# An Overview of the NET Playground - A Heterogeneous, Multi-Functional
Network Test Bed

Paul Schwenteck1, Sandra Zimmermann1, Caspar von Lengerke1, Giang T. Nguyen25,
Christian Scheunert3, Frank H. P. Fitzek15
1 Deutsche Telekom Chair of Communication Networks, TU Dresden, Germany
2 Haptic Communication Systems, TU Dresden, Germany
3 Chair of Communication Theory, TU Dresden, Germany
5 Centre for Tactile Internet with Human-in-the-Loop (CeTI)
E-mails<EMAIL_ADDRESS>
###### Abstract

This paper provides an overview of the hardware and software components used
in our test bed project the NET Playground. All source information is stored
in the GitLab repository in [1]. In the Hardware section, we present sketches
and 3D views of mechanical parts and technical drawings of printed boards. The
Software section discusses relay control using shell scripts and the
utilization of Ansible for automation. We also introduce a C++ framework for
connecting with the INA231 energy sensor. This paper serves as a reference for
understanding and replicating our project’s hardware and software components.

###### Index Terms:

Hardware test bed, energy measurement, distributed architectures

##  I Introduction

The successful execution of any project relies on the effective integration of
hardware and software components. This paper presents a detailed overview of
the hardware and software aspects employed in our GitLab repository [1], the
NET Playground test bed, providing valuable insights into its design and
functionality. The NET Playground is a heterogeneous, multi-functional network
testbed with 128 single-board computers connected in an extensive network. All
data in the repository is freely available to all and will continue to be
updated. So the content of this overview may still change in the future.

The hardware section showcases our meticulous approach to gathering and
organizing essential information. We present a comprehensive collection of
sketches, 3D views, and dimension drawings for the individual components
utilized in our project. The mechanical parts, including the sturdy frame,
metal components, and transparent plexiglass plates, are presented with
detailed 3D views. Additionally, we document the technical drawings and
pictures of the printed boards, such as the level shifter, relay board, and
INA231 energy sensor board.

The software section delves into two critical aspects: relay control and
orchestration with Ansible. We introduce a shell script specifically developed
for relay control, highlighting its role as an interface between software and
hardware components. The script defines the Odroid pins and their connection
to a level shifter, which enables efficient switching of the relays and
precise regulation of the power supply. We discuss the development of
individual shell scripts for each relay, tailored to their specific roles
within the project, and a combined version for simultaneous control.
Additionally, we explore the utilization of Ansible, an automation tool, in
conjunction with an inventory file and ansible-playbooks. We explain how host
groups are defined based on IP addresses, and we provide examples of general-
purpose ansible-playbooks for powering on/off Odroids and
installing/configuring IPFS on designated host groups.

Furthermore, we introduce a C++ framework for connecting with the INA231
energy sensor. This framework facilitates TCP-based communication,

##  II Hardware

In the Hardware Section, we have gathered a comprehensive collection of
sketches for the individual components of our project. To ensure organized
data management, we have classified the information into two distinct groups:

i) Mechanical Parts: This group comprises various components, such as the
sturdy frame, all the essential metal parts, and the transparent plexiglass
plates. We have meticulously prepared detailed 3D views of these mechanical
elements, providing a comprehensive understanding of their structure,
dimensions, and interconnections. These visual representations offer valuable
insights into the overall design and facilitate accurate hardware assembly.

ii) Printed Boards: In this category, we have focused on the intricate
circuitry of our project. Specifically, we have documented the technical
drawings and schematics of critical boards, including the level shifter, relay
board, and the board housing the INA231 energy sensor. These detailed drawings
allow for a precise understanding of the circuit connections, component
placement, and proper positioning of the relevant electronic elements.
Additionally, we have provided dimension drawings for the metal parts
associated with the printed boards, enabling a comprehensive understanding of
their exact measurements.

##  III Software

###  III-A Relay Control

Here, we have included the shell script responsible for controlling the relays
that govern the power supply of our devices. The script is a crucial interface
between the software and hardware components, allowing us to manage the
relay’s operation effectively.

We have defined the pins on the odroid within the script, our chosen
microcontroller for relay control. These pins are connected to a level
shifter, which facilitates the conversion of the output voltage to 5 V. This 5
V output is then utilized to effectively switch the relays, enabling us to
accurately regulate the power supply to our devices.

To provide comprehensive control over the relay system, we have developed
individual shell scripts for each of the four relays. This approach allows us
to tailor the behavior of each relay based on its specific role within the
project. These individual scripts provide fine-grained control, Whether
activating or deactivating a particular device or managing power flow to
specific components.

Additionally, we have also created a combined version of the shell script.
This unified script allows us to control multiple relays simultaneously,
streamlining the management process and simplifying the overall control of the
power supply.

###  III-B Ansible

Ansible [2] works with a combination of an inventory file and ansible-
playbooks. The inventory defines host groups depending on their IP addresses.
A group consists of a [name] and an IP address range. In our case, we have a
group called [odroids-testgroup] that consists of devices with IP addresses
ranging from 192.168.1.1 to 192.168.1.16. In addition to the address range, we
can define the ssh password that Ansible should use when connecting to the
device, which is, in our case, odroid. When executing ansible-playbooks, the
path to the inventory needs to be defined with the -i flag.

⬇

# The inventory file for the NET PLayground

[odroids-testgroup]

192.168.1.[1:16] ansible_ssh_pass=odroid

[odroids-testgroup-consumer]

192.168.1.1 ansible_ssh_pass=odroid

[odroids-control]

192.168.[1:8].42 ansible_ssh_pass=odroid

We divide our ansible-playbooks into two categories—one for general purposes
and one for specific use cases. The two ansible-playbooks for general purpose
are odroids_power and ipfs_init. Generally usable ansible-playbook. The
ansible-playbook odroids_power copies the gpio.sh, script to all control-
droids and runs the script on them. The host group odroid-control is defined
in the inventory file. The sh-script defines gpio pins for the shifter board
connected to the relays. Periodically all pins are activated, which switches
the relays and powers the individual odroids. The control-odroid must not
switch all nodes simultaneously, as that would lead to voltage spikes in the
power supply and damage it. In the following, we show the code for the
playbook, which includes the host group odroids-control defined in the
inventory, the option become for executing the tasks on a sudo level, and the
two tasks. It is also possible to define variables like power that can be
defined when executing the ansible-playbook.

⬇

—

# This playbook turns on all odroids;

# control-odroids need to be connected

- hosts: odroids-control

become: yes

tasks:

- name: copy swr file to remote host

copy:

src: ./gpio.sh

dest: /home/odroid/

- name: switch odroids {{ power }}

shell:

| bash /home/odroid/gpio.sh {{ power }}

The ansible-playbook ipfs_init installs and configures Interplanetary File
System (IPFS) [3, 4] on the defined host group. For the script to work, we
first need to build IPFS on the destination device and copy the built version
of IPFS into the directory of the ansible-playbook. The playbook then copies
the build version to all device and initiate it. Additionally, it created an
IPFS service file, so it starts when the device is booted. Logs from the IPFS
service are stored in /var/ on the device. The service file is included in the
directory.

The individual playbooks are divided according to the experiments. We have
only one experiment that measures energy consumption in an IPFS peer-to-peer
network. The individual playbooks are divided according to the experiments. We
have only one experiment that measures energy consumption in an IPFS peer-to-
peer network. Here we have defined scripts that define the link properties
like delay and bandwidth on one. Fixed links are essential for a deterministic
measurement process. Another playbook deletes all added content in IPFS and
resets all traffic control settings. This way, the odroids can be used for
another experiment without previous presets.

###  III-C INA231

For the connection with the INA231 sensor, we have developed a framework in
C++. Installing the framework on a device can be connected to any other device
with a connected INA231. The connection is established via TCP. The measured
current of the INA231 is sent over the established TCP connection. Thus, a
central device can retrieve the measurement data from all devices in the
network. Installing can be done with the install.sh file included in the
install files. Additional help is offered in the help.txt.

For the connection with the INA231 sensor, we have developed a robust and
flexible framework in C++. This framework facilitates seamless integration of
the INA231 sensor with other devices in our project. Installing the framework
on a device can easily connect to any other device equipped with an INA231
sensor, enabling efficient data communication and measurement retrieval.

The connection between devices is established using TCP, a reliable and
widely-used communication protocol. Through this TCP connection, the measured
current values from the INA231 sensor are transmitted, allowing for real-time
monitoring and data collection. This approach enables a central device to
retrieve measurement data from multiple devices in the network, providing a
centralized and comprehensive view of the energy consumption across the
project.

To facilitate the installation process, we have included an installation
script, install.sh, in the install files. This script streamlines the
framework’s setup on the target device, ensuring a smooth and hassle-free
installation experience. Additionally, we provide a detailed help file,
help.txt, which offers comprehensive instructions and guidance for configuring
and utilizing the framework effectively.

## Acknowledgment

Funded in part by the German Research Foundation (DFG, Deutsche
Forschungsgemeinschaft) as part of Germany’s Excellence Strategy – EXC 2050/1
– Project ID 390696704 – Cluster of Excellence ”Centre for Tactile Internet
with Human-in-the-Loop” (CeTI) of Technische Universität Dresden as well as by
the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under
Project ID 450566247 and the Federal Ministry of Education and Research of
Germany in the programme of ”Souverän. Digital. Vernetzt.” – Joint project
6G-life – projectID: 16KISK001K and 16KISK002

## References

  * [1] P. Schwenteck, “Net playground,” https://gitlab.com/Paulteck/net-playground, 2023.
  * [2] L. Hochstein and R. Moser, _Ansible: Up and Running: Automating configuration management and deployment the easy way_. ” O’Reilly Media, Inc.”, 2017.
  * [3] J. Benet, “Ipfs - content addressed, versioned, p2p file system,” 2014.
  * [4] D. Trautwein, A. Raman, G. Tyson, I. Castro, W. Scott, M. Schubotz, B. Gipp, and Y. Psaras, “Design and evaluation of ipfs: A storage layer for the decentralized web,” in _Proceedings of the ACM SIGCOMM 2022 Conference_. New York, NY, USA: Association for Computing Machinery, 2022, p. 739–752. [Online]. Available: https://doi.org/10.1145/3544216.3544232

  *[IPFS]: Interplanetary File System


# Certifying optimality for convex quantum
channel optimization problems

Bryan Coutts  Institute for Quantum Computing
University of Waterloo, Canada  School of Computer Science
University of Waterloo, Canada  Mark Girard  Institute for Quantum Computing
University of Waterloo, Canada  John Watrous  Institute for Quantum Computing
University of Waterloo, Canada  School of Computer Science
University of Waterloo, Canada  Canadian Institute for Advanced Research
Toronto, Canada

(December 5, 2018)

###### Abstract

We identify necessary and sufficient conditions for a quantum channel to be
optimal for any convex optimization problem in which the optimization is taken
over the set of all quantum channels of a fixed size. Optimality conditions
for convex optimization problems over the set of all quantum measurements of a
given system having a fixed number of measurement outcomes are obtained as a
special case. In the case of linear objective functions for measurement
optimization problems, our conditions reduce to the well-known
Holevo–Yuen–Kennedy–Lax measurement optimality conditions. We illustrate how
our conditions can be applied to various state transformation problems having
non-linear objective functions based on the fidelity, trace distance, and
quantum relative entropy.

##  1 Introduction

Several problems and settings that arise in quantum information theory can be
expressed as _optimization problems_ in which a real-valued function, defined
for a class of quantum channels or measurements, is either minimized or
maximized. The problem of _minimum error quantum state discrimination_ [BC09],
in which a quantum state randomly selected from a known ensemble of states is
to be identified with the smallest possible probability of error by means of a
measurement, provides a well-known example. This problem is naturally
expressed as the optimization of a real-valued linear function defined on the
set of all measurements with a fixed number of outcomes. Other examples arise
in the study of _quantum cloning_ [SIGA05] and the closely related notion of
_quantum money_ [AFG+12], where one is generally interested in knowing how
well an optimally selected quantum channel can transform a single copy of a
given state into multiple copies of the same state, with respect to a number
of different figures of merit. Another example can be found in quantum
complexity theory, in which _two-message quantum interactive proof systems_
[JUW09] are naturally analyzed as optimization problems in which the objective
function describes the probability that a given verifier accepts, and where
the optimization is over all quantum channels of a fixed size, which describe
the possible actions of a prover.

Concerning the optimization of _linear functions_ defined on the set of all
measurements with a fixed number of outcomes, necessary and sufficient
conditions for optimality were identified by Holevo [Hol73b, Hol73a] and Yuen,
Kennedy, and Lax [YKL70, YKL75]. These conditions, which are described
explicitly later in this paper, are relatively easy to check; the problem of
actually finding or approximating an optimal measurement, while efficiently
solvable through the use of semidefinite programming [JvF02, Ip03, EMV03], is
in general a more computationally involved task. These optimality conditions
can be easily extended to obtain optimality conditions for real-valued linear
functions defined on the set of all quantum channels transforming one quantum
system to another.

We prove a generalization of these results to convex optimization problems
whose objective functions are not necessarily linear. To be more precise, we
consider optimization problems of the form

minimize $\displaystyle f(\Phi)$ (1) subject to
$\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}),$

where

$f:\mathrm{C}(\mathcal{X},\mathcal{Y})\rightarrow\mathbb{R}\cup\\{\infty\\}$
(2)

is a convex function. Here, $\mathrm{C}(\mathcal{X},\mathcal{Y})$ denotes the
set of all channels (i.e., completely positive and trace-preserving linear
maps) from an input system to an output system having associated complex
Euclidean spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. A channel
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is said to be _optimal_ for the
problem (1) if it is the case that $f(\Phi)\leq f(\Psi)$ for all
$\Psi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$. In this paper we do not consider
the difficulty of finding or approximating an optimal channel $\Phi$ for a
given function $f$, but instead we focus only on the task of _verifying_ that
a given channel $\Phi$ is indeed optimal. The optimality conditions we obtain
can be easily checked for differentiable functions $f$, and can also be used
to verify optimality of channels for some non-differentiable functions. We
stress that our optimality conditions are not generic optimality conditions
that hold for all convex optimization problems, but rather rely on a specific
structure that arises when the optimization is over all quantum channels of a
fixed size.

There are, of course, situations in which one would prefer a method to find an
optimal channel $\Phi$ for a chosen function $f$, as opposed to simply
verifying the optimality of a given $\Phi$, but the task of verifying
optimality nevertheless has value for multiple reasons. For instance, one
might hypothesize that a particular channel $\Phi$ is optimal based upon an
intuition concerning the function $f$, or upon a heuristic method, making the
task of verifying optimality essentially important. The computational task of
finding an optimal solution for a chosen function $f$ might also be expensive,
time-consuming, or delegated to an untrusted computer, but once this task has
been performed the optimality of the solution can be verified, allowing anyone
who performs the verification to trust in the optimality of the solution.
Finally, there are situations in which the function $f$ could be indeterminate
in some respect, eliminating the possibility that a numerical computation
could reveal an optimal solution, but potentially allowing for an optimal
solution to be expressed and checked analytically. (The results of Bacon,
Childs, and van Dam [BCvD05] on hidden subgroup algorithms for certain groups
provide a striking example of this potential.)

Our optimality conditions are applicable to convex optimization problems in
which the optimization is over all measurements having a fixed number of
outcomes, as opposed to being over all channels of a fixed size. This is done
through a standard correspondence between measurements and _quantum-to-
classical channels_ , described in the section following this one. We observe
that Holevo [Hol73b, Hol73a] also derived optimality conditions for
optimizations over measurements having differentiable (but not necessarily
convex) objective functions. Holevo proved that these conditions are necessary
for optimality, but did not prove they are sufficient (as they are not
sufficient in general). When one restricts their attention to convex objective
functions, our optimality conditions are equivalent to a set of intermediate
conditions identified by Holevo, but not to the final set of conditions he
identified.

We provide a few examples of how our optimality conditions can be applied to
interesting categories of optimization problems. As a simple warm-up, we first
explain how our conditions imply the Holevo–Yuen–Kennedy–Lax conditions for
the optimality of measurements for minimum error state discrimination, which
are easily extended to channel optimization problems having linear objective
functions. We then discuss optimization problems relating to state
transformations having objective functions based on fidelity, trace distance,
and quantum relative entropy. We note that optimization problems of the form
we consider have recently been studied in the context of _recovery measures_
(i.e., the _fidelity of recovery_ and generalizations of this measure) [FR15,
SW15, BHOS15, CHM+16, BT16, BFT17]. Expositions of this topic can be found in
[Sut18] and Chapter 12 of [Wil17].

##  2 Background and notation

This section summarizes some concepts from convex analysis and optimization
theory, narrowly focused on their applications to this paper. Further
information on these topics can be found in [Roc70], [BL06], [BV04], and
[MN13], for instance. We assume the reader is familiar with quantum
information theory, which is covered in the books [NC00], [Wil17], and
[Wat18], among others. We will, however, summarize the notion of the Choi
operator of a channel, clarify its basic connection to the sorts of
optimization problems we consider, and discuss the correspondence between
measurements and quantum-to-classical channels, as these topics are essential
to an explanation of our results. It will also be helpful to begin the section
by establishing some basic notation and terminology concerning linear algebra.

#### Linear algebra notation and terminology

When we refer to a _complex Euclidean space_ , we mean $\mathbb{C}^{n}$ for
some positive integer $n$, or more generally the complex vector space
consisting of vectors indexed by an arbitrary finite set in place of the index
set $\\{1,\ldots,n\\}$. The elementary unit vectors of the space
$\mathbb{C}^{n}$ are denoted $e_{1},\ldots,e_{n}$. Complex Euclidean spaces
will be denoted by capital calligraphic letters such as $\mathcal{X}$,
$\mathcal{Y}$, and $\mathcal{Z}$.

For a complex Euclidean space $\mathcal{X}$, the space of linear operators on
$\mathcal{X}$ is denoted $\mathrm{L}(\mathcal{X})$, and the identity operator
on $\mathcal{X}$ is denoted $\mathds{1}_{\mathcal{X}}$. For indices
$j,k\in\\{1,\ldots,n\\}$, the operator $E_{j,k}\in\mathrm{L}(\mathbb{C}^{n})$
is defined as $E_{j,k}=e_{j}e_{k}^{\ast}$. Equivalently, with respect to the
basis $\\{e_{1},\ldots,e_{n}\\}$, the matrix representation of $E_{j,k}$ has a
1 in the $(j,k)$ entry, with all other entries 0.

The real vector space of Hermitian operators acting on a complex Euclidean
space $\mathcal{X}$ is denoted $\mathrm{Herm}(\mathcal{X})$; the cone of
positive semidefinite operators acting on $\mathcal{X}$ is denoted
$\mathrm{Pos}(\mathcal{X})$; the set of positive definite operators acting on
$\mathcal{X}$ is denoted $\mathrm{Pd}(\mathcal{X})$; and the set of density
operators (i.e., positive semidefinite operators having unit trace) is denoted
$\mathrm{D}(\mathcal{X})$. The Hilbert–Schmidt inner product of two Hermitian
operators $X,Y\in\mathrm{Herm}(\mathcal{X})$ is given by

$\langle X,Y\rangle=\operatorname{Tr}(XY).$ (3)

For a subspace $\mathcal{V}\subseteq\mathcal{X}$ of a complex Euclidean space
$\mathcal{X}$, we write $\Pi_{\mathcal{V}}\in\mathrm{Pos}(\mathcal{X})$ to
denote the (orthogonal) projection operator that projects onto the subspace
$\mathcal{V}$. Finally, whenever we refer to the _inverse_ of a positive
semidefinite operator $X\in\mathrm{Pos}(\mathcal{X})$, it should be understood
that we are referring to the _Moore–Penrose pseudo-inverse_ of $X$ (i.e., the
operator that acts as the inverse of $X$ on the image of $X$ and zero on the
kernel of $X$).

#### Convex functions taking real or infinite values

Let $\mathcal{X}$ be a complex Euclidean space and let

$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$ (4)

be a function mapping each Hermitian operator to either a real number or to
positive infinity. The _domain_ of $f$ is defined as

$\operatorname{dom}(f)=\\{X\in\mathrm{Herm}(\mathcal{X}):f(X)\in\mathbb{R}\\}.$
(5)

For any function of the form $f:\mathcal{C}\rightarrow\mathbb{R}$ defined only
on a subset $\mathcal{C}\subseteq\mathrm{Herm}(\mathcal{X})$, one may
naturally extend $f$ to a function of the form (4) by defining $f(X)=\infty$
whenever $X\not\in\mathcal{C}$.

A function $f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
is _convex_ if $\operatorname{dom}(f)$ is a convex set and $f$ is convex on
$\operatorname{dom}(f)$. More explicitly, $f$ is convex if for all
$X,Y\in\operatorname{dom}(f)$ and $\lambda\in[0,1]$, it is the case that
$\lambda X+(1-\lambda)Y\in\operatorname{dom}(f)$ and

$f(\lambda X+(1-\lambda)Y)\leq\lambda f(X)+(1-\lambda)f(Y).$ (6)

A function $f$ of the form (4) is _proper_ if
$\operatorname{dom}(f)\neq\emptyset$.

The _indicator function_ of a set
$\mathcal{C}\subseteq\mathrm{Herm}(\mathcal{X})$ is the function

$I_{\mathcal{C}}:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
(7)

defined as

$I_{\mathcal{C}}(X)=\begin{cases}0&X\in\mathcal{C}\\\
\infty&X\notin\mathcal{C}\end{cases}$ (8)

for all $X\in\mathrm{Herm}(\mathcal{X})$. It is evident that
$\operatorname{dom}(I_{\mathcal{C}})=\mathcal{C}$ for every set
$\mathcal{C}\subseteq\mathrm{Herm}(\mathcal{X})$, and if $\mathcal{C}$ is a
convex set then $I_{\mathcal{C}}$ is a convex function.

#### Subdifferentials

Let $f$ be a proper function of the form (4) and let
$X\in\operatorname{dom}(f)$. The _subdifferential_ of $f$ at $X$ is the set
defined as

$\partial f(X)=\\{Z\in\mathrm{Herm}(\mathcal{X})\,:\,f(Y)-f(X)\geq\langle
Z,Y-X\rangle\;\text{for all}\;Y\in\operatorname{dom}(f)\\}.$ (9)

A key property of the subdifferential of a proper function, which follows
trivially from the definition of the subdifferential, is its relation to
global minima: $X\in\operatorname{dom}(f)$ is a global minimizer of $f$ if and
only if $0\in\partial f(X)$. Two additional properties of subdifferentials
that are relevant to this paper are the following:

  1. 1.

If $f$ is convex and differentiable at $X\in\operatorname{dom}(f)$ then
$\partial f(X)=\\{\nabla f(X)\\}$.

  2. 2.

If $\lVert\mspace{1.0mu}\cdot\mspace{1.0mu}\rVert$ is any norm on
$\mathrm{Herm}(\mathcal{X})$ and
$f(X)=\lVert\mspace{1.0mu}X\mspace{1.0mu}\rVert$ for all
$X\in\mathrm{Herm}(\mathcal{X})$, then

$\partial f(X)=\\{Y:\langle Y,X\rangle=\lVert X\rVert,\,\lVert Y\rVert_{*}\leq
1\\},$ (10)

where $\lVert\mspace{1.0mu}Y\mspace{1.0mu}\rVert_{\ast}=\sup\\{\langle
Y,X\rangle:\lVert\mspace{1.0mu}X\mspace{1.0mu}\rVert\leq 1\\}$ is the dual
norm to $\lVert\mspace{1.0mu}\cdot\mspace{1.0mu}\rVert$.

Finally, we will make use of the following theorem, which presents a variant
of the _chain rule for subdifferentials_. In the statement of this theorem,
$\operatorname{relint}$ denotes the _relative interior_ of a set, and for a
linear map
$\Lambda:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathrm{Herm}(\mathcal{X})$, the
map

$\Lambda^{\ast}:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$
(11)

denotes the adjoint map to $\Lambda$, which is the uniquely determined linear
map that satisfies

$\langle X,\Lambda(Y)\rangle=\langle\Lambda^{\ast}(X),Y\rangle$ (12)

for all $X\in\mathrm{Herm}(\mathcal{X})$ and $Y\in\mathrm{Herm}(\mathcal{Y})$.

######  Theorem 1.

Let $\mathcal{X}$ and $\mathcal{Y}$ be complex Euclidean spaces, let
$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$ be a
convex function, and let
$g:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathrm{Herm}(\mathcal{X})$ be an
affine linear map, meaning that

$g(Y)=\Lambda(Y)+A$ (13)

for all $Y\in\mathrm{Herm}(\mathcal{Y})$, for some choice of a linear map
$\Lambda:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathrm{Herm}(\mathcal{X})$ and
an operator $A\in\mathrm{Herm}(\mathcal{X})$.

  1. 1.

For every $Y\in\mathrm{Herm}(\mathcal{Y})$, it is the case that

$\partial(f\circ g)(Y)\supseteq\Lambda^{\ast}(\partial f(g(Y))).$ (14)

  2. 2.

If
$\operatorname{im}(\Lambda)\cap\operatorname{relint}(\operatorname{dom}(f))\not=\varnothing$,
then

$\partial(f\circ g)(Y)=\Lambda^{\ast}(\partial f(g(Y)))$ (15)

for every $Y\in\mathrm{Herm}(\mathcal{Y})$.

#### Convex optimization problems

Convex optimization problems in quantum information theory often have the
following general form, for some choice of a convex function
$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$ and a
convex set $\mathcal{C}\subseteq\mathrm{Herm}(\mathcal{X})$:

minimize $\displaystyle f(X)$ (16) subject to $\displaystyle X\in\mathcal{C}.$

An operator $X\in\mathcal{C}\cap\operatorname{dom}(f)$ is said to be _optimal_
for the optimization problem (16) if $f(X)\leq f(Y)$ for all
$Y\in\mathcal{C}$. As we will see below, convenient conditions for optimality
can be given if it is the case that

$\operatorname{relint}\bigl{(}\operatorname{dom}(f)\bigr{)}\cap\operatorname{relint}(\mathcal{C})\neq\emptyset.$
(17)

A (constrained) convex optimization problem of the form (16) can be considered
as an unconstrained convex optimization problem by minimizing
$f(X)+I_{\mathcal{C}}(X)$ over all Hermitian operators
$X\in\mathrm{Herm}(\mathcal{X})$. The domain of the function
$f+I_{\mathcal{C}}$ is given by

$\operatorname{dom}(f+I_{\mathcal{C}})=\operatorname{dom}(f)\cap\mathcal{C}.$
(18)

As was mentioned previously, an operator
$X\in\operatorname{dom}(f)\cap\mathcal{C}$ is a global minimizer of
$f(X)+I_{\mathcal{C}}(X)$ if and only if $0\in\partial(f+I_{\mathcal{C}})(X)$.
If the condition in (17) holds and both $f$ and $\mathcal{C}$ are convex, a
subdifferential sum-rule implies that

$\partial(f+I_{\mathcal{C}})(X)=\partial f(X)+\partial I_{\mathcal{C}}(X)$
(19)

for all $X\in\operatorname{dom}(f)\cap\mathcal{C}$. For this reason, a
characterization of the subdifferential set $\partial I_{\mathcal{C}}(X)$ for
a convex set $\mathcal{C}$ can be useful in identifying necessary and
sufficient conditions for optimality.

#### Optimizing over Choi operators of channels

For complex Euclidean spaces $\mathcal{X}$ and $\mathcal{Y}$, the set of
completely positive, trace-preserving linear maps (i.e., channels) from
$\mathrm{L}(\mathcal{X})$ to $\mathrm{L}(\mathcal{Y})$ is denoted
$\mathrm{C}(\mathcal{X},\mathcal{Y})$. Given a linear map
$\Phi:\mathrm{L}(\mathcal{X})\rightarrow\mathrm{L}(\mathcal{Y})$, its Choi
representation $J(\Phi)\in\mathrm{L}(\mathcal{Y}\otimes\mathcal{X})$ is
defined as

$J(\Phi)=\sum_{j,k=1}^{n}\Phi(E_{j,k})\otimes E_{j,k},$ (20)

assuming $\mathcal{X}=\mathbb{C}^{n}$. (An analogous definition is used for
index sets other than $\\{1,\ldots,n\\}$.) Through this representation, the
set of channels is isomorphic to the set

$J(\mathrm{C}(\mathcal{X},\mathcal{Y}))=\\{X\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})\,:\,\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}}\\}\subseteq\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X}).$
(21)

This set is convex, and it is helpful to observe that its relative interior is

$\operatorname{relint}\bigl{(}J(\mathrm{C}(\mathcal{X},\mathcal{Y}))\bigr{)}=\\{X\in\mathrm{Pd}(\mathcal{Y}\otimes\mathcal{X})\,:\,\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}}\\}.$
(22)

The action of a map $\Phi$ can be recovered from its Choi representation by
the relation

$\Phi(X)=\operatorname{Tr}_{\mathcal{X}}((\mathds{1}_{\mathcal{Y}}\otimes
X^{\scriptscriptstyle\mathsf{T}})J(\Phi))$ (23)

for all operators $X\in\mathrm{L}(\mathcal{X})$, where
$X^{\scriptscriptstyle\mathsf{T}}$ denotes the transpose of $X$.

An optimization problem of the form

minimize $\displaystyle g(\Phi)$ (24) subject to
$\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}),$

where

$g:\mathrm{C}(\mathcal{X},\mathcal{Y})\rightarrow\mathbb{R}\cup\\{\infty\\}$
(25)

is a given function, can equivalently be expressed as

minimize $\displaystyle f(X)$ (26) subject to $\displaystyle X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y})),$

where

$f:J(\mathrm{C}(\mathcal{X},\mathcal{Y}))\rightarrow\mathbb{R}\cup\\{\infty\\}$
(27)

is defined as $f(J(\Phi))=g(\Phi)$ for all
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$. Although the formulations (24)
and (26) are equivalent, it will be convenient for us to focus primarily on
the formulation (26). The results we obtain can, however, easily be adapted to
the formulation (24).

#### Subdifferentials of the indicator function of the set of Choi operators
of channels

The following proposition provides a characterization of the subdifferential
of the indicator function of $J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$ at every
point $X\in J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$.

######  Proposition 2.

For complex Euclidean spaces $\mathcal{X}$ and $\mathcal{Y}$, and for $X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$, it holds that

$\partial
I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}(X)=\\{-Y-\mathds{1}_{\mathcal{Y}}\otimes
Z\,:\,Z\in\mathrm{Herm}(\mathcal{X}),\;Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}),\;YX=0\\}.$
(28)

###### Proof.

Define
$\mathcal{K}=\\{K\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\,:\,\operatorname{Tr}_{\mathcal{Y}}(K)=\mathds{1}_{\mathcal{X}}\\}$.
It is the case that

$J(\mathrm{C}(\mathcal{X},\mathcal{Y}))=\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})\cap\mathcal{K},$
(29)

and therefore

$I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}=I_{\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})}+I_{\mathcal{K}}.$
(30)

As $\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$ and $\mathcal{K}$ are both
convex and

$\operatorname{relint}(\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}))\cap\operatorname{relint}(\mathcal{K})=\\{P\in\mathrm{Pd}(\mathcal{Y}\otimes\mathcal{X})\,:\,\operatorname{Tr}_{\mathcal{Y}}(P)=\mathds{1}_{\mathcal{X}}\\}\not=\varnothing,$
(31)

one has that

$\begin{multlined}\partial
I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}(X)=\partial
I_{\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})}(X)+\partial
I_{\mathcal{K}}(X)\\\\[2.84526pt]
=\\{-Y\,:\,Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}),\;XY=0\\}+\\{\mathds{1}_{\mathcal{Y}}\otimes
Z\,:\,Z\in\mathrm{Herm}(\mathcal{X})\\},\end{multlined}\partial
I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}(X)=\partial
I_{\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})}(X)+\partial
I_{\mathcal{K}}(X)\\\\[2.84526pt]
=\\{-Y\,:\,Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}),\;XY=0\\}+\\{\mathds{1}_{\mathcal{Y}}\otimes
Z\,:\,Z\in\mathrm{Herm}(\mathcal{X})\\},$ (32)

which implies the proposition. ∎

#### The Lagrange dual problem for channel optimization

Consider a channel optimization problem expressed in the following form that
is equivalent to (26):

minimize $\displaystyle f(X)$ (33) subject to
$\displaystyle\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}},$
$\displaystyle X\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}).$

One may then formulate the associated Lagrange dual problem:

maximize $\displaystyle g(Y,Z)$ (34) subject to $\displaystyle
Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X}),$ $\displaystyle
Z\in\mathrm{Herm}(\mathcal{X}),$

where

$g(Y,Z)=\operatorname{Tr}(Z)+\inf_{X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})}\bigl{(}f(X)-\bigl{\langle}X,Y+\mathds{1}_{\mathcal{Y}}\otimes
Z\bigr{\rangle}\bigr{)}$ (35)

for all $Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$ and
$Z\in\mathrm{Herm}(\mathcal{X})$.

The optimal value of the Lagrange dual problem (34) is a lower-bound for the
optimal value of the original problem (33) (even if the function $f$ is not
convex), which is a property known as weak duality. _Slater’s theorem_ implies
that if $f$ is convex, and there exists a positive definite operator
$X\in\mathrm{Pd}(\mathcal{Y}\otimes\mathcal{X})$ such that
$\operatorname{Tr}_{\mathcal{Y}}(X)=\mathds{1}_{\mathcal{X}}$ and
$X\in\operatorname{relint}(\operatorname{dom}(f))$, then the problems (33) and
(34) have the same optimal value.

#### Quantum measurements as channels

Optimizations over the set of all measurements having a fixed number of
outcomes can be expressed as optimizations over channels, as we now explain.
Consider first a measurement on a complex Euclidean space $\mathcal{X}$ that
has $m$ possible measurement outcomes, and is described by measurement
operators $\\{P_{1},\ldots,P_{m}\\}\subseteq\mathrm{Pos}(\mathcal{X})$.
Letting $\mathcal{Y}=\mathbb{C}^{m}$, this measurement can be represented by
the channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ defined as

$\Phi(X)=\sum_{k=1}^{m}\langle P_{k},X\rangle\,E_{k,k}$ (36)

for all $X\in\mathrm{L}(\mathcal{X})$. Any channel expressible in this form is
called a _quantum-to-classical channel_. An equivalent condition to a channel
taking the form (36) is that its Choi operator takes the form

$J(\Phi)=\sum_{k=1}^{m}E_{k,k}\otimes P_{k}^{{\scriptscriptstyle\mathsf{T}}}.$
(37)

Next, for each $k\in\\{1,\ldots,m\\}$, define a linear map
$\Xi_{k}:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{X})$
as

$\Xi_{k}(X)=\bigl{(}(e_{k}^{\ast}\otimes\mathds{1}_{\mathcal{Y}})X(e_{k}\otimes\mathds{1}_{\mathcal{Y}})\bigr{)}^{{\scriptscriptstyle\mathsf{T}}},$
(38)

and observe that one has

$P_{k}=\Xi_{k}(J(\Phi))$ (39)

for the quantum-to-classical channel $\Phi$ given by (36). In words, for a
measurement represented by operators $\\{P_{1},\ldots,P_{m}\\}$, the linear
map $\Xi_{k}$ allows for the recovery of the operator $P_{k}$ from the Choi
operator $J(\Phi)$ of the quantum-to-classical channel associated with that
measurement. A function $g(P_{1},\ldots,P_{m})$ of these measurement operators
can therefore be expressed as a function

$f(J(\Phi))=g\bigl{(}\Xi_{1}(J(\Phi)),\ldots,\Xi_{m}(J(\Phi))\bigr{)}$ (40)

of the Choi operator $J(\Phi)$. Note that if
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is an arbitrary (i.e., not
necessarily quantum-to-classical) channel, then the operators
$P_{1},\ldots,P_{m}$ defined by (39) will still necessarily satisfy
$P_{1},\ldots,P_{m}\in\mathrm{Pos}(\mathcal{X})$ and
$P_{1}+\cdots+P_{m}=\mathds{1}_{\mathcal{X}}$, and therefore represent a valid
measurement. (In general, the same measurement is given by a continuum of
channels $\Phi$, including the quantum-to-classical channel described
earlier.) An optimization problem of the form

minimize $\displaystyle g(P_{1},\ldots,P_{m})$ (41) subject to $\displaystyle
P_{1},\ldots,P_{m}\in\mathrm{Pos}(\mathcal{X})$ $\displaystyle
P_{1}+\cdots+P_{m}=\mathds{1}_{\mathcal{X}}$

can therefore be expressed equivalently as follows:

minimize $\displaystyle f(X)$ (42) subject to $\displaystyle X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$

for $f$ defined from $g$ as in (40).

##  3 Optimality conditions for convex channel optimization

In this section, we present our main general result regarding optimality
conditions for convex optimization problems over quantum channels. As
suggested in the previous section, it is convenient to associate quantum
channels with their Choi representations, and to consider optimization
problems of the form

minimize $\displaystyle f(X)$ (43) subject to $\displaystyle X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$

for various choices of convex functions

$f:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}.$
(44)

Optimality conditions for such problems can be translated to optimality
conditions for problems of the form (1), as will be illustrated in the section
following this one.

######  Theorem 3.

Let
$f:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
be a convex function such that

$\operatorname{relint}\bigl{(}\operatorname{dom}(f)\bigr{)}\cap\operatorname{relint}\bigl{(}J\bigl{(}\mathrm{C}(\mathcal{X},\mathcal{Y})\bigr{)}\bigr{)}\not=\emptyset,$
(45)

and let $X\in J\bigl{(}\mathrm{C}(\mathcal{X},\mathcal{Y})\bigr{)}$ be the
Choi representation of a channel such that $X\in\operatorname{dom}(f)$. The
following statements are equivalent:

  1. 1.

The operator $X$ is optimal for the optimization problem (43).

  2. 2.

There exists an operator $H\in\partial f(X)$ satisfying

$\operatorname{Tr}_{\mathcal{Y}}(HX)\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HX).$ (46)

###### Proof.

An operator $X$ is optimal for the problem (43) if and only if

$0\in\partial(f+I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))})(X)=\partial
f(X)+\partial I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}(X).$ (47)

By the characterization of the subdifferential of the indicator function
$I_{J(\mathrm{C}(\mathcal{X},\mathcal{Y}))}$ given by Proposition 2, it
follows that $X$ is optimal for the optimization problem (43) if and only if
there exist operators $Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$,
$Z\in\mathrm{Herm}(\mathcal{X})$, and $H\in\partial f(X)$ satisfying

$YX=0\quad\text{and}\quad H-Y-\mathds{1}_{\mathcal{Y}}\otimes Z=0.$ (48)

Assume first that statement 2 holds, and define
$Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$ and
$Z\in\mathrm{Herm}(\mathcal{X})$ as

$Y=H-\mathds{1}_{\mathcal{Y}}\otimes Z\quad\text{and}\quad
Z=\operatorname{Tr}_{\mathcal{Y}}(HX).$ (49)

As $X$ and $Y$ are both positive operators and

$\langle Y,X\rangle=\langle
H-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HX),X\rangle=\langle
H,X\rangle-\langle\operatorname{Tr}_{\mathcal{Y}}(HX),\mathds{1}_{\mathcal{X}}\rangle=0,$
(50)

it follows that $YX=0$. The conditions in (48) are therefore satisfied,
implying that statement 1 holds.

Now assume that statement 1 holds, so that there exist operators
$Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$,
$Z\in\mathrm{Herm}(\mathcal{X})$, and $H\in\partial f(X)$ such that $YX=0$ and
$H-Y-\mathds{1}_{\mathcal{Y}}\otimes Z=0$. It follows that
$HX=(\mathds{1}_{\mathcal{Y}}\otimes Z)X$ and therefore

$\operatorname{Tr}_{\mathcal{Y}}(HX)=\operatorname{Tr}_{\mathcal{Y}}\bigl{(}(\mathds{1}_{\mathcal{Y}}\otimes
Z)X\bigr{)}=Z\operatorname{Tr}_{\mathcal{Y}}(X)=Z,$ (51)

which is a Hermitian operator. Moreover, one has
$H-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HX)=Y$,
which is positive semidefinite by assumption. The conditions in (46) are
therefore satisfied, which implies that statement 2 holds, completing the
proof. ∎

To make use of Theorem 3, one requires that the relative interior of the
domain of the objective function contains at least one point in the relative
interior of the set of channels. This requirement is precisely Slater’s
condition for this optimization problem, which guarantees strong duality with
the corresponding dual problem (34). This regularity condition is
automatically satisfied if $f$ is continuous at at least one point in the set
of Choi representations of channels. In particular, if $f$ is differentiable
at $X$ then one may take the operator $H$ in Theorem 3 to be $H=\nabla f(X)$.

######  Corollary 4.

Let
$f:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
be a convex function, let $X\in J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$ be the
Choi representation of a channel, and assume that $f$ is differentiable at
$X$. The operator $X$ is an optimal solution to the optimization problem (43)
if and only if

$\operatorname{Tr}_{\mathcal{Y}}(\nabla
f(X)X)\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad\nabla
f(X)\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(\nabla
f(X)X).$ (52)

###### Proof.

As $f$ is differentiable at $X$, one has that $\partial f(X)=\\{\nabla
f(X)\\}$. Furthermore, $f$ must be finite in some neighborhood around $X$ and
thus
$\operatorname{relint}(\operatorname{dom}(f))\cap\operatorname{relint}(J(\mathrm{C}(\mathcal{X},\mathcal{Y})))\neq\emptyset$
must hold. The result now follows from Theorem 3. ∎

We remark that the optimality conditions represented by this corollary appear
to be a special feature of problems involving an optimization over channels.
In essence, in differentiable convex quantum channel optimization problems,
optimal dual problem solutions $Z=\operatorname{Tr}_{\mathcal{Y}}(\nabla
f(X)X)$ and $Y=\nabla
f(X)-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(\nabla
f(X)X)$ are uniquely determined by optimal primal solutions $X$.

It is natural to ask if there is an approximate version of the implication
that statement 2 implies statement 1 in Theorem 3. That is, if the
requirements (46) hold approximately for some $H\in\partial f(X)$, then is $X$
necessarily close to being optimal? The following theorem demonstrates that
this is indeed the case, which allows for bounds to be placed on the optimal
value of such optimization problems in the case the conditions in (46) cannot
be verified exactly.

######  Theorem 5.

Let
$f:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
be a function, let $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ be a channel
such that $J(\Phi)\in\operatorname{dom}(f)$, and let $H\in\partial f(X)$. It
is the case that

$f(J(\Phi))\leq\inf_{\Psi\in\mathrm{C}(\mathcal{X},\mathcal{Y})}f(J(\Psi))+\varepsilon\,\mathrm{dim}(\mathcal{X}),$
(53)

where

$\varepsilon=\inf_{P\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})}\bigl{\lVert}\mspace{1.0mu}H-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))-P\mspace{1.0mu}\bigr{\rVert}_{\infty}.$
(54)

###### Proof.

Because the spectral norm is a continuous function, a compactness argument
implies that there must exist a positive semidefinite operator
$P\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$ for which

$\varepsilon=\bigl{\lVert}\mspace{1.0mu}H-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))-P\mspace{1.0mu}\bigr{\rVert}_{\infty}.$
(55)

We will let such a $P$ be fixed for the remainder of the proof.

Define a Hermitian operator

$Z=\frac{1}{2}\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))+\frac{1}{2}\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))^{\ast}-\varepsilon\,\mathds{1}_{\mathcal{X}},$
(56)

and define

$Y=H-\mathds{1}_{\mathcal{Y}}\otimes Z.$ (57)

Also define

$A=H-\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))-P,$
(58)

so that $\lVert\mspace{1.0mu}A\mspace{1.0mu}\rVert_{\infty}=\varepsilon$, and
therefore

$\biggl{\lVert}\mspace{1.0mu}\frac{1}{2}A+\frac{1}{2}A^{\ast}\mspace{1.0mu}\biggr{\rVert}_{\infty}\leq\varepsilon.$
(59)

It is the case that

$Y=P+\frac{A+A^{\ast}}{2}+\varepsilon\,\mathds{1}_{\mathcal{Y}}\otimes\mathds{1}_{\mathcal{X}}\geq
P,$ (60)

and therefore $Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{X})$.

Now consider the Lagrange dual problem (34) associated with the minimization
of $f$ over all channels in $\mathrm{C}(\mathcal{X},\mathcal{Y})$. As $Z$ is
Hermitian and $Y$ is positive semidefinite, $(Y,Z)$ is a dual feasible
solution to this dual problem. As $H\in\partial f(J(\Phi))$, one has that

$f(X)-\langle H,X\rangle\geq f(J(\Phi))-\langle H,J(\Phi)\rangle$ (61)

for every $X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$. Therefore, when
the dual objective function $g$ is evaluated at $(Y,Z)$, we find that

$\displaystyle g(Y,Z)$
$\displaystyle=\operatorname{Tr}(Z)+\inf_{X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})}\bigl{(}f(X)-\bigl{\langle}X,Y+\mathds{1}_{\mathcal{Y}}\otimes
Z\bigr{\rangle}\bigr{)}$ (62) $\displaystyle=\langle
H,J(\Phi)\rangle-\varepsilon\,\mathrm{dim}(\mathcal{X})+\inf_{X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})}\bigl{(}f(X)-\langle
H,X\rangle\bigr{)}$ $\displaystyle\geq
f(J(\Phi))-\varepsilon\,\mathrm{dim}(\mathcal{X}).$

The theorem follows by weak duality. ∎

We note that the previous theorem does not require the function $f$ to be
convex, or for the associated optimization problem to satisfy the conditions
of Slater’s theorem. However, having knowledge of the subdifferential
$\partial f(X)$ at an operator $X$ requires knowledge of the global behavior
of the function $f$ when $f$ is not convex, so the usefulness of Theorem 5 may
be limited when $f$ is not convex.

##  4 Applications

In this section we apply the optimality conditions given by Theorem 3 to a few
categories of examples, including the simple case of channel optimization
problems having linear objective functions and three variants of problems
involving quantum state transformations. In these examples we will make use of
various facts concerning differentiation for functions mapping between spaces
of Hermitian operators; a short discussion of this topic, along with a lemma
that is needed for one of the examples, can be found in the section following
this one.

###  4.1 Linear objective functions

We will begin by considering the simple case in which the objective function
$f$ in Theorem 3 is linear. In this situation, the optimization problem (43)
may be rewritten as

minimize $\displaystyle\langle H,X\rangle$ (63) subject to $\displaystyle X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y}))$

for some choice of a Hermitian operator
$H\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$. The subdifferential of the
function $f(X)=\langle H,X\rangle$ is given by $\partial f(X)=\\{H\\}$ for all
$X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$. By Theorem 3 it follows
that the Choi operator $X=J(\Phi)$ of a channel
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is optimal for the problem (63)
if and only if

$\operatorname{Tr}_{\mathcal{Y}}(HX)\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HX).$ (64)

We observe that this optimality criterion can alternatively be obtained
through semidefinite programming duality and complementary slackness. (See,
for instance, Exercise 3.5 of [Wat18], observing that the inequality is
reversed in that exercise because the optimization problem is expressed as a
maximization rather than a minimization.)

The problem of _minimum error state discrimination_ , which was mentioned in
the introduction, is a special case in which the function $f$ in the
optimization problem (43) is linear. Consider an ensemble of states, which
represents the random selection of one of a finite number of quantum states
according to a given probability distribution. Formally speaking, an ensemble
is described by a collection
$\\{\rho_{1},\ldots,\rho_{n}\\}\subseteq\mathrm{D}(\mathcal{X})$ of density
operators together with a probability vector $p=(p_{1},\ldots,p_{n})$. The
problem of minimum-error state discrimination seeks a measurement on the
system represented by the space $\mathcal{X}$ that identifies, with the
minimum possible probability of error, a state chosen randomly according to
this ensemble.

For a given choice of a measurement, represented by operators
$\\{P_{1},\ldots,P_{n}\\}\subseteq\mathrm{Pos}(\mathcal{X})$, the error
probability incurred by this measurement can be expressed as

$\sum_{k=1}^{n}\bigl{\langle}\rho-p_{k}\rho_{k},P_{k}\bigr{\rangle},$ (65)

where

$\rho=\sum_{k=1}^{n}p_{k}\rho_{k}$ (66)

is the average state of the ensemble. A minimization of the error probability
(65) over all measurements $\\{P_{1},\ldots,P_{n}\\}$ can be represented as an
optimization of the form (63) by letting $Y=\mathbb{C}^{n}$ and setting

$H=\sum_{k=1}^{n}E_{k,k}\otimes(\rho-
p_{k}\rho_{k})^{{\scriptscriptstyle\mathsf{T}}}.$ (67)

The measurement described by $\\{P_{1},\ldots,P_{n}\\}$, which may
alternatively be represented by a quantum-to-classical channel whose Choi
operator is

$X=\sum_{k=1}^{n}E_{k,k}\otimes P_{k}^{{\scriptscriptstyle\mathsf{T}}},$ (68)

is optimal for the minimization of the error probability (65) if and only of
the conditions (64) hold. These conditions can be simplified by first
calculating that

$\operatorname{Tr}_{\mathcal{Y}}(HX)=\sum_{k=1}^{n}(\rho-
p_{k}\rho_{k})^{{\scriptscriptstyle\mathsf{T}}}P_{k}^{{\scriptscriptstyle\mathsf{T}}}=\rho^{{\scriptscriptstyle\mathsf{T}}}-\sum_{k=1}^{n}p_{k}\rho_{k}^{{\scriptscriptstyle\mathsf{T}}}P_{k}^{{\scriptscriptstyle\mathsf{T}}},$
(69)

then observing that this operator is Hermitian if and only if the operator

$\sum_{k=1}^{n}p_{k}P_{k}\rho_{k}$ (70)

is Hermitian, and finally noting that the inequality
$H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HX)$ is
equivalent to

$\sum_{j=1}^{n}E_{j,j}\otimes
p_{j}\rho_{j}\leq\mathds{1}_{\mathcal{Y}}\otimes\sum_{k=1}^{n}p_{k}P_{k}\rho_{k},$
(71)

which may alternatively be expressed as

$\sum_{k=1}^{n}p_{k}P_{k}\rho_{k}\geq p_{j}\rho_{j}\quad\text{(for all
$j=1,\ldots,n$)}.$ (72)

The conditions that the operator (70) is Hermitian and satisfies the
inequalities (72) comprise the Holevo–Yuen–Kennedy–Lax conditions for
measurement optimality [Hol73b, Hol73a, YKL70, YKL75].

###  4.2 State transformation

The second category of channel optimization problems we consider involves
channels that transform one state into another in such a way that the distance
to a target state is minimized. One may consider any number of specific
measures of distance in such a problem; we will analyze measures based on the
fidelity, trace distance, and quantum relative entropy. For each of these
measures, we will consider the situation in which two bipartite states,
$\rho\in\mathrm{D}(\mathcal{X}\otimes\mathcal{Z})$ and
$\sigma\in\mathrm{D}(\mathcal{Y}\otimes\mathcal{Z})$, for complex Euclidean
spaces $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, are given. The
optimization problem to be considered is to minimize the distance (or maximize
the similarity) between the states
$\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)$ and
$\sigma$, with respect to the measure under consideration, over all possible
channels $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$.

Optimization problems of this sort were suggested in [SW15] (see Remark 6) in
the special case that $\mathcal{Y}=\mathcal{X}\otimes\mathcal{W}$ and
$\rho=\operatorname{Tr}_{\mathcal{W}}(\sigma)$, so that the aim of the channel
$\Phi$ is to “recover” the original state $\sigma$ from the portion of it
represented by $\rho$. The resulting quantities are known as the _fidelity of
recovery_ , _relative entropy of recovery_ , and so on. These and related
measures have been studied in a number of recent papers (such as [FR15,
BHOS15, CHM+16, BT16, BFT17]), and the more general situation in which $\rho$
and $\sigma$ are arbitrary states arises naturally in this study. (For
example, [BT16] uses the term _generalized fidelity of recover_ in the
situation mentioned above in the case of the fidelity.)

When analyzing optimality conditions for these problems, it will be helpful to
refer to the _evaluation map_ corresponding to the operator
$\rho\in\mathrm{D}(\mathcal{X}\otimes\mathcal{Z})$. This is the uniquely
determined completely positive map
$\Psi_{\rho}\in\mathrm{CP}(\mathcal{X},\mathcal{Z})$ that satisfies

$\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}\bigr{)}(J(\Phi))=\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)$
(73)

for every complex Euclidean space $\mathcal{Y}$ and every channel
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$. The relationship between the map
$\Psi_{\rho}$ and the state $\rho$ is closely related to the Choi
representation of maps: assuming for the moment that
$\mathcal{X}=\mathbb{C}^{n}$, one has that

$\rho=\sum_{j,k=1}^{n}E_{j,k}\otimes\Psi_{\rho}(E_{j,k}).$ (74)

That is, up to swapping the tensor factors corresponding to $\mathcal{X}$ and
$\mathcal{Z}$, the state $\rho$ is the Choi operator of the map $\Psi_{\rho}$.

#### Objective functions based on the fidelity

For positive semidefinite operators $P,Q\in\mathrm{Pos}(\mathcal{X})$, one
defines the fidelity between $P$ and $Q$ as

$\operatorname{F}(P,Q)=\Bigl{\lVert}\mspace{1.0mu}\sqrt{P}\sqrt{Q}\mspace{1.0mu}\Bigr{\rVert}_{1}=\operatorname{Tr}\sqrt{\sqrt{P}Q\sqrt{P}}.$
(75)

The first variant of the optimal state transformation problem we will consider
is as follows:

minimize
$\displaystyle-\operatorname{F}\bigl{(}\sigma,\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\bigr{)}$
(76) subject to $\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}).$

The fidelity function is jointly concave, and therefore is concave in each of
its arguments, from which it follows that this problem is a convex
optimization problem. It is possible to express this optimization problem as a
semidefinite program, as is demonstrated in [BFT17].

The following theorem establishes optimality conditions for a channel
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ in the optimization problem (76)
under the assumption that the operator $\operatorname{Tr}_{\mathcal{X}}(\rho)$
is positive definite. We note that the theorem statement does not actually
require $\rho$ and $\sigma$ to have unit trace—they can be arbitrary positive
semidefinite operators, but we nevertheless use the letters $\rho$ and
$\sigma$ to make the connection to the optimization problem (76) clear.

######  Theorem 6.

Let $\rho\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Z})$ and
$\sigma\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{Z})$ be positive
semidefinite operators, for complex Euclidean spaces $\mathcal{X}$,
$\mathcal{Y}$, and $\mathcal{Z}$, and assume that
$\operatorname{Tr}_{\mathcal{X}}(\rho)$ is a positive definite operator. A
channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is optimal for the
optimization problem (76) if and only if the following two conditions are met:

  1. 1.

$\operatorname{im}(\sigma)\subseteq\operatorname{im}\bigl{(}\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\bigr{)}$.

  2. 2.

The operator

$H=-\frac{1}{2}(\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast})\Bigl{(}\sqrt{\sigma}\mspace{1.0mu}\bigl{(}\sqrt{\sigma}\,(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\sqrt{\sigma}\bigr{)}^{-\frac{1}{2}}\sqrt{\sigma}\Bigr{)}$
(77)

satisfies

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)).$
(78)

(As per the convention mentioned in Section 2, the inverse in (77) refers to
the Moore–Penrose pseudo-inverse in case $\sigma$ does not have full rank.)

######  Remark 7.

The theorem assumes that $\operatorname{Tr}_{\mathcal{X}}(\rho)$ has full
rank, as this assumption allows for a cleaner theorem statement. We note in
particular that this assumption is equivalent to the condition that
$\operatorname{im}(\Psi_{\rho})=\mathrm{Herm}(\mathcal{Z})$ when $\Psi_{\rho}$
is regarded as a linear map from $\mathrm{Herm}(\mathcal{X})$ to
$\mathrm{Herm}(\mathcal{Z})$. It is, however, straightforward to apply the
theorem to a situation in which $\operatorname{Tr}_{\mathcal{X}}(\rho)$ does
not have full rank. Specifically, for an arbitrary choice of $\rho$ and
$\sigma$, one may take $B\in\mathrm{L}(\mathcal{V},\mathcal{Z})$ to be an
isometry for which $BB^{\ast}$ is the projection onto the image of
$\operatorname{Tr}_{\mathcal{X}}(\rho)$, and then observe that by replacing
$\rho$ and $\sigma$ with $(\mathds{1}_{\mathcal{X}}\otimes
B^{\ast})\rho(\mathds{1}_{\mathcal{X}}\otimes B)$ and
$(\mathds{1}_{\mathcal{Y}}\otimes
B^{\ast})\sigma(\mathds{1}_{\mathcal{Y}}\otimes B)$, respectively, an
equivalent problem is obtained that satisfies the assumptions of the theorem.

###### Proof of Theorem 6.

Let $r=\operatorname{rank}(\sigma)$, let $\mathcal{W}=\mathbb{C}^{r}$, and let
$A\in\mathrm{L}(\mathcal{W},\mathcal{Y}\otimes\mathcal{Z})$ be any isometry
for which $AA^{\ast}=\Pi_{\operatorname{im}(\sigma)}$ (the projection onto the
image of $\sigma$). Define a function

$g:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})\rightarrow\mathbb{R}\cup\\{\infty\\}$
(79)

as

$g(Y)=\begin{cases}-\operatorname{F}(A^{\ast}\sigma
A,A^{\ast}YA)&\text{if}\;A^{\ast}YA\in\mathrm{Pos}(\mathcal{W})\\\
\infty&\text{otherwise},\end{cases}$ (80)

for all $Y\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})$, and observe that
$g(Y)=-\operatorname{F}(\sigma,Y)$ for every
$Y\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{Z})$.

For a given operator $Y\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})$
satisfying $A^{\ast}YA\in\mathrm{Pos}(\mathcal{W})$, there are two cases for
the subdifferential $\partial g(Y)$.

  * Case 1: $A^{\ast}YA$ is positive definite. In this case $g$ is differentiable at $Y$, and

$\nabla
g(Y)=-\frac{1}{2}\sqrt{\sigma}\Bigl{(}\sqrt{\sigma}\,Y\sqrt{\sigma}\Bigr{)}^{-\frac{1}{2}}\sqrt{\sigma},$
(81)

which follows from Lemma 11, stated and proved in the section following this
one. It therefore follows that

$\partial
g(Y)=\biggl{\\{}-\frac{1}{2}\sqrt{\sigma}\Bigl{(}\sqrt{\sigma}\,Y\sqrt{\sigma}\Bigr{)}^{-\frac{1}{2}}\sqrt{\sigma}\biggr{\\}}.$
(82)

  * Case 2: $A^{\ast}YA$ is not positive definite. In this case, $\partial g(Y)=\varnothing$, which also follows from Lemma 11.

Next, define

$\Lambda=\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho},$ (83)

and observe that

$(g\circ\Lambda)(J(\Phi))=-\operatorname{F}\bigl{(}\sigma,\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\bigr{)}$
(84)

for every channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$. When $\Lambda$
is regarded as a linear map from
$\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$ to
$\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})$, one has

$\operatorname{im}(\Lambda)=\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})$ (85)

by virtue of the assumption that $\operatorname{Tr}_{\mathcal{X}}(\rho)$ is
positive definite. It follows that

$\operatorname{im}(\Lambda)\cap\operatorname{relint}(\operatorname{dom}(g))=\mathrm{Pd}(\mathcal{Y}\otimes\mathcal{Z})\not=\varnothing,$
(86)

and therefore, by Theorem 1,

$\partial(g\circ\Lambda)(X)=\Lambda^{\ast}(\partial g(\Lambda(X)))$ (87)

for every $X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$.

The theorem now follows from Theorem 3. In greater detail, if $\Phi$ is
optimal for the problem (76), then it must hold that there exists an operator
$H\in\partial(g\circ\Lambda)(J(\Phi))$ such that

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)).$
(88)

Case 1 described above must therefore hold when $Y=\Lambda(J(\Phi))$, for
otherwise the subdifferential $\partial(g\circ\Lambda)(J(\Phi))$ would be
empty. It follows that

$H=-\frac{1}{2}(\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast})\Bigl{(}\sqrt{\sigma}\mspace{1.0mu}\bigl{(}\sqrt{\sigma}\,(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\sqrt{\sigma}\bigr{)}^{-\frac{1}{2}}\sqrt{\sigma}\Bigr{)},$
(89)

from which the second condition in the statement of the theorem follows.

Conversely, if the two conditions in the statement of the theorem hold, then
it follows that $H\in\partial(g\circ\Lambda)(J(\Phi))$, and moreover

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)),$
(90)

and one concludes by Theorem 3 that $\Phi$ is optimal for the optimization
problem (76). ∎

An interesting special case of the optimization problem (76) is when the
states $\rho$ and $\sigma$ take the form

$\rho=\sum_{k=1}^{n}p_{k}\rho_{k}\otimes
E_{k,k}\quad\text{and}\quad\sigma=\sum_{k=1}^{n}p_{k}\sigma_{k}\otimes
E_{k,k}$ (91)

for a probability vector $(p_{1},\ldots,p_{n})$ and collections of states
$\\{\rho_{1},\ldots,\rho_{n}\\}\subseteq\mathrm{D}(\mathcal{X})$ and
$\\{\sigma_{1},\ldots,\sigma_{n}\\}\subseteq\mathrm{D}(\mathcal{Y})$. The
objective function simplifies in this case to

$-\sum_{k=1}^{n}p_{k}\operatorname{F}(\sigma_{k},\Phi(\rho_{k})),$ (92)

and therefore the optimization concerns the average fidelity with which a
channel $\Phi$ maps each $\rho_{k}$ to $\sigma_{k}$. The operator

$H=-\frac{1}{2}\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast}\bigr{)}\Bigl{(}\sqrt{\sigma}\Bigl{(}\sqrt{\sigma}\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\sqrt{\sigma}\Bigr{)}^{-\frac{1}{2}}\sqrt{\sigma}\Bigr{)}$
(93)

simplifies in this case to

$H=-\frac{1}{2}\sum_{k=1}^{n}p_{k}\sqrt{\sigma_{k}}\Bigl{(}\sqrt{\sigma_{k}}\mspace{1.0mu}\Phi(\rho_{k})\sqrt{\sigma_{k}}\Bigr{)}^{-\frac{1}{2}}\sqrt{\sigma_{k}}\otimes\rho_{k}^{{\scriptscriptstyle\mathsf{T}}}.$
(94)

An analogous criterion for optimality can also be found when the figure of
merit is the square of the fidelity, which is also jointly concave. The
corresponding optimization problem is the following:

minimize
$\displaystyle-\sum_{k=1}^{m}p_{k}\operatorname{F}\bigl{(}\sigma_{k},\Phi(\rho_{k})\bigr{)}^{2}$
(95) subject to $\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}).$

By differentiating the objective function in (95), one finds that a channel
$\Phi$ is optimal precisely when the same conditions in Theorem 6 are met, but
with $H\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$ given by

$H=-\sum_{k=1}^{n}p_{k}\operatorname{F}\bigl{(}\sigma_{k},\Phi(\rho_{k})\bigr{)}\sqrt{\sigma}\Bigl{(}\sqrt{\sigma}\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\sqrt{\sigma}\Bigr{)}^{-\frac{1}{2}}\sqrt{\sigma}\otimes\rho_{k}^{{\scriptscriptstyle\mathsf{T}}}.$
(96)

#### Objective functions based on trace distance

Next we consider an optimization problem that is analogous to (76), but based
on the trace distance rather than the fidelity:

minimize
$\displaystyle\bigl{\lVert}\mspace{1.0mu}\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\mspace{1.0mu}\bigr{\rVert}_{1}$
(97) subject to $\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}).$

The objective function of this optimization problem is convex (with respect to
$\Phi$), so we may use Theorem 3 to obtain optimality conditions for this
problem. We note that, similar to its variant based on the fidelity described
above, this optimization problem can be represented as a semidefinite program.

For this optimization problem, the optimality conditions we obtain may not be
efficiently checkable. We do, however, obtain an efficiently checkable
condition that is sufficient for optimality, and we conjecture that this
condition is also a necessary for optimality.

######  Theorem 8.

Let $\rho\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Z})$ and
$\sigma\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{Z})$ be positive
semidefinite operators, for complex Euclidean spaces $\mathcal{X}$,
$\mathcal{Y}$, and $\mathcal{Z}$, and let
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ be a channel. The channel $\Phi$
is optimal for the state transformation problem in (97) if and only if there
exists an operator $Y\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{Z})$ with
$\lVert\mspace{1.0mu}Y\mspace{1.0mu}\rVert_{\infty}=1$ such that the following
conditions are satisfied:

  1. 1.

It is the case that
$\bigl{\lVert}\mspace{1.0mu}\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\mspace{1.0mu}\bigr{\rVert}_{1}=\bigl{\langle}Y,\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{\rangle}$.

  2. 2.

The operator

$H=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast}\bigr{)}(Y)$
(98)

satisfies

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}\bigl{(}HJ(\Phi)\bigr{)}.$
(99)

The proof of Theorem 8 follows directly from Theorem 3 and applying the rules
of subdifferentiation presented in Section 2. Indeed, operators of the form in
(98) are precisely the elements of the subdifferential of the objective
function in (97) at $J(\Phi)$. As a generalization, one can replace the trace
norm $\lVert\cdot\rVert_{1}$ in the statement of Theorem 8 with any other norm
on operators, and replace $\lVert\cdot\rVert_{\infty}$ with the corresponding
dual norm.

In the event that the operator
$\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)$ arising in
Theorem 8 has no zero eigenvalues, there is a unique choice of $Y$ for which
the first condition the theorem holds. Specifically, if

$\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)=\sum_{k=1}^{m}\lambda_{k}\Pi_{k}$
(100)

is a spectral decomposition where each $\lambda_{k}$ is nonzero, then the
unique operator $Y$ satisfying condition 1 in Theorem 8 is given by

$Y=\sum_{k=1}^{m}\operatorname{sign}(\lambda_{k})\,\Pi_{k}.$ (101)

In this case it is sufficient for the second condition to be checked for this
unique choice of $Y$, yielding an efficiently checkable optimality criterion.
However, if it is the case that
$\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)$ has one or
more zero eigenvalues, then the first condition holds for a continuum of
choices of $Y$, and from Theorem 8 we conclude only that the optimality of
$\Phi$ is equivalent to the existence of at least one such choice of $Y$ for
which the second statement in the theorem hold.

It is reasonable, though, to view the operator $Y$ defined by (101), where now
it is to be understood that $\operatorname{sign}(0)=0$, as a natural selection
of an operator through which optimality may be verified. We conjecture, based
on numerical evidence, that this choice yields an efficiently checkable
necessary and sufficient optimality condition.

######  Conjecture 9.

Let $\rho\in\mathrm{D}(\mathcal{X}\otimes\mathcal{Z})$ and
$\sigma\in\mathrm{D}(\mathcal{Y}\otimes\mathcal{Z})$ be density operators, for
complex Euclidean spaces $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, and
let $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ be a channel. Let

$\sigma-(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)=\sum_{k=1}^{m}\lambda_{k}\Pi_{k}$
(102)

be a spectral decomposition, and define

$Y=\sum_{k=1}^{m}\operatorname{sign}(\lambda_{k})\,\Pi_{k},$ (103)

where $\operatorname{sign}(\alpha)=1$ and $\operatorname{sign}(-\alpha)=-1$
for all $\alpha>0$ and $\operatorname{sign}(0)=0$. The channel $\Phi$ is
optimal for the state transformation problem in (97) if and only if the
operator

$H=\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast}\bigr{)}(Y)$
(104)

satisfies

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}\bigl{(}HJ(\Phi)\bigr{)}.$
(105)

#### Objective functions based on relative entropy

Finally, we consider a variant of the optimal state transformation problem
based on the quantum relative entropy. For positive semidefinite operators
$P,Q,\in\mathrm{Pos}(\mathcal{X})$, the quantum relative entropy of $P$ with
respect to $Q$ is defined as

$\mathrm{D}(P\lVert
Q)=\begin{cases}\operatorname{Tr}(P\log(P))-\operatorname{Tr}(P\log(Q))&\text{if}\;\operatorname{im}(P)\subseteq\operatorname{im}(Q)\\\
\infty&\text{otherwise}.\end{cases}$

The specific variant of the problem to be considered is

minimize
$\displaystyle\mathrm{D}\bigl{(}\sigma\mspace{1.0mu}\big{\|}\mspace{1.0mu}(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{)}$
(106) subject to $\displaystyle\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y}).$

The relative entropy is jointly convex, which implies that it is convex in its
second argument, and therefore the problem above is a convex optimization
problem. Because the relative entropy can be approximated through the use of
semidefinite programming [FF18, FSP18], it is possible to efficiently
approximate the optimization problem (106) on a computer.

######  Theorem 10.

Let $\rho\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Z})$ and
$\sigma\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{Z})$ be positive
semidefinite operators, for complex Euclidean spaces $\mathcal{X}$,
$\mathcal{Y}$, and $\mathcal{Z}$, and assume that
$\operatorname{Tr}_{\mathcal{X}}(\rho)$ is a positive definite operator. A
channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is optimal for the
optimization problem (106) if and only if the following two conditions are
met:

  1. 1.

$\operatorname{im}(\sigma)\subseteq\operatorname{im}\bigl{(}(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{)}$.

  2. 2.

The operator

$H=-(\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast})\bigl{(}D\log\bigl{(}\Pi\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\Pi\bigr{)}(\sigma)\bigr{)}$
(107)

satisfies

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)).$
(108)

Here, $\Pi$ denotes the projection onto the image of $\sigma$ and $D\log(P)$
denotes the differential operator of the logarithm function at the operator
$P$ (as described in (156) at the end of Section 5).

###### Proof.

If the first condition does not hold for a given channel $\Phi$, then the
objective function in (106) takes an infinite value. However, by the
assumption that $\operatorname{Tr}_{\mathcal{X}}(\rho)$ is positive definite,
one has that the channel

$\Omega(X)=\frac{\operatorname{Tr}(X)\mathds{1}_{\mathcal{Y}}}{\mathrm{dim}(\mathcal{Y})}$
(109)

yields a finite value for the same objective function, implying that $\Phi$ is
not optimal. If $\Phi$ is optimal, the first condition must therefore hold. It
remains to prove that if $\Phi$ satisfies the first condition, then $\Phi$ is
optimal if and only if the second condition holds.

Let $r$ be the rank of $\sigma$, let $\mathcal{W}=\mathbb{C}^{r}$, and let
$A\in\mathrm{L}(\mathcal{W},\mathcal{Y}\otimes\mathcal{Z})$ be an isometry
that satisfies $AA^{\ast}=\Pi_{\operatorname{im}(\sigma)}$. Define a linear
map
$\Xi:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{W})$
as

$\Xi(X)=A^{\ast}\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}\bigr{)}(X)A$
(110)

for all $X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$. For every channel
$\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ it is the case that

$\mathrm{D}\bigl{(}\sigma\mspace{1.0mu}\big{\|}\mspace{1.0mu}(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{)}=\mathrm{D}\bigl{(}A^{\ast}\sigma
A\mspace{1.0mu}\big{\|}\mspace{1.0mu}\Xi(J(\Phi))\bigr{)}.$ (111)

With this observation in mind, define a function
$f:\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
as

$f(X)=\begin{cases}\mathrm{D}\bigl{(}A^{\ast}\sigma
A\mspace{1.0mu}\big{\|}\mspace{1.0mu}\Xi(X)\bigr{)}&\text{if}\;\Xi(X)\in\mathrm{Pd}(\mathcal{W})\\\
\infty&\text{otherwise,}\end{cases}$ (112)

so that a given channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is
optimal for the problem (106) if and only if $J(\Phi)$ is optimal for the
problem

minimize $\displaystyle f(X)$ (113) subject to $\displaystyle X\in
J(\mathrm{C}(\mathcal{X},\mathcal{Y})).$

The function $f$ is differentiable at every operator
$X\in\mathrm{Herm}(\mathcal{Y}\otimes\mathcal{X})$ in its domain, with its
gradient being

$\begin{split}\nabla f(X)&=-\Xi^{\ast}\bigl{(}D\log(\Xi(X))(A^{\ast}\sigma
A)\bigr{)}\\\
&=-\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast}\bigr{)}\bigl{(}D\log\bigl{(}\Pi\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}\bigr{)}(X)\Pi\bigr{)}(\sigma)\bigr{)}.\end{split}$
(114)

As $f$ is differentiable at every point in its domain, it follows that

$\partial f(J(\Phi))=\\{\nabla f(J(\Phi))\\}$ (115)

for every $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ for which
$\operatorname{im}(\sigma)\subseteq\operatorname{im}\bigl{(}(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{)}$.
For a given channel $\Phi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ for which
$\operatorname{im}(\sigma)\subseteq\operatorname{im}\bigl{(}(\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})})(\rho)\bigr{)}$,
it therefore follows from Theorem 3 that $\Phi$ is optimal if and only if the
operator

$H=-\bigl{(}\mathds{1}_{\mathrm{L}(\mathcal{Y})}\otimes\Psi_{\rho}^{\ast}\bigr{)}\bigl{(}D\log\bigl{(}\Pi\bigl{(}\Phi\otimes\mathds{1}_{\mathrm{L}(\mathcal{Z})}\bigr{)}(\rho)\Pi\bigr{)}(\sigma)\bigr{)}$
(116)

satisfies

$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\mathrm{Herm}(\mathcal{X})\quad\text{and}\quad
H\geq\mathds{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)),$
(117)

which is the second condition in the statement of the theorem. ∎

##  5 Gradients and subdifferentials of functions on matrices

Results in the previous section have required the computation of gradients and
subdifferentials for various functions mapping Hermitian operators to the real
numbers. In this section we provide details on these computations.

#### Definitions and basic results

It is appropriate to begin with some basic definitions. Throughout this
discussion, $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are arbitrary
complex Euclidean spaces.

Suppose that

$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$ (118)

is a partial function, meaning that it may only be defined on some subset of
inputs $X\in\mathrm{Herm}(\mathcal{X})$. The function $f$ is _(Fréchet)
differentiable_ at $X\in\mathrm{Herm}(\mathcal{X})$ if it is continuous in a
neighborhood of $X$ and there exists a linear map

$\Phi:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$ (119)

for which the equation

$\lim_{Z\rightarrow
0}\frac{\bigl{\lVert}\mspace{1.0mu}f(X+Z)-f(X)-\Phi(Z)\mspace{1.0mu}\bigr{\rVert}}{\lVert\mspace{1.0mu}Z\mspace{1.0mu}\rVert}=0$
(120)

is satisfied. If there does exist such a map, it must be unique, and we denote
it by $Df(X)$. Whenever $f$ is differentiable at $X$, it must be the case that

$Df(X)(Z)=\frac{\textup{d}}{\textup{d}t}f(X+tZ)\>\Big{|}_{t=0}$ (121)

for all choices of $Z\in\mathrm{Herm}(\mathcal{X})$. In the special case that
$\mathcal{Y}=\mathbb{C}$, which is equivalent to $f$ taking the form

$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R},$ (122)

one has that

$Df(X)(Z)=\big{\langle}\nabla f(X),Z\big{\rangle}$ (123)

for all $Z\in\mathrm{Herm}(\mathcal{X})$, assuming $f$ is differentiable at
$X$.

The _chain rule_ for differentiation states that if

$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})\quad\text{and}\quad
g:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathrm{Herm}(\mathcal{Z}),$ (124)

$f$ is differentiable at $X$, and $g$ is differentiable at $Y=f(X)$, then

$D(g\circ f)(X)(Z)=Dg(f(X))(Df(X)(Z))$ (125)

for all $Z\in\mathrm{Herm}(\mathcal{X})$.

#### Affine linear functions

A function

$f:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$ (126)

is _affine linear_ if there exists a linear map

$\Phi:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$ (127)

and an operator $Y\in\mathrm{Herm}(\mathcal{Y})$ such that

$f(X)=\Phi(X)+Y$ (128)

for all $X\in\mathrm{Herm}(\mathcal{X})$. Every such function is
differentiable at every $X\in\mathrm{Herm}(\mathcal{X})$, with its derivative
given by

$Df(X)=\Phi.$ (129)

For an arbitrary function

$g:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathrm{Herm}(\mathcal{Z}),$ (130)

one therefore finds that

$D(g\circ f)(X)(Z)=Dg(f(X))(\Phi(Z)),$ (131)

provided that $g$ is differentiable at $f(X)$, and if $g$ takes the form

$g:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathbb{R},$ (132)

then it is the case that

$\nabla(g\circ f)(X)=\Phi^{\ast}(\nabla g(f(X))).$ (133)

#### Real-valued functions extended to Hermitian operators

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is a function, then it may be extended
to a function of the form

$g:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{X})$ (134)

in a standard way: for any choice of $X\in\mathrm{Herm}(\mathcal{X})$, one
considers the spectral decomposition

$X=\sum_{k=1}^{m}\lambda_{k}\Pi_{k}$ (135)

of $X$, then defines

$g(X)=\sum_{k=1}^{m}f(\lambda_{k})\Pi_{k}.$ (136)

(It is typical that this extended function is given the same name as the
original function on the real numbers, but for the sake of clarity we have
introduced a distinct name for the extended function.) Naturally, if $f$ is
defined only on a subset of $\mathbb{R}$, then $g$ is defined for all $X$
whose eigenvalues are contained in the domain of $f$.

The function $g$ is differentiable at every Hermitian operator whose
eigenvalues correspond to differentiable points of the function $f$. The
derivative of $g$ can be described explicitly by first defining a function

$h(\alpha,\beta)=\begin{cases}\frac{f(\alpha)-f(\beta)}{\alpha-\beta}&\text{if}\;\alpha\not=\beta\\\
f^{\prime}(\alpha)&\text{if}\;\alpha=\beta\end{cases}$ (137)

for every pair of points $(\alpha,\beta)$ for which $f$ is differentiable at
both $\alpha$ and $\beta$. (The function $h$ is sometimes called the _first
divided difference_ of $f$, although this terminology is sometimes limited to
the case that $\alpha\not=\beta$.) In terms of this function, the derivative
of $g$ at an operator $X$ having a spectral decomposition (135) is

$Dg(X)(Z)=\sum_{j,k=1}^{m}h(\lambda_{j},\lambda_{k})\Pi_{j}Z\Pi_{k}$ (138)

for every $Z\in\mathrm{Herm}(\mathcal{X})$.

#### Gradients of functions involving the fidelity

Let $\mathcal{Y}$ be a complex Euclidean space, and consider the function
$f:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathbb{R}\cup\\{\infty\\}$ defined as

$f(Y)=\begin{cases}-\operatorname{Tr}\sqrt{Y}&\text{if}\;Y\in\mathrm{Pos}(\mathcal{Y})\\\
\infty&\text{otherwise.}\end{cases}$ (139)

This function is differentiable at every positive definite operator
$Y\in\mathrm{Pd}(\mathcal{Y})$, with its gradient being

$\nabla f(Y)=-\frac{1}{2}Y^{-\frac{1}{2}}.$ (140)

One way to verify this expression is to first consider the function
$g(Y)=\sqrt{Y}$, defined for every positive semidefinite operator
$Y\in\mathrm{Pos}(\mathcal{Y})$, and to use the formula (138) to conclude that

$Dg(Y)(Z)=\sum_{j,k=1}^{m}\frac{\Pi_{j}Z\Pi_{k}}{\sqrt{\lambda_{j}}+\sqrt{\lambda_{k}}},$
(141)

provided that $Y$ is positive definite and has spectral decomposition

$Y=\sum_{k=1}^{m}\lambda_{k}\Pi_{k}.$ (142)

The equation (140) follows from the chain rule.

######  Lemma 11.

Let $\mathcal{X}$ be a complex Euclidean space, let
$P\in\mathrm{Pd}(\mathcal{X})$ be a positive definite operator, and define a
function $g:\mathrm{Herm}(\mathcal{X})\rightarrow\mathbb{R}\cup\\{\infty\\}$
as

$g(X)=\begin{cases}-\operatorname{F}(P,X)&\text{if}\;X\in\mathrm{Pos}(\mathcal{X})\\\
\infty&\text{otherwise}.\end{cases}$ (143)

For every $X\in\mathrm{Pd}(\mathcal{X})$, the function $g$ is differentiable
at $X$, and

$\nabla
g(X)=-\frac{1}{2}\sqrt{P}\Bigl{(}\sqrt{P}X\sqrt{P}\Bigr{)}^{-\frac{1}{2}}\sqrt{P}.$
(144)

For every operator $X\in\mathrm{Pos}(\mathcal{X})$ that is not positive
definite, it is the case that $\partial g(X)=\varnothing$.

###### Proof.

Define a linear map
$\Lambda:\mathrm{Herm}(\mathcal{X})\rightarrow\mathrm{Herm}(\mathcal{Y})$ as

$\Lambda(X)=\sqrt{P}X\sqrt{P}$ (145)

for all $X\in\mathrm{Herm}(\mathcal{X})$. It is the case that
$g=f\circ\Lambda$, where $f$ is as defined in (139) above. By the chain rule
for differentiation, one has

$\nabla g(X)=-\frac{1}{2}\Lambda^{\ast}(\nabla
f(\Lambda(X)))=-\frac{1}{2}\sqrt{P}\Bigl{(}\sqrt{P}X\sqrt{P}\Bigr{)}^{-\frac{1}{2}}\sqrt{P},$
(146)

provided that

$\sqrt{P}X\sqrt{P}\in\mathrm{Pd}(\mathcal{Y}),$ (147)

which is equivalent to $X\in\mathrm{Pd}(\mathcal{X})$.

Now suppose that $X\in\mathrm{Pos}(\mathcal{X})$ is not positive definite, and
let $\Delta$ be the projection onto the kernel of $\sqrt{P}X\sqrt{P}$, which
is nonzero by the assumption that $X$ is not positive definite. Consider the
operator

$Y=X+\lambda P^{-\frac{1}{2}}\Delta P^{-\frac{1}{2}}$ (148)

for an arbitrary choice of $\lambda>0$. It is the case that

$g(Y)-g(X)=\operatorname{Tr}\sqrt{\sqrt{P}X\sqrt{P}}-\operatorname{Tr}\sqrt{\sqrt{P}X\sqrt{P}+\lambda\Delta}=-\sqrt{\lambda}\operatorname{Tr}(\Delta).$
(149)

Thus, if there were to exist an element $Z\in\partial g(X)$, one would have

$g(Y)-g(X)\geq\langle Z,Y-X\rangle$ (150)

for all $Y\in\operatorname{dom}(f)$, including the operator (148) for every
$\lambda>0$. It would then follow that

$\lambda\Bigl{\langle}Z,P^{-\frac{1}{2}}\Delta
P^{-\frac{1}{2}}\Bigr{\rangle}=\langle Z,Y-X\rangle\leq
g(Y)-g(X)=-\sqrt{\lambda}\operatorname{Tr}(\Delta),$ (151)

or equivalently

$\frac{\Bigl{\langle}Z,P^{-\frac{1}{2}}\Delta
P^{-\frac{1}{2}}\Bigr{\rangle}}{\operatorname{Tr}(\Delta)}\leq-\frac{1}{\sqrt{\lambda}},$
(152)

for every $\lambda>0$, which is impossible given that the left-hand side is a
finite value independent of $\lambda$ and the right-hand side approaches
$-\infty$ as $\lambda$ approaches 0. ∎

#### Gradients of functions involving the quantum relative entropy

Let $\mathcal{Y}$ be a complex Euclidean space, let
$P\in\mathrm{Pd}(\mathcal{Y})$ be a positive definite operator, and consider
the function
$f:\mathrm{Herm}(\mathcal{Y})\rightarrow\mathbb{R}\cup\\{\infty\\}$ defined as

$f(Y)=\begin{cases}\mathrm{D}(P\|Y)&\text{if}\;Y\in\mathrm{Pd}(\mathcal{Y})\\\
\infty&\text{otherwise.}\end{cases}$ (153)

This function is differentiable at every positive definite operator
$Y\in\mathrm{Pd}(\mathcal{Y})$, with its gradient being

$\nabla f(Y)=-D\log(Y)(P),$ (154)

where $D\log(Y)$ is the derivative of the logarithm function at $Y$. If

$Y=\sum_{k=1}^{m}\lambda_{k}\Pi_{k}$ (155)

is the spectral decomposition of $Y$, then by means of the expression (138)
this function can be described explicitly as

$D\log(Y)(Z)=\sum_{k=1}^{m}\frac{1}{\lambda_{k}}\Pi_{k}Z\Pi_{k}+\sum_{j\not=k}\frac{\log(\lambda_{j})-\log(\lambda_{k})}{\lambda_{j}-\lambda_{k}}\Pi_{j}Z\Pi_{k}.$
(156)

## Acknowledgments

We thank Jamie Sikora for helpful comments and discussions and Mark Wilde for
valuable suggestions. This research was supported by Canada’s NSERC.

## References

  * [AFG+12] Scott Aaronson, Edward Farhi, David Gosset, Avinatan Hassidim, Jonathan Kelner, and Andrew Lutomirski.  Quantum money.  Communications of the ACM, 55(8):84–92, 2012.
  * [BC09] Stephen Barnett and Sarah Croke.  Quantum state discrimination.  Advances in Optics and Photonics, 1(2):238–278, 2009.
  * [BCvD05] Dave Bacon, Andrew Childs, and Wim van Dam.  From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups.  In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 469–478, 2005.
  * [BFT17] Mario Berta, Omar Fawzi, and Marco Tomamichel.  On variational expressions for quantum relative entropies.  Letters in Mathematical Physics, 107(12):2239–2265, 2017.
  * [BHOS15] Fernando Brandão, Aram Harrow, Jonathan Oppenheim, and Sergii Strelchuk.  Quantum conditional mutual information, reconstructed states, and state redistribution.  Physical Review Letters, 115(5):050501, 2015.
  * [BL06] Jonathan Borwein and Adrian Lewis.  Convex Analysis and Nonlinear Optimization: Theory and Examples.  Springer, 2006.
  * [BT16] Mario Berta and Marco Tomamichel.  The fidelity of recovery is multiplicative.  IEEE Transactions on Information Theory, 62(4):1758–1763, 2016\.
  * [BV04] Stephen Boyd and Lieven Vandenberghe.  Convex Optimization.  Cambridge University Press, 2004.
  * [CHM+16] Tom Cooney, Christoph Hirche, Ciara Morgan, Jonathan Olson, Kaushik Seshadreesan, John Watrous, and Mark Wilde.  Operational meaning of quantum measures of recovery.  Physical Review A, 94(2):022310, 2016.
  * [EMV03] Yonina Eldar, Alexandre Megretski, and George Verghese.  Designing optimal quantum detectors via semidefinite programming.  IEEE Transactions on Information Theory, 49(4):1007–1012, 2003\.
  * [FF18] Hamza Fawzi and Omar Fawzi.  Efficient optimization of the quantum relative entropy.  Journal of Physics A: Mathematical and Theoretical, 51(15):154003, 2018.
  * [FR15] Omar Fawzi and Renato Renner.  Quantum conditional mutual information and approximate Markov chains.  Communications in Mathematical Physics, 340(2):575–611, 2015.
  * [FSP18] Hamza Fawzi, James Saunderson, and Pablo Parrilo.  Semidefinite approximations of the matrix logarithm.  Foundations of Computational Mathematics, 2018.
  * [Hol73a] Alexander Holevo.  Statistical decision theory for quantum systems.  Journal of Multivariate Analysis, 3:337–394, 1973.
  * [Hol73b] Alexander Holevo.  Statistical problems in quantum physics.  In Proceedings of the Second Japan–USSR Symposium on Probability Theory, volume 330 of Lecture Notes in Mathematics, pages 104–119. Springer, 1973.
  * [Ip03] Lawrence Ip.  Shor’s algorithm is optimal.  Unpublished manuscript, available from http://www.lawrenceip.com/papers/hspsdp.pdf, 2003.
  * [JUW09] Rahul Jain, Sarvagya Upadhyay, and John Watrous.  Two-message quantum interactive proofs are in PSPACE.  In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pages 534–543, 2009.
  * [JvF02] Miroslav Ježek, Jaroslav Řeháček, and Jaromír Fiurášek.  Finding optimal strategies for minimum-error quantum-state discrimination.  Physical Review A, 65(6):060301, 2002.
  * [MN13] Boris Mordukhovich and Nguyen Mau Nam.  An Easy Path to Convex Analysis and Applications.  Morgan & Claypool Publishers, 2013.
  * [NC00] Michael Nielsen and Isaac Chuang.  Quantum Computation and Quantum Information.  Cambridge University Press, 2000.
  * [Roc70] R. Tyrell Rockafellar.  Convex Analysis.  Princeton University Press, 1970.
  * [SIGA05] Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin, and Antonio Acín.  Quantum cloning.  Reviews of Modern Physics, 77(4):1225–1256, 2005.
  * [Sut18] David Sutter.  Approximate Quantum Markov Chains, volume 28 of SpringerBriefs in Mathematical Physics.  Springer, 2018.
  * [SW15] Kaushik Seshadreesan and Mark Wilde.  Fidelity of recovery, squashed entanglement, and measurement recoverability.  Physical Review A, 92(4):042321, 2015.
  * [Wat18] John Watrous.  Theory of Quantum Information.  Cambridge University Press, 2018.
  * [Wil17] Mark Wilde.  Quantum Information Theory.  Cambridge University Press, second edition, 2017.
  * [YKL70] Horace Yuen, Robert Kennedy, and Melvin Lax.  On optimal quantum receivers for digital signal detection.  Proceedings of the IEEE, 58(10):1770–1773, 1970.
  * [YKL75] Horace Yuen, Robert Kennedy, and Melvin Lax.  Optimum testing of multiple hypotheses in quantum detection theory.  IEEE Transactions on Information Theory, 21(2):125–134, 1975.