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根据提供的文件信息，我们可以提取以下与信号处理相关的知识点： 1. 量子信号处理：文章提到了量子数据压缩（Quantum Data Compression），这是量子信号处理的一部分。量子信号处理涉及对量子信息源进行压缩，以减少在存储或传输量子信息时所需的量子比特（qubits）数量。在量子信息理论的早期突破之一中，Schumacher证明了无记忆量子信息源的冯·诺依曼熵（von Neumann entropy）是进行无损压缩的最佳速率。 2. 一次量子率失真编码（One-Shot Quantum Rate Distortion Coding）：文件中提到了一种框架，旨在确定压缩量子信息所需的最小量子比特数，这与压缩后解压缩时失真超过特定水平的概率有关。这涉及到量子率失真理论，这是一种确定压缩数据到特定失真水平所需的最小比特率的技术。 3. 平滑最大信息（Smooth Max-Information）：文档提到了一个量——平滑最大信息，该量之前已用于一次性量子反香农定理（One-Shot Quantum Reverse Shannon Theorem）。最大信息可能是描述量子通道容量或者量子编码方案性能的一个重要概念。 4. 纠缠辅助量子率失真编码（Entanglement-Assisted Quantum Rate-Distortion Coding）：文章提到了纠缠辅助编码，它可能利用量子纠缠来改善通信或数据压缩的性能。在量子信息处理中，纠缠是一种重要的资源，它允许在两个或多个量子系统之间建立一种特殊的关联。 5. 有限块长度特性（Finite Block Length Characterization）：文档强调了对纠缠辅助最小量子比特压缩大小进行紧致、有限块长度的特性化的重要性。在实际的信号处理中，了解有限数据集的特性对于设计实际可操作的算法至关重要。 6. 平均符号失真约束（Average Symbol-Wise Distortion Constraint）：这是关于数据压缩的另一个重要概念，涉及到对量子比特源的平均符号失真进行约束。在实际应用中，需要在保持信息内容质量的前提下，对压缩率和失真水平进行权衡。 7. 信息论指数术语：文件中提到了包括纠缠辅助、假设检验（Hypothesis Testing）、相对熵（Relative Entropy）、最大信息、最小和最大熵（Min-and Max-Entropy）、量子率失真等指数术语，这些都是信息论中的核心概念，并且在信号处理领域中占有重要地位。 8. 香农定理（Shannon Theorem）与量子香农定理（Quantum Shannon Theorem）：传统的香农定理讨论了经典通信中的数据压缩和信道容量问题，而量子香农定理则拓展到量子信息处理领域。这些理论对于理解信号处理的极限和可能性提供了基本框架。综合上述信息，我们可以看出文件涉及的信号处理知识点主要是量子信息理论及其在通信领域的应用。这些概念和方法在无线通信、量子计算、量子加密以及量子网络等领域中都有广泛的应用。量子信息处理作为IT领域一个非常前沿的分支，正在逐步解决传统经典信息技术所面临的许多难题，为信息的存储和传输提供了新的可能性。

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 12, DECEMBER 2013 8057

One-Shot Lossy Quantum Data Compression

Nilanjana Datta, Joseph M. Renes, Renato Renner, and Mark M. Wilde, Senior Member, IEEE

Abstract—We provide a framework for one-shot quantum rate

distortion coding, in which the goal is to determine the minimum

number of qubits required to compress quantum information as

a function of the probability that the distortion incurred upon

decompression exceeds some speciﬁed level. We obtain a one-shot

characterization of the minimum qubit c ompression size for an

entanglement-assisted quantum rate-distortion code in terms of

the smooth max-information, a quantity previously employed in

the one-shot quantum reverse Shannon theorem. Next, we show

how this characterization converges to the known expression for

the entanglement-assisted quantum rate distortion function for

asymptotically many copies of a memoryless q uantum information

source.Finally,wegiveatight,ﬁnite blocklength characterization

for the entanglement-assisted minimum qubit compression size

of a memoryless isotropic qubit source subject to an average

symbolwise distortion constraint.

Index Terms—Entanglement assistance, hypothesis testing, rel-

ative entropy, lossy quantum data compression, max-information,

min- and max-entropy, quantum rate distortion.

I. INTRODUCTION

HE reliable compression of data is essential for the

efﬁcient use of available storage or communication

resources. In one of the ﬁrst breakthroughs of quantum infor-

mation th eory, Schumacher [1] proved that the v on Neumann

entropy of a memoryless quantum information source is the op-

timal rate at which we can compress it. This data com pressio n

limit was evaluated under the requirement that the compres-

sion–decompression sch eme is asymptotically lossless,inthe

sense that the information em itted by the source is r ecovered

with arbitrarily good accuracy in the limit of asymptotically

many copies of the source.

Manuscript received April 19, 2013; accepted August 12, 2013. Date of

publication September 26, 2013; date of current version November 19, 2013.

J. M. Renes and R. Renner were supported in part by the Sw iss National

Science Foundation through the National Centre of Competence in Research

“Quantum Science and Technology” under Grant 200020-135048 and in part

by the European Research Council under Grant 258932. M. M. Wilde was

supported by the Centre de Recherches Mathématiques and b y the hospitality

of the Statistical Laboratory at th e University of Cambridge and the Pauli

Center for Theoretical Studies (ETH Zurich) during a research visit in January

and February of 2013, when the majority of this work was com pleted.

N. Datta is with the Statistical Laborator y, University of Cambridge, Cam-

bridge CB30WB, U.K. (e-mail: N.Datta@statslab.cam.ac.uk).

J. M. Renes and R. Renner are with the I nstitute for Theor etical Physics, ETH

Zurich, 8093 Zürich, S witzerland (e-mail: joerenes@gmail.c om; renner@phys .

ethz.ch).

M. M. Wilde was with the School of Compu ter Science, McGill University,

Montreal, QC H3A 2A7, Canada . He is now with the Hearne In stitute for Theo -

retical Physics, Department of Physics and Astronomy, and the Center for Com-

putation and Technology, Louisian a State Universit y, Baton Rouge, LA 70803

USA (e-mail: mwilde@ lsu.edu).

Communicated by A. Holevo, Associate Editor for Quantum Infor mation

Theory.

Dig

ital Object Identiﬁer 10.1109/TIT.2013.2283723

However, one could envisage scenarios in which som e im-

perfection in the recovered information would be tolerable or

even necessary. The characterization of the tradeoff betw een

an allowed distortion and the compression rate is the subject of

quantum rate distortion the or y. Its classical counterpart was de-

veloped by Shannon [2], and the tradeoff is g iven by a rate-dis-

tortion functio n ,whichisdeﬁned as the minimum rate of c om -

pression for a given distortion, w ith r e spect to a s ui tabl y d eﬁ ned

distortion measure. To our know le dge, there are at least two im-

portant reasons for developing the theory of lossy quantum data

compression:

1) O ne might need to com press a quantum information source

at a rate smaller than its von Neumann entropy. This is

necessary, for example, in the case where there is insuf-

ﬁcient storage available, or if one needs to transmit infor-

mation emitted by a source over a channel whose quantum

capacity is sm aller than the von Neumann entropy of the

source. The strong converse to Schumacher’s theorem im-

plies that there is no tradeoff possible betw een the rate

of compression and the error incurred in recovery in the

asymptotic limit (see [3, Th. I.1 9]) . T hat is, there cannot

be a “rate-error” tradeoff because if one compresses at a

rate below the von N eum a nn entropy, then the ﬁ delity be-

tween the initial and recovered state approaches zero ex-

ponentially in the number of cop ies of the source. In spite

of th is “no-go” theorem, the theory of quantum rate dis-

tortion shows that there can be a f undamental tradeoff b e-

tween rate and distor tion for a suitably deﬁned distortion

measure.

2) Allowing a ﬁnite distortion in the recovered data is es-

sential for some continu ous-variable quantum inform ation

sources (see [4] and references therein) for w hich the re-

quirement of arbitrarily good accuracy becomes meaning-

less.

That is, we would like to have a theory that char-

acterizes the compression of analog quantum information

into digital quantum information along with the distortion

incurred in doing so.

The ﬁrst paper to discuss rate distortion in the quantum realm

was by Barnum [5]. He introduced a deﬁnition of the quan tum

rate-distortion function as the lowest rate at which a sender can

compress a memoryless quantum sou rce u nder som e distortio n

constraint. Th e main result of his paper is a lower bound on the

quantum rate distorti on function in terms of a well-known en -

tropic quantity, n amel y, the coherent informa tio n. Even though

Barnum’s result was t he ﬁrst in quantum rate distortion theory,

it is unsatisfactory since the bound is ob vio usly loose—the c o-

herent information can be negative, whereas the quantum rate

An important exception here is the case of a bosonic thermal source, w hich

has a discrete representation in the orthonormal photon-number basis. Thus,

Schumacher compression of a bosonic thermal source is indeed possible, even

though its representation in the coherent-state basis is continuous.

8058 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 12, DECEMBER 2013

distortion function i s deﬁned operationally to be nonnegative.

Tighter, nonnegative lower b oun ds were found in later work, by

allowing for assisting resources such as entanglement assistance

[6] or a side classical channel [7].

Even though classical rate distortion theory has been an area

of active research, i ts quantum analog had received very little at-

tention, there being only a few results o n it since Barnum’s work

[8]–[10]. In the past few years, however, there has been a revival

of interest in quantum rate distortion theory, an d quite a f ew new

results have been o btained [6], [11], [7]. T hese later works found

various expressions for quantum rate distortion functions, both

in the absence and presence of auxiliary resources, which can

be exploited in the data co mp ressio n task.

In all prior work on quantum rate distortion th eory, t he

rate-distortion functions were evaluated in the li mi t of asymp-

totically many copies of a memoryless quantu m information

source. Since the data compression rates in those works were

achieved using block codes, this corresponds to the limit

,where denotes the length of the block code. These

results then give us eful bounds in an idealized setting, but they

are not part icularly helpful in cha racterizing t he rate -di s tor tion

tradeoff f or more realistic settings, such as the ﬁnite block-

length setting or one in which the source is not memoryless.

A more f und amen tal problem, of both theoretical and prac-

tical interest, is to ﬁnd bounds on rate distortion functions for a

given distortion

and an “excess-distortion” probability

. For example, consider the classical case. Let a source

be described by a random variable

taking values in a ﬁnite

alphabet

. We wou ld like t o ﬁnd the minimum number of bits

to which we can compress this source, such that the probability

of exceeding a distortion level

is no larger than some small

(1)

where

denotes the indicator functio n, is a distortion

measure, and

and are the respective encoder and decoder for

the scheme. We could then evaluate such a bound for a source

that is invoked a ﬁnite number of times. In the classical case,

in cer tain ap pli cations, relativ ely shor t blo cklengths are in f act

common, bo th due to delay and com plexity constraints, and we

would expect similar constraints to apply in the quantum case.

In this vein, Kostina and Verdú recently obtained bounds on

the minimum achievable rate of classical data compression as

a function of blocklength

and e xcess distortion probability

[12].

II. O

VERVIEW OF RESULTS

In this paper, we co ntr ibute the follo win g r e sult s:

1) We ﬁ rst establish a fram ework for one-shot quan tum rate-

distortion theory. T his includes some basic deﬁnitions and

the notion of a n excess-distortion projector,whichisde-

rived from a distortion observable.

The deﬁnitions apply

A distortio n observable is a gen eralization of the d istortion measu re used in

classical rate-distortion theory.

in settings wher e e ither there is no assisting resource or en-

tanglement assistance is available.

2) We obtain two lower (converse) bound s (Propositions 7

and 9) on the minimum qubit com pressio n size, which is

the minimum number of qubits needed to comp ress the

source state such that a receiver can recover it up to some

speciﬁed excess-distortion probability. T he bounds apply

in the entanglement-assisted setting, and as such, they

apply in the unassisted case as well. These bounds are

given in terms of quantities deﬁned in the smooth-entropy

framework of one-shot information theory (see [13 ]– [16],

and references therein) and are proved by employing ideas

from qu antu m hypothesis testing (see, e.g., [15] and refer-

ences therein). One of our converse boun ds (Proposition

9) can in fact be viewed as a generalization of a converse

bound proved in the classical case by Kostina and Verdú

[12].

3) Achievability boun ds in Sections VI-B and VI-C are

proved using a one-shot version of the quantum reverse

Shannon (channel simu lation) theorem [17]. A chann el

simulation theorem provides bounds on the minim um

number of qubits that a sender (say, Alice) needs to send

to a receiver (say, Bob) in order to simulate a quantum

channeluptoaﬁnite accuracy. A channel sim ulation

strategy then leads to bounds on the one-shot entan-

glement-assisted quantum rate-distortion function by

choosing the simulated channel to depend on the distor-

tion measure.

4) Th eorem 11 uniﬁes the abov e results, demonstrating th at

the smooth max-information from [17] provides a charac-

terization of the one-shot entanglement-assisted rate dis-

tortionfunctionuptologarithmiccorrectionterms.

5) Th e bounds obtained in the one-sho t setting readily yield

bounds on th e minimum qu bit compression rate for ﬁ-

nite blocklength, for a memoryless quantum info rmation

source. In t he limit of asymptotically large blocklengt h

(

), these bo unds converge independently to the

known single-letter expression for the entang lement-as-

sisted quantum rate distortion function [6], given in terms

of the quantum mutual information.

6) We demonstrate how a good ch annel simulation protocol,

in which the simulated channel depends on the distortion

measure, leads to a rate distortion protocol that performs

well with respect to the excess-distortion prob abil ity cr it e-

rion (see Lemma 19 for details).

7) Our ﬁnal contributi ons in S ection X are 1) to evaluate

one of the aforementioned converse bounds for the spe-

cial case of an isotropic qubit source and an entangle-

ment-ﬁdelity-based distortion measure an d 2 ) to outline a

quantum teleportation strategy t hat nearly meets this con-

verse bound in the ﬁnite blocklength regime. Even thoug h

this lat ter strategy is rather simple, it represent s the ﬁrst ex-

ample in quantum rate distortion theory where a strategy

other than channel simulation is used to achieve nontrivial

compression rates.

This paper is organized as follows. In the next section, we

introduce necessary notation and deﬁnitions, especially for t he

entropic quantities arising in the statements of the t heorem s. The

DATTA et al.: ONE-SHOT LOSSY QUANTUM DATA COMPRESSION 8059

rest of the paper proceeds in the order of the results mentioned

above, and then we end with a conclusion that summarizes our

results and points to open questions for future research.

III. N

OTATION AND DEFINITIONS

Let denote the algebra of linear operators acting on

a ﬁnite-dimensional Hilbert space

,let denote the

set of positiv e semideﬁnite operators on

,andlet

denote the set of density operators (or states), i.e., p os-

itive semideﬁnite operators of unit trace. Furtherm ore, we de-

ﬁne the set of subnormalized states

. Throughout th is paper, for simplicity, we restrict our

considerations to ﬁnite-dimensional Hilbert spaces, and we de-

note the d imension of a Hi lber t space

as .

For states ,thequantumﬁde lity is deﬁned as

(2)

where

. Uhlm ann characterized the ﬁdelity

as the maximal overlap between any two puriﬁcations

and

of and , respectively [19]:

Thus, the square of the ﬁdelity has an operational interpreta-

tion as the optimal probability with which a puriﬁcation of

would pass a test for being a puriﬁcation of [20]. Since all

puriﬁcations are related by an isometric operation on the puri-

fying system, Uhlmann’s characterization is equivalent to the

following one:

(3)

where

and are now two ﬁxed puriﬁcations of and ,

respectively, and the optimization is over all isometries acting

on the purifying system . The fact that (2) is eq ual to (3) i s k nown

as Uhlmann’s theorem . The trace distance between two states

and is deﬁned as follows: ,andtheﬁdelity and

trace distance are related by the Fuchs-van-de-Graaf inequali-

ties [21]:

(4)

Moreover, for

let denote the general-

ized ﬁdeli ty [22]:

(5)

Observe that the generalized ﬁdelity reduces to the standard ﬁ-

delity in (2) if at least one of the two states is normalized. The

puriﬁed distance quantiﬁes the distance between any two sub-

normalized states

[22]:

(6)

However, note that none of our bounds depend on the dimension of the input

space, so that our results may easily be generalized to cases where the data to

be compre ssed are inﬁnite dimensional. We leave t h i s consideration for future

work, where one should be able to use the methods from [18].

We deno te a quantum channel, i.e., a com pletely positive

trace-preserving (CPTP) map

simply as

. Sim ilarly, we denote an i sometry

simply as .

The von Neumann entropy of a state

is given

. Throughout this paper, we take

the logarith m to base 2. For a b ipar ti te state

the conditional entropy of

given , and the quantum mutual

information between

and are respectively given by

(7)

(8)

where

denotes the von Neumann entropy of the reduced

state

. Furthermore, for and

,suchthat , the quantum relative en-

tropy is deﬁned as

(9)

We also m a ke use of several other entropic quantities havin g

their o rig in in the work of Renner [13]. The max-relative en-

tropy of a subn ormalized state

and an operator

is deﬁned as [23]

(10)

For a ny

, the smooth max-relative en tropy is given by

where denotes a ball of subnormalized states around

(11)

The conditional min-entropy of

given for

is deﬁned as

(12)

If the system

is tri v ial , then this reduces to

,where denotes the op erator n orm . Th e

max-information that

has abou t for is

deﬁned as [17]

(13)

For any

, the smooth versions of the above quantities

are deﬁned as follows:

(14)

(15)

For sequences of tensor power states, the (condition al) von

Neumann entropy and the quantum m utual inform ation are

equal to the smooth entropy quantities deﬁnedaboveinan

asymptotic limit [24], [14 ]. That is, for a sequence of states

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