pt] \cdots(\langle
b_{1}\rvert\otimes\mathds{1}_{\mathcal{Z}_{1}})U_{1}(\mathds{1}_{\mathcal{Z}_{0}}\otimes\lvert
a_{1}\rangle).$ (38)

By considering the action of the strategy represented by $v$ and
$U_{1},\ldots,U_{n}$, then performing the required operator-vector
multiplications required to evaluate the expression $\langle
uu^{\ast},X\rangle$ when $X=J(\Xi_{n})$ for $\Xi_{n}$ given by (35), one
concludes that

$\langle
uu^{\ast},X\rangle=\lVert\mspace{1.0mu}Av\mspace{1.0mu}\rVert^{2}=\langle
vv^{\ast},A^{\ast}A\rangle.$ (39)

Next we turn to the reversed interaction. To obtain a strategy operator $Y$
satisfying the requirements of the theorem, we consider the strategy obtained
by initializing the register $\mathsf{Z}_{n}$ to a particular choice of a pure
state $w$, which will be selected later, then applying in sequence the
invertible isometric channels corresponding to the operators
$U_{n}^{{\scriptscriptstyle\mathsf{T}}},\ldots,U_{1}^{{\scriptscriptstyle\mathsf{T}}}$.
(The bottom picture in Figure 4 illustrates this for the case $n=3$.) That is,
for

$\begin{multlined}V=(\mathds{1}_{\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{2}}\otimes
U_{1}^{{\scriptscriptstyle\mathsf{T}}})\cdots(U_{n}^{{\scriptscriptstyle\mathsf{T}}}\otimes\mathds{1}_{\mathcal{Y}_{n-1}\otimes\cdots\otimes\mathcal{Y}_{1}})\\\\[2.84526pt]
\in\mathrm{U}(\mathcal{Z}_{n}\otimes\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1},\mathcal{Z}_{0}\otimes\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{1}),\end{multlined}V=(\mathds{1}_{\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{2}}\otimes
U_{1}^{{\scriptscriptstyle\mathsf{T}}})\cdots(U_{n}^{{\scriptscriptstyle\mathsf{T}}}\otimes\mathds{1}_{\mathcal{Y}_{n-1}\otimes\cdots\otimes\mathcal{Y}_{1}})\\\\[2.84526pt]
\in\mathrm{U}(\mathcal{Z}_{n}\otimes\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1},\mathcal{Z}_{0}\otimes\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{1}),$
(40)

we consider the channel
$\Theta_{n}\in\mathrm{C}(\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1},\mathcal{X}_{n}\otimes\cdots\otimes\mathcal{X}_{1})$
defined as

$\Theta_{n}(Z)=\operatorname{Tr}_{\mathcal{Z}_{0}}\bigl{(}V(ww^{\ast}\otimes
Z)V^{\ast}\bigr{)}$ (41)

for all $Z\in\mathrm{L}(\mathcal{Y}_{n}\otimes\cdots\otimes\mathcal{Y}_{1})$.
It is evident from the specification of this channel, irrespective of the
choice of the pure state $w$, that
$Y=J(\Theta_{n})\in\mathcal{S}_{n}(\mathcal{Y}_{n},\ldots,\mathcal{Y}_{1};\mathcal{X}_{n},\ldots,\mathcal{X}_{1})$.
By considering the action of this strategy, a similar calculation to the one
above reveals that

$\langle
Wuu^{\ast}W^{\ast},Y\rangle=\lVert\mspace{1.0mu}A^{{\scriptscriptstyle\mathsf{T}}}w\mspace{1.0mu}\rVert^{2}=\bigl{\langle}ww^{\ast},\overline{A}A^{{\scriptscriptstyle\mathsf{T}}}\bigr{\rangle}.$
(42)

The nonzero eigenvalues of $A^{\ast}A$ and
$\overline{A}A^{{\scriptscriptstyle\mathsf{T}}}$ are equal, and therefore by
choosing $w$ to be an eigenvector corresponding to the largest eigenvalue of
$\overline{A}A^{{\scriptscriptstyle\mathsf{T}}}$ one obtains

$\langle
Wuu^{\ast}W^{\ast},Y\rangle=\bigl{\langle}ww^{\ast},\overline{A}A^{{\scriptscriptstyle\mathsf{T}}}\bigr{\rangle}\geq\langle
vv^{\ast},A^{\ast}A\rangle=\langle uu^{\ast},X\rangle.$ (43)

If it holds that
$\dim(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})\leq\dim(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$,
then $\dim(\mathcal{Z}_{0})\leq\dim(\mathcal{Z}_{n})$, which implies that the
inequality in (43) may be taken as an equality for an appropriate choice of a
pure state $w$. This completes the proof.

##  3 Application to min- and max-entropy

In this section we connect the main result proved in the previous section to
the conditional min- and max-entropy functions. These function, which were
first introduced in [6], may be defined as follows. First, one defines the
max- and min-relative entropy of $P$ with respect to $Q$, for positive
semidefinite operators $P$ and $Q$ (acting on the same space), as follows:

$\displaystyle\operatorname{D}_{\text{max}}(P\,\|\,Q)$
$\displaystyle=\log\bigl{(}\min\\{\lambda\geq 0\,:\,P\leq\lambda
Q\\}\bigr{)},$ (44) $\displaystyle\operatorname{D}_{\text{min}}(P\,\|\,Q)$
$\displaystyle=-\log\bigl{(}\operatorname{F}(P,Q)^{2}\bigr{)}.$ (45)

Then, with respect to a given state
$\rho\in\mathrm{D}(\mathcal{X}\otimes\mathcal{Y})$ of a pair of registers
$(\mathsf{X},\mathsf{Y})$, one defines

$\displaystyle\operatorname{H}_{\text{min}}(\mathsf{X}|\mathsf{Y})=-\inf_{\sigma\in\mathrm{D}(\mathcal{Y})}\operatorname{D}_{\text{max}}\bigl{(}\rho\,\big{\|}\,\mathds{1}_{\mathcal{X}}\otimes\sigma\bigr{)},$
(46)
$\displaystyle\operatorname{H}_{\text{max}}(\mathsf{X}|\mathsf{Y})=-\inf_{\sigma\in\mathrm{D}(\mathcal{Y})}\operatorname{D}_{\text{min}}\bigl{(}\rho\,\big{\|}\,\mathds{1}_{\mathcal{X}}\otimes\sigma\bigr{)}.$
(47)

It is known that these two quantities are related in the following way: with
respect to any pure state $uu^{\ast}$ of a triple of registers
$(\mathsf{X},\mathsf{Y},\mathsf{Z})$, one has that

$\operatorname{H}_{\text{min}}(\mathsf{X}|\mathsf{Y})=-\operatorname{H}_{\text{max}}(\mathsf{X}|\mathsf{Z}).$
(48)

(Indeed, in [12] the conditional max-relative entropy of a state of
$(\mathsf{X},\mathsf{Z})$ is _defined_ by the equation (48), which does not
depend on which purification of this state is chosen, and is then proved to
agree with the definition stated previously.)

Consider any unit vector
$u\in\mathcal{X}\otimes\mathcal{Y}\otimes\mathcal{Z}$, which defines a pure
state $uu^{\ast}$ of a triple of registers
$(\mathsf{X},\mathsf{Y},\mathsf{Z})$. We will consider two optimization
problems defined by $u$, the first of which is as follows:

maximize: $\displaystyle\quad\bigl{\langle}uu^{\ast},X\bigr{\rangle}$ (49)
subject to: $\displaystyle\quad
X\in\mathcal{S}_{2}(\mathcal{Y},\mathcal{Z};\mathcal{X},\mathbb{C}).$

This optimization problem is illustrated in Figure 5.

$uu^{\ast}$$\Phi_{1}$$\Phi_{2}$$\mathsf{Y}$$\mathsf{Z}$$\mathsf{X}$$\mathsf{W}$
Figure 5: The optimization problem (49) corresponds to a maximization of the
linear functions defined by $uu^{\ast}$ over all strategies given by channels
$\Phi_{1}$ and $\Phi_{2}$, for an arbitrary choice of a register $\mathsf{W}$.

In this case, the channel $\Phi_{2}$ takes registers $\mathsf{Z}$ and
$\mathsf{W}$ as input and outputs nothing (which is equivalent to outputting
the unique state $1\in\mathrm{D}(\mathbb{C})$ of a one-dimensional system).
That is, $\Phi_{2}$ must be the trace mapping. One may therefore simplify this
problem, obtaining the following semidefinite program:

Primal problem

maximize:
$\displaystyle\langle\operatorname{Tr}_{\mathcal{Z}}(uu^{\ast}),X\rangle$
subject to:
$\displaystyle\operatorname{Tr}_{\mathcal{X}}(X)=\mathds{1}_{\mathcal{Y}},$
$\displaystyle X\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Y}).$

Dual problem

minimize: $\displaystyle\operatorname{Tr}(Y)$ subject to:
$\displaystyle\mathds{1}_{\mathcal{X}}\otimes
Y\geq\operatorname{Tr}_{\mathcal{Z}}(uu^{\ast}),$ $\displaystyle
Y\in\mathrm{Herm}(\mathcal{X}).$

By examining the dual problem, one sees that the optimal value of this
semidefinite program is

$2^{-\operatorname{H}_{\text{min}}(\mathsf{X}|\mathsf{Y})}$ (50)

with respect to the state $uu^{\ast}$ of $(\mathsf{X},\mathsf{Y},\mathsf{Z})$.
König, Renner, and Schaffner [12] observed that the primal problem coincides
with the value represented by the expression (50), which is consistent with
the observation that strong duality always holds for this semidefinite program
(which may be verified through Slater’s theorem, for instance).

The second optimization problem we consider is the time-reversal of the first,
and may be stated as follows:

maximize: $\displaystyle\quad\bigl{\langle}Wuu^{\ast}W^{\ast},Y\bigr{\rangle}$
(51) subject to: $\displaystyle\quad
Y\in\mathcal{S}_{2}(\mathbb{C},\mathcal{X};\mathcal{Z},\mathcal{Y}).$

Figure 6 illustrates the interaction corresponding to this optimization
problem.

$Wuu^{\ast}W^{\ast}$$\Psi_{1}$$\Psi_{2}$$\mathsf{X}$$\mathsf{Z}$$\mathsf{Y}$$\mathsf{W}$
Figure 6: The optimization problem (51) corresponds to a maximization of the
linear functions defined by $Wuu^{\ast}W^{\ast}$ over all strategies given by
channels $\Psi_{1}$ and $\Psi_{2}$, for an arbitrary choice of a register
$\mathsf{W}$.

The inclusion
$X\in\mathcal{S}_{2}(\mathbb{C},\mathcal{X};\mathcal{Z},\mathcal{Y})$, for a
given operator
$X\in\mathrm{Pos}(\mathcal{Z}\otimes\mathcal{Y}\otimes\mathcal{X})$, is
equivalent to the condition that
$\operatorname{Tr}_{\mathcal{Y}}(X)=\sigma\otimes\mathds{1}_{\mathcal{X}}$ for
some $\sigma\in\mathrm{D}(\mathcal{Z})$. After re-ordering tensor factors, we
obtain the following semidefinite program:

Primal problem

maximize: $\displaystyle\langle uu^{\ast},X\rangle$ subject to:
$\displaystyle\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}}\otimes\sigma,$
$\displaystyle
X\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Y}\otimes\mathcal{Z}),$
$\displaystyle\sigma\in\mathrm{D}(\mathcal{Z}).$

Dual problem

minimize: $\displaystyle\lambda$ subject to: $\displaystyle
Y\otimes\mathds{1}_{\mathcal{Y}}\geq uu^{\ast},$
$\displaystyle\lambda\mathds{1}_{\mathcal{Z}}\geq\operatorname{Tr}_{\mathcal{X}}(Y),$
$\displaystyle Y\in\mathrm{Herm}(\mathcal{X}\otimes\mathcal{Z}),$
$\displaystyle\lambda\in\real.$

An examination of the primal problem reveals (through Uhlmann’s theorem) that
the optimal value of this semidefinite program is

$2^{\operatorname{H}_{\text{max}}(\mathsf{X}|\mathsf{Z})}.$ (52)

By our main theorem, it follows that the two optimization problems have the
same optimal value, and therefore we obtain an alternative proof that with
respect to every pure state of a triple or registers
$(\mathsf{X},\mathsf{Y},\mathsf{Z})$ one has

$\operatorname{H}_{\text{min}}(\mathsf{X}|\mathsf{Y})=-\operatorname{H}_{\text{max}}(\mathsf{X}|\mathsf{Z}).$
(53)

It is natural to ask if the connections among min-entropy, max-entropy, and
optimization problems involving three-message strategies have interesting
implications or generalizations for interactions involving four or more
messages. As a partial answer to this question, we observe that when our main
result is applied to the four-message interaction depicted in Figure 7, it
reveals the identity

$\max_{\Phi\in\mathrm{C}(\mathcal{Y},\mathcal{X})}\operatorname{F}\bigl{(}\operatorname{Tr}_{\mathcal{W}}(uu^{\ast}),J(\Phi)\otimes\mathds{1}_{\mathcal{Z}}\bigr{)}=\max_{\Psi\in\mathrm{C}(\mathcal{W},\mathcal{Z})}\operatorname{F}\bigl{(}\operatorname{Tr}_{\mathcal{X}}(uu^{\ast}),\mathds{1}_{\mathcal{Y}}\otimes
J(\Psi)\bigr{)}$ (54)

for all vectors
$u\in\mathcal{X}\otimes\mathcal{Y}\otimes\mathcal{Z}\otimes\mathcal{W}$.

$uu^{\ast}$$\Phi_{1}$$\Phi_{2}$$\mathsf{Y}$$\mathsf{Z}$$\mathsf{X}$$\mathsf{W}$
Figure 7:  Maximizing the linear function defined by $uu^{\ast}$ over all
four-message strategies of the form depicted yields the left-hand side of
(54). By reversing time, the right-hand side of that equation is obtained, and
the equality of the two is implied by the main theorem.

This identity is appealing in its simplicity and symmetry, and by taking
$\mathcal{W}=\mathbb{C}$ (or $\mathcal{Y}=\mathbb{C}$) a statement equivalent
to (53) for all pure states of $(\mathsf{X},\mathsf{Y},\mathsf{Z})$ is
obtained. We do not know, however, if the quantity represented by either side
of the identity has any direct operational significance.

Other identities may be obtained through a similar methodology, although they
become increasingly complex as the number of messages is increased.

##  4 Online pure state entanglement manipulation

The following three statements are equivalent for a given operator
$X\in\mathrm{L}(\mathcal{Y}\otimes\mathcal{X})$:

  1. 1.

$X\in\mathcal{S}_{1}(\mathcal{X};\mathcal{Y})$. (Equivalently,
$X\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$ and
$\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}}$.)

  2. 2.

$X=(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{X})})(\operatorname{vec}(\mathds{1}_{\mathcal{X}})\operatorname{vec}(\mathds{1}_{\mathcal{X}})^{\ast})$
for some channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$.

  3. 3.

$X=(\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi)(\operatorname{vec}(\mathds{1}_{\mathcal{Y}})\operatorname{vec}(\mathds{1}_{\mathcal{Y}})^{\ast})$
for some completely positive and unital map
$\Psi\in\mathrm{CP}(\mathcal{Y},\mathcal{X})$.

(Here and throughout this section, vec refers to the vectorization mapping,
which is the mapping obtained by extending the transformation $\lvert
a\rangle\langle b\rvert\mapsto\lvert a\rangle\lvert b\rangle$ for standard
basis states to arbitrary operators by linearity. In particular,
$\operatorname{vec}(\mathds{1}_{\mathcal{X}})$ is a non-normalized vector
proportional to the canonical maximally entangled pure state corresponding to
two identical copies of a system whose state space is $\mathcal{X}$.) The maps
$\Phi$ and $\Psi$ uniquely determine one another, and it is reasonable to view
these maps as being related by transposition (with respect to the standard
basis): $\Psi=\Phi^{{\scriptscriptstyle\mathsf{T}}}$ and
$\Phi=\Psi^{{\scriptscriptstyle\mathsf{T}}}$. To obtain a Kraus representation
for $\Psi$, for instance, one may simply take a Kraus representation of $\Phi$
and transpose each of the Kraus operators. (The transpose of an arbitrary map
can be defined in a manner that is consistent with these statements, but it is
sufficient for our needs to focus on channels and completely positive unital
maps.)

A generalization of the equivalence mentioned above to the quantum strategy
framework may also be verified. For an operator
$X\in\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$,
these three statements are equivalent:

  1. 1.

$X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$.

  2. 2.

There exist complex Euclidean spaces
$\mathcal{Z}_{1},\ldots,\mathcal{Z}_{n-1}$ (and $\mathcal{Z}_{0}=\mathbb{C}$
and $\mathcal{Z}_{n}=\mathbb{C}$), along with channels
$\Phi_{1},\ldots,\Phi_{n}$ having the form

$\Phi_{k}\in\mathrm{C}(\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k},\mathcal{Y}_{k}\otimes\mathcal{Z}_{k}),$
(55)

such that the channel
$\Xi_{n}\in\mathrm{C}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n},\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})$
defined as

$\Xi_{n}=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1})}\otimes\Phi_{n}\bigr{)}\cdots\bigl{(}\Phi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n})}\bigr{)}$
(56)

satisfies

$X=\bigl{(}\Xi_{n}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})}\bigr{)}(\operatorname{vec}(\mathds{1}_{\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}})^{\ast}).$
(57)

  3. 3.

There exist complex Euclidean spaces
$\mathcal{Z}_{1},\ldots,\mathcal{Z}_{n-1}$ (and $\mathcal{Z}_{0}=\mathbb{C}$
and $\mathcal{Z}_{n}=\mathbb{C}$), along with completely positive and unital
maps $\Psi_{1},\ldots,\Psi_{n}$ having the form

$\Psi_{k}\in\mathrm{C}(\mathcal{Y}_{k}\otimes\mathcal{Z}_{k},\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k}),$
(58)

such that the unital map
$\Lambda_{n}\in\mathrm{CP}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n},\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$
defined as

$\Lambda_{n}=\bigl{(}\Psi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n})}\bigr{)}\cdots\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1})}\otimes\Psi_{n}\bigr{)}$
(59)

satisfies

$X=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})}\otimes\Lambda_{n}\bigr{)}(\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})^{\ast}).$
(60)

Through this equivalence, for a given state
$\rho\in\mathrm{D}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$,
one arrives at an alternative interpretation of the semidefinite program

maximize: $\displaystyle\langle\rho,X\rangle$ (61) subject to: $\displaystyle
X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$

that concerns an online variant of entanglement manipulation, as is explained
shortly. The term “online” in this context refers to a situation in which a
quantum state must be manipulated in multiple turns, where an output is
required immediately after each input system arrives and prior to the next
input system being made available, similar to an online process.

By the equivalence of the third statement above to the first, a maximization
over all
$X\in\mathcal{S}_{n}(\mathcal{X}_{1},\ldots,\mathcal{X}_{n};\mathcal{Y}_{1},\ldots,\mathcal{Y}_{n})$
is equivalent to a maximization over all operators

$\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})}\otimes\Lambda_{n}\bigr{)}(\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})^{\ast})$
(62)

for

$\Lambda_{n}=\bigl{(}\Psi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n})}\bigr{)}\cdots\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1})}\otimes\Psi_{n}\bigr{)}$
(63)

and $\Psi_{1},\ldots,\Psi_{n}$ being completely positive and unital maps of
the form

$\Psi_{k}\in\mathrm{C}(\mathcal{Y}_{k}\otimes\mathcal{Z}_{k},\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k}).$
(64)

The value of the objective function $\langle\rho,X\rangle$ may therefore be
expressed as

$\bigl{\langle}(\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})}\otimes\Lambda_{n}^{\ast})(\rho),\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})^{\ast}\bigr{\rangle},$
(65)

which is $\dim(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})$ times the
squared fidelity between the maximally entangled state
$\tau\in\mathrm{D}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}\otimes\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})$
given by

$\tau=\frac{\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n}})^{\ast}}{\dim(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n})}$
(66)

and the state obtained by applying the channel $\Lambda_{n}^{\ast}$ to the
portion of $\rho$ corresponding to the spaces
$\mathcal{X}_{1},\ldots,\mathcal{X}_{n}$. In the case that $n=1$, König,
Renner, and Schaffner [12] refer to this quantity as the _quantum
correlation_. This situation is illustrated for the case $n=3$ in Figure 8.

$\rho$$\Psi_{1}^{\ast}$$\Psi_{2}^{\ast}$$\Psi_{3}^{\ast}$$\mathsf{X}_{1}$$\mathsf{X}_{2}$$\mathsf{X}_{3}$$\mathsf{Y}_{1}$$\mathsf{Y}_{2}$$\mathsf{Y}_{3}$$\mathsf{Z}_{1}$$\mathsf{Z}_{2}$$\mathsf{Y}_{1}$$\mathsf{Y}_{1}$$\mathsf{Y}_{2}$$\mathsf{Y}_{2}$$\mathsf{Y}_{3}$$\mathsf{Y}_{3}$
Figure 8: The channel
$\Lambda_{3}^{\ast}=(\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2})}\otimes\Psi_{3}^{\ast})(\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1})}\otimes\Psi_{2}^{\ast}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{3})})(\Psi_{1}^{\ast}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\mathcal{X}_{3})})$
is applied to registers $(\mathsf{X}_{1},\mathsf{X}_{2},\mathsf{X}_{3})$ of a
state
$\rho\in\mathrm{D}(\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}\otimes\mathcal{Y}_{3}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3})$
with the aim of maximizing the fidelity of the output state with the canonical
maximally entangled state.
$\rho$$\Phi_{3}$$\Phi_{2}$$\Phi_{1}$$\mathsf{Y}_{3}$$\mathsf{Y}_{2}$$\mathsf{Y}_{1}$$\mathsf{X}_{3}$$\mathsf{X}_{2}$$\mathsf{X}_{1}$$\mathsf{Z}_{2}$$\mathsf{Z}_{1}$$\mathsf{X}_{1}$$\mathsf{X}_{1}$$\mathsf{X}_{2}$$\mathsf{X}_{2}$$\mathsf{X}_{3}$$\mathsf{X}_{3}$
Figure 9: A similar process to the one illustrated in Figure 8, but with
channels applied to $\mathsf{Y}_{3}$, $\mathsf{Y}_{2}$, $\mathsf{Y}_{1}$
rather than $\mathsf{X}_{1}$, $\mathsf{X}_{2}$, $\mathsf{X}_{3}$.

By Theorem 1, one finds that when $\rho$ is pure, the same optimal value is
achieved when the ordering of the channels and the registers on which they act
is reversed, as illustrated in Figure 9 for the case $n=3$. That is, when
$\rho$ is a pure state, the optimal value of the semidefinite program (61)
represents the value

$\bigl{\langle}(\Xi_{n}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})})(\rho),\operatorname{vec}(\mathds{1}_{\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}})\operatorname{vec}(\mathds{1}_{\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n}})^{\ast}\bigr{\rangle},$
(67)

maximized over all channels
$\Xi_{n}\in\mathrm{C}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n},\mathcal{X}_{1}\otimes\cdots\otimes\mathcal{X}_{n})$
of the form

$\Xi_{n}=\bigl{(}\Phi_{1}\otimes\mathds{1}_{\mathrm{L}(\mathcal{X}_{2}\otimes\cdots\otimes\mathcal{X}_{n})}\bigr{)}\cdots\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y}_{1}\otimes\cdots\otimes\mathcal{Y}_{n-1})}\otimes\Phi_{n}\bigr{)}$
(68)

for channels $\Phi_{1},\ldots,\Phi_{n}$ taking the form

$\Phi_{k}\in\mathrm{C}(\mathcal{Y}_{k}\otimes\mathcal{Z}_{k},\mathcal{Z}_{k-1}\otimes\mathcal{X}_{k})$
(69)

and for $\mathcal{Z}_{2},\ldots,\mathcal{Z}_{n-1}$ arbitrary complex Euclidean
spaces (along with $\mathcal{Z}_{0}=\mathbb{C}$ and
$\mathcal{Z}_{n}=\mathbb{C}$).

##  5 Conclusion

We have identified a time-reversal property for rank-one quantum strategy
functions, explained its connection to conditional min- and max-entropy, and
described an alternative view of this property through an online variant of
pure state entanglement manipulation. An obvious question arises: are there
interesting applications or implications of this property beyond those we have
mentioned?

### Acknowledgments

Yuan Su was supported in part by the Army Research Office (MURI award
W911NF-16-1-0349) and the National Science Foundation (grant 1526380). John
Watrous acknowledges the support of Canada’s NSERC. We thank Frédéric Dupuis,
James R. Garrison, Brian Swingle, Penghui Yao, Ronald de Wolf, and Māris Ozols
for helpful discussions, and we thank the anonymous referees for their
comments and suggestions.

## References

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# Comment on “Strong Quantum Darwinism and Strong Independence are Equivalent
to Spectrum Broadcast Structure”

Alexandre Feller1 Benjamin Roussel1 Irénée Frérot2,3 Pascal Degiovanni4 (1)
European Space Agency - Advanced Concepts Team, ESTEC, Keplerlaan 1, 2201 AZ
Noordwijk, The Netherlands.  (2) ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3,
08860 Barcelona, Spain  (3) Max-Planck-Institut für Quantenoptik, D-85748
Garching, Germany  (4) Univ Lyon, Ens de Lyon, Université Claude Bernard Lyon
1, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

In a recent Letter [Phys. Rev. Lett. 122, 010403 (2019)], an equivalence is
proposed between the so-called Spectrum Broadcast Structure for a system-
multienvironment quantum state, and the conjunction of two information-theory
notions: (a) Strong Quantum Darwinism; and (b) Strong Independence. Here, we
show that the mathematical formulation of condition (b) by the authors
(namely, the pairwise independence of the fragments of the environment,
conditioned on the system), is necessary but not sufficient to ensure the
equivalence. We propose a simple counter-example, together with a strengthened
formulation of condition (b), ensuring the equivalence proposed by the
authors.

In their paper Castro-2019 , the authors introduce the notions of: (a) “strong
quantum Darwinism”; and (b) “strong independence”. As the main result of the
paper, it is proposed that, taken together, conditions (a) and (b) are
equivalent to the so-called “spectrum broadcast structure” (SBS) for the
system–environment quantum state $\rho_{{\cal S}{\cal E}}$ (throughout this
Comment, we follow the notations and definitions used in the paper). By
definition, if $\rho_{{\cal S}{\cal E}}$ has a SBS, the state of ${\cal
E}={\cal E}_{1}\cdots{\cal E}_{F}$ relative to ${\cal S}$ is fully factorized,
and has no correlations whatsoever; and the authors aimed at capturing this
fully-factorized structure by an information-theory criterion of “strong
independence”. In this Comment, we show that condition (b), while ensuring
pairwise independence of the fragments ${\cal E}_{i}$ when conditioning on the
system (a necessary condition for having the SBS), does not rule out the
existence of higher-order correlations, and as such is insufficient to imply a
SBS. We first construct a (completely classical) counter-example of a state
with no SBS yet satisfying conditions (a) and (b). We then propose a stronger
condition (b’), ensuring the equivalence of conditions (a) and (b’) with SBS.
At a conceptual level, the main conclusion of the paper, namely the
equivalence between, on the one hand, the SBS, and on the other, the
conjunction of “strong quantum Darwinism” (as defined by the authors) and
“strong independence” (as now defined by (b’)), remains therefore unaltered.

Counter-example. We consider a qubit system ${\cal
S}:\\{|0\rangle,|1\rangle\\}$, and the environment ${\cal E}={\cal E}_{1}{\cal
E}_{2}{\cal E}_{3}$, with three qutrits fragments ${\cal
E}_{i}:\\{|0_{i}\rangle,|1_{i}\rangle,|2_{i}\rangle\\}$ ($i\in\\{1,2,3\\}$).
We define the projectors $\Pi_{a}^{({\cal S})}=|a\rangle\langle a|$ and
$\Pi_{a}^{(i)}=|a_{i}\rangle\langle a_{i}|$. Finally, we define the projectors
$\Pi_{abc}=\Pi_{a}^{(1)}\otimes\Pi_{b}^{(2)}\otimes\Pi_{c}^{(3)}$. Our
counter-example is the state ($0\leq p\leq 1$):

$\rho_{{\cal S}{\cal E}}=(1-p)~{}\Pi_{0}^{({\cal S})}\otimes\rho_{{\cal
E}}^{(0)}+p~{}\Pi_{1}^{({\cal S})}\otimes\rho_{{\cal E}}^{(1)}~{},$ (1)

with the relative states:

$\displaystyle\rho_{{\cal E}}^{(0)}$ $\displaystyle=$ $\displaystyle\Pi_{000}$
(2) $\displaystyle\rho_{{\cal E}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[\Pi_{111}+\Pi_{122}+\Pi_{212}+\Pi_{221}]$ (3)
$\displaystyle\rho_{{\cal E}_{i}{\cal E}_{j}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]\otimes\frac{1}{2}[\Pi_{1}^{(j)}+\Pi_{2}^{(j)}]$
(4) $\displaystyle\rho_{{\cal E}_{i}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]$ (5)

As the relative states $\rho_{{\cal E}_{i}}^{(0)}=\Pi_{0}^{(i)}$ and
$\rho_{{\cal E}_{i}}^{(1)}=[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]/2$ have disjoint
supports, it is clear that each fragment has full access to the system’s
state. Therefore: $I({\cal S}:{\cal E}_{i})=I_{\rm acc}({\cal S}:{\cal
E}_{i})=\chi({\cal S}^{\Pi}:{\cal E}_{i})=H({\cal S})$ [condition (a)]. The
state also fulfills condition (b) (pairwise-independence conditioned on ${\cal
S}$). Indeed, the relative states $\rho_{{\cal E}_{i}{\cal
E}_{j}}^{(0)}=\Pi_{0}^{(i)}\otimes\Pi_{0}^{(j)}$ and $\rho_{{\cal E}_{i}{\cal
E}_{j}}^{(1)}$ [Eq. (4)] are product states, which implies that $I({\cal
E}_{i}:{\cal E}_{j}|{\cal S})=0$ (as can be checked by direct computation on
$\rho_{{\cal S}{\cal E}_{i}{\cal E}_{j}}$). Yet, the state $\rho_{{\cal
S}{\cal E}_{1}{\cal E}_{2}{\cal E}_{3}}$ does not admit a SBS. Indeed, the
relative state $\rho_{{\cal E}}^{(1)}$ [Eq. (3)] does not factorize (while
pairs of fragments are uncorrelated, the parity of the total number of
fragments in state $|1_{i}\rangle$ is odd, yielding genuine 3-partite
correlations). This concludes our proof that the conditions (a) and (b) are
insufficient to imply a SBS.

Proposed correct formulation of the theorem. The above counter-example
suggests to introduce a stronger condition (b’) ensuring the complete
factorization of the fragments when conditioning on the system, which is a
defining property of the SBS. Noticing that a multipartite state is
factorized, $\rho_{{\cal E}_{1},\dots\cal E_{F}}=\otimes_{i=1}^{F}\rho_{{\cal
E}_{i}}$, iff the multipartite mutual information vanishesYang-2009 : $I({\cal
E}_{1},\dots\cal E_{F}):=\sum_{i=1}^{F}H(\rho_{{\cal E}_{i}})-H(\rho_{{\cal
E}_{1},\dots\cal E_{F}})=0$, we are led to propose the following:
Theorem. The state $\rho_{{\cal S}{\cal E}}$ has the SBS iff:

  * (a1)

$I({\cal S}:{\cal E})=\chi({\cal S}^{\Pi},{\cal E})$ (classical–quantum
state),

  * (a2)

$I_{\rm acc}({\cal S}:{\cal E}_{i})=H({\cal S})$ for all $i$ (the information
about ${\cal S}$ can be fully recovered by measuring any fragment),

  * (b’)

$I({\cal E}_{1},\cdots{\cal E}_{F}|{\cal S})=0$ (totally-factorized relative
states).

###### Proof.

If the state has the SBS, it is clear that conditions (a1), (a2) and (b’) are
fulfilled. Conversely, condition (a1) (vanishing discord) implies that the
system–environment state is of the form $\rho_{{\cal S}{\cal
E}}=\sum_{s}p_{s}\Pi_{s}^{({\cal S})}\otimes\rho_{\cal E}^{(s)}$ with
$\\{\Pi_{s}^{({\cal S})}\\}$ forming mutually orthogonal projectors for the
system. Condition (b’) then implies that the relative states factorize:
$\rho_{\cal E}^{(s)}=\otimes_{i=1}^{F}\rho_{{\cal E}_{i}}^{(s)}$. Finally,
condition (a2) implies that for each $i$, the states $\rho_{{\cal
E}_{i}}^{(s)}$ are pairwise orthogonal. ∎

###### Acknowledgements.

We thank Thao P. Le and Alexandra Olaya-Castro for constructive feedback on
our manuscript. This work has been supported by the European Space Agency
(Ariadna study 1912-01). IF acknowledges support from the Government of Spain
(FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundació Cellex and
Fundació Mir-Puig through an ICFO-MPQ Postdoctoral Fellowship, Generalitat de
Catalunya (CERCA, AGAUR SGR 1381 and QuantumCAT).

## References

  * (1) Thao P. Le and Alexandra Olaya-Castro, _Strong quantum darwinism and strong independence are equivalent to spectrum broadcast structure_ , Phys. Rev. Lett. 122 (2019), 010403.
  * (2) Dong Yang, Karol Horodecki, Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, and Wei Song, _Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof_ , IEEE Transactions on Information Theory 55 (2009), no. 7, 3375–3387.


# Automated Measurement of Pericoronary Adipose Tissue
Attenuation and Volume in CT Angiography

###### Abstract

Pericoronary adipose tissue (PCAT) is the deposition of fat in the vicinity of
the coronary arteries. It is an indicator of coronary inflammation and
associated with coronary artery disease. Non-invasive coronary CT angiography
(CCTA) is presently used to obtain measures of the thickness, volume, and
attenuation of fat deposition. However, prior works solely focus on measuring
PCAT using semi-automated approaches at the right coronary artery (RCA) over
the left coronary artery (LCA). In this pilot work, we developed a fully
automated approach for the measurement of PCAT mean attenuation and volume in
the region around both coronary arteries. First, we used a large subset of
patients from the public ImageCAS dataset (n = 735) to train a 3D full
resolution nnUNet to segment LCA and RCA. Then, we automatically measured PCAT
in the surrounding arterial regions. We evaluated our method on a held-out
test set of patients (n = 183) from the same dataset. A mean Dice score of 83%
and PCAT attenuation of -73.81 ± 12.69 HU was calculated for the RCA, while a
mean Dice score of 81% and PCAT attenuation of -77.51 ± 7.94 HU was computed
for the LCA. To the best of our knowledge, we are the first to develop a fully
automated method to measure PCAT attenuation and volume at both the RCA and
LCA. Our work underscores how automated PCAT measurement holds promise as a
biomarker for identification of inflammation and cardiac disease.

##  1 Introduction

Coronary artery disease is the leading cause of mortality worldwide,
originating from the formation of atherosclerotic plaque within the coronary
arteries [1]. Studies have shown that the rupture of non-obstructive and
highly inflamed plaques have led to $\sim$60% of all myocardial infarctions
(heart attacks) [2, 3]. Therefore, finding biomarkers that can indicate
inflammation and can be used as a predictor for the risk of cardiac disease
are of clinical interest. In recent years, the fat deposition around the
coronary arteries called pericoronary adipose tissue (PCAT) has been
identified as a potential biomarker for early diagnosis and intervention [2,
3, 4, 5, 6].

Presently, only semi-automated methods targeted towards PCAT measurement exist
[7, 6, 8]. Furthermore, a disproportionate number of past research has focused
on the RCA at the expense of the LCA for evaluation of PCAT. Measurement of
PCAT surrounding the left main (LM), left anterior descending (LAD), and left
circumflex (LCX) arteries may offer a broader perspective with respect to the
distinct hormonal signaling pathways between fat and plaque [2]. Given the
morphological and functional variability in the coronary arteries and their
branches [5, 4, 2], the development of a fully automated method for PCAT
attenuation and volume measurements across all these arteries remains elusive.

Fig. 1: Summary of automated measurement of pericoronary adipose tissue (PCAT)
in coronary CT angiography (CCTA) volumes. In the training phase, a 3D nnUNet
framework was trained on CCTA volumes and ground truth (GT) right coronary
artery (RCA, red) and left coronary artery (LCA, green) masks. At inference
time, the model was run on volumes from the test dataset. Finally, PCAT was
automatically measured at the RCA (RPCAT, blue) and PCAT at the LCA (LPCAT,
yellow).

In this pilot work, we developed a fully automated approach for the
measurement of PCAT mean attenuation and volume in the region around both left
and right coronary arteries. We used a large subset of patients from the
public ImageCAS dataset (n = 735) to train a 3D full resolution nnUNet model
segmentation of the LCA and RCA. Next, we automatically measured the PCAT in
the vicinity of the RCA and LCA. We evaluated our pipeline on a held-out test
set of patients (n = 183) from the same dataset.

##  2 Methods

###  2.1 Data

The ImageCAS dataset [9] consists of 1000 3D CCTA volumes acquired from 1000
patients. There were 586 males and 414 females with average ages of 60 and 57
years, respectively. These patients had a history of ischemic stroke, ischemic
heart attack and/or peripheral artery disease. A single Siemens CT scanner
(128-slice, dual-source) was used for imaging, and the coronary arteries in
this dataset were labeled by three radiologists. The labels consisted of the
right coronary artery (RCA), left main coronary artery (LM), left anterior
descending artery (LAD), left circumflex artery (LCX), and other branches.
These arteries branched out of the aorta, but the label mask for the aorta was
not provided in this dataset. We segmented the aorta with the help of a
publicly available tool called TotalSegmentator [10].

Next, we ran connected component analysis to separate the arteries on the
right side of the heart from the left side. As shown in Figs. 1 and 2, we
obtained two distinct labels; one for the RCA (label 1), and another for the
LCA (label 2). The LCA consisted of the combined masks of the LM, LAD, and
LCX. This resulted in 918 patient volumes with an effective split of the
coronary arteries into LCA and RCA, and the exclusion of 82 volumes where the
connected component analysis failed. This division was manually verified, and
the dataset was split into a training set (n = 735 studies) and a held-out
test set (n = 183 studies).

Fig. 2: Illustration of the automated pericoronary adipose tissue (PCAT)
measurement at the right coronary artery (RCA) and left coronary artery (LCA),
which includes the left main (LM), left anterior descending (LAD), and left
circumflex (CX) arteries. Dashed centerlines (black) were extracted from each
artery (cyan). They were used to determine the segment length for PCAT
measurement, and the bifurcation point of the LM into the LCX + LAD. PCAT at
the RCA (RPCAT) was measured 10 mm distal to the ostium and for a 40 mm
segment length. PCAT at the LCA (LPCAT) was measured along the full length of
the LM, but only for a 40 mm segment from the bifurcation point.

###  2.2 Model

We trained a 3D full resolution nnUNet model [11] to segment the left and
right coronary arteries. The nnUNet model is a self-configuring segmentation
framework that can adapt to various datasets and modalities, including CT. It
is currently the de-facto standard for segmentation tasks. It automatically
determines the optimal hyper-parameters for training a segmentation model and
learns to segment target structures of interest. During training, our 3D full-
resolution nnUNet took input CCTA volumes and the corresponding ground-truth
masks for the LCA and RCA (n = 735). The model learned to segment the target
structures of interest in the CT volume and iteratively refined it via a loss
function. This loss function computed a segmentation error that measured the
overlap between the prediction and ground-truth.

The framework executed a 5-fold cross-validation scheme, and used different
subsets of training and validation data for training the model in each fold.
At inference time, the 3D nnUNet predicted the segmentation masks for the LCA
and RCA in the held-out test CT volumes. The best model with the lowest loss
from each of the 5 folds was used for inference on the test CT volume, and
predictions from these five folds were ensembled together (via majority
voting).

###  2.3 Measurement of PCAT

Once the LCA and RCA were segmented, the PCAT regions around them were
measured. The perivascular fat region is defined as the adipose tissue within
a 5 mm radial distance from the outer vessel wall that is within the [-190,
-30] Hounsfield units (HU) range. To identify this region, the centerlines for
the LCA and RCA were first extracted from the corresponding segmentations. We
used the skeletonize function in Python [12] for this task.

Right PCAT (RPCAT). Consistent with prior work [8, 6], we commenced the PCAT
measurement at a distance of 10 mm distal to the ostium to avoid proximity to
the aortic wall. As seen in Fig. 2, at each point on the centerline of the
RCA, we estimated the closest distance to the vessel wall. From this distance,
the approximate diameter of the vessel was ascertained. Then, a sphere of
radius equal to the vessel diameter was placed on each point along the
centerline, and this resulted in a 3D cylinder that encompassed the region of
the PCAT and the segmented vessel. All the voxels that fell within the
aforementioned HU range were averaged together to provide the mean attenuation
of the RPCAT. The RPCAT volume was computed by multiplying the number of
voxels by the volumetric spacing.

Left PCAT (LPCAT). To locate the left PCAT area, we first identified the
primary bifurcation point closest to the aorta that indicated the end of the
LM artery and start of the LAD and LCX. The centerline voxel that had 3 or
more neighboring centerline voxels was considered the bifurcation point. As
shown in Fig. 2, without the location of the primary bifurcation, the length
of the segment to measure the PCAT along the LAD and LCX would be unknown.
From this bifurcation point, a 40 mm segment was considered, such that the LCX
and LAD were included. With these segments ascertained, we estimated the
vessel diameter at each point along the centerlines for the LM, LAD, and LCX.
Similar to the previous section, a sphere with radius equal to the vessel
diameter was placed at each ceneterline voxel to obtain the PCAT region. The
voxels in the specified HU range within this region were used to calculate the
LPCAT mean attenutation and volume.

##  3 Experiments & Results

###  3.1 Implementation

The 3D nnUNet model was trained using 5-fold cross-validation with the
different initialization of trainable parameters for a total of 1000 epochs.
The loss function used by the model was an equally weighted combination of
binary cross-entropy and soft Dice losses. It was optimized using the
Stochastic Gradient Descent (SGD) optimizer with an initial learning rate of
$10^{-3}$ and a batch size of 1. Each CT volume in the test split was passed
to the model from each fold, and predictions from five folds were ensembled
together. All experiments were done on a workstation running Ubuntu 22.04 LTS
with a NVIDIA Tesla V100 GPU.

###  3.2 Results

The 3D nnUNet model was run on the held-out test split (n = 183 studies). Fig.
1 provides examples of the segmentation output of the model for the left and
right arteries. Fig. 3(a) shows box plots of the segmentation dice scores for
the left and right coronary arteries respectively. The model achieved median
and mean DICE scores of 84% and 83% ± 7% for the RCA, and 83% and 81% ± 6% for
the LCA respectively. The quantitative results were similar for both arteries,
and are also consistent with prior work [9] on coronary artery segmentation.
From a qualitative standpoint, we noticed that the segmentations of the
proximal RCA and LCA were successful, but there were some under-segmentations
of the smaller distal branches of these arteries.

Box plots of the mean attenuation and volume of the LPCAT and RPCAT are also
shown in Figs. 3(b) and 3(c). Our results indicated that the RPCAT had a mean
attenuation of -73.81 ± 12.69 HU and mean volume of 0.15 ± 0.07 ml,
respectively. Similarly, the LPCAT had a mean attenuation of -77.51 ± 7.94 HU
and mean volume of 0.58 ± 0.35 ml, respectively. LPCAT mean volume was
impacted by the precision of bifurcation point detection. There was a stronger
correlation between RPCAT attenuation and volume ($R^{2}$ = 0.35) than that of
LPCAT ($R^{2}$ = 0.04) as shown in Fig. 4(b). The HU distribution for the
RPCAT and LPCAT are also seen in Figs. 3(d) and 3(e). Fig. 4(a) shows the
correlation between the mean attenuations of the RPCAT and LPCAT regions with
a corresponding $R^{2}$ = 0.23, indicating a weak relationship between the two
measurements.

(a)

(b)

(c)

(d)

(e)

Fig. 3: Box plots for: (a) Dice scores for RCA and LCA respectively, (b)
Attenuation of the RPCAT and LPCAT regions, and (c) Volume of the RPCAT and
LPCAT regions. (d)-(e) HU attenuation distributions for RPCAT and LPCAT.

(a)

(b)

Fig. 4: (a) Correlation between the HU attenuations of the LPCAT and RPCAT
regions. (b) Correlation between mean attenuation and PCAT volume for RPCAT
(cyan) and LPCAT (orange).

##  4 Discussion

The 3D nnUNet model segmented the RCA and LCA with mean dice scores of 83% and
81% respectively, and the results were consistent with previous work [9]. Our
framework was also able to automatically measure the attenuation and volume of
the PCAT regions surrounding these arteries. Despite successful segmentation
of the coronary arteries, we qualitatively observed under-segmentations of
smaller distal arterial branches. However, since we were primarily interested
in only the proximal 40 mm segment of the RCA, LAD, and LCX, these under-
segmentations had very little to no impact on the PCAT measurements.

It was crucial to find the bifurcation point of the LM as it determined the
segmental length for PCAT measurement. Without a priori knowledge of the
correct segmental length, measurement of PCAT would be difficult and
inconsistent. However, differences in anatomy among patients can lead to
variations in the length of the LM. This can also have an effect on the
detection of the bifurcation, leading to wider standard deviations in the
LPCAT volume as seen in Fig. 3(c).

These challenges were unexplored previously [6, 8] as a fully automated
technique for PCAT measurement had not been used. The automated method in our
pilot work can delineate this critical juncture of the LCA and simultaneously
use the RCA for PCAT measurement. From Fig. 3(b), the mean attenuation of the
LPCAT was fairly close to that of RPCAT. Both hold promise to be used as
biomarkers. Importantly, our measurement is repeatable and not subject to
annotator biases as in previous semi-automated approaches [6, 8].

The mean attenuation for RPCAT that we computed in this work is also
consistent with prior work [2, 4, 13, 14, 8, 6, 15, 16]. The PCAT attenuation
at the RCA was higher than that of the LCA by 3.7 HU (-77.5 vs. -73.8). Weak
attenuation correlation between RPCAT and LPCAT suggests that the two may have
independent inflammatory pathways with regard to coronary artery disease.
Although the patients in the ImageCAS dataset had some form of cardiac-related
disease, the dataset did not publicly provide the information related to
underlying cardiovascular risk and disease status. Therefore, it was not
possible to correlate our PCAT measurements against the patient
characteristics.

The volume of the PCAT measured in this work may also prove to be beneficial
in stratifying patients with cardiac disease, but this measure has not been
standardized in the past [2]. We have presented the first method to also
standardize the assessment of volumetric PCAT measures. Past literature
suggests that the density (attenuation) of fat deposition in the vicinity
surrounding the coronary arteries holds more promise as a biomarker [2]. In
particular, associating the PCAT mean attenuation with the total burden of
plaque and non-calcified plaque in the heart may be of increased diagnostic
interest [2, 4, 15, 16].

To our knowledge, we are the first to develop a fully automated framework for
the repeatable measurement of PCAT mean attenuation and volume in the region
around the right and left coronary arteries. The method is in contrast to
previous semi-automated methods, which are subject to annotator biases. Our
framework holds promise for the PCAT measurements to be used as predictive
biomarkers for identifying inflammation and the risk of coronary artery
disease.

##  5 ACKNOWLEDGEMENTS

This work was supported by the Intramural Research Program of the NIH Clinical
Center (project number 1Z01 CL040004). This work utilized the computational
resources of the NIH HPC Biowulf cluster.

##  6 Compliance with Ethical Standards

This study was approved by the Institutional Review Board (IRB) at the NIH and
performed with retrospectively acquired patient data. The need for informed
consent was waived.

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