Table Of ContentLossless quantum prefix compression for communication channels
that are always open
Markus Mu¨ller
∗
Max Planck Institute for Mathematics in the Sciences
Inselstraße 22, D-04103 Leipzig, Germany
Caroline Rogers and Rajagopal Nagarajan
† ‡
Department of Computer Science, University of Warwick
Coventry, CV47AL, United Kingdom
(Dated: November18, 2008)
Wedescribeamethodforlosslessquantumcompressioniftheoutputoftheinformationsourceis
9
not known. We compute the best possible compression rate, minimizing the expected base length
0
of the output quantum bit string (the base length of a quantum string is the maximal length in
0
the superposition). This complements work by Schumacher and Westmoreland who calculated the
2
corresponding rate for minimizing theoutput’saverage length.
n Our compressed code words are prefix-free indeterminate-length quantum bit strings which can
a beconcatenatedinthecaseofmultiplesources. Therefore,wegeneralizetheknowntheoryofprefix-
J
free quantum codes to the case where strings have indeterminate length. Moreover, we describe a
9 communication model which allows the lossless transmission of the compressed code words. The
1 benefit of compression is then thereduction of transmission errors in thepresence of noise.
] PACSnumbers: 03.67.-a,03.67.Hk
h
p
-
t I. INTRODUCTION viously stated [3, 4, 5, 6, 7] that lossless compression
n
of an ensemble = p , ψ of quantum states is in
a E { i | ii}
u One of the main aims of information theory is to de- general impossible, if the value i of the state |ψii to be
q compressed is unknown. A related objection [7] is that
termine the most efficient way to compress messages.
[ prefix-free codes are also useless in the quantum situa-
The solution to this problem often reveals relations to
tion: a prefix-freecode wordcarriesits ownlengthinfor-
3 entropy-like quantities, as in Shannon’s noiseless coding
mation. If it is transmitted over a channel, that length
v theorem[1], where the entropyof the informationsource
3 information must be read out to see when the transmis-
determines the best possible compression rate.
0 sion is over and the channel can be closed. Again, if the
0 The situation in quantum information theory is quite code word is in a superposition of different lengths, this
2 similar. The most popular example is Schumacher’s reading-out measurement disturbs the code word.
. noiselesscodingtheorem[2],showingthatthebestpossi-
8 In this paper, we show that the aforementioned prob-
blecompressionrateinthequantumcaseisgivenbyvon
0 lemsdonotappearifoneusesachannelinsteadwhichis
8 Neumann entropy. “Compression” here means that the always open. Inthiscase,thereisnoneedtodecidewhen
0 numberofqubitsthathavetobetransmittedtofaithfully
the transmission is finished. Even in the case of such a
: exchange a quantum state is minimized. This definition
v channel, compression can be beneficial: it can help to
i shows that the compression of quantum information is reduce transmission errors.
X
automatically related to the problem of communication:
To better understand the purpose of this paper, it
r once the compression is accomplished, then how can the
makes sense to think about the compressionof quantum
a
compressed code words be transmitted to a receiver?
information as taking place in severalsteps:
This question addresses an important difficulty in the
Step 1. First, the quantum state is compressed, typically
quantum situation which does not arise in classical in-
yieldinganoutputcodewhichisinasuperposition
formation theory: if a variable-length code is used for
of different lengths.
quantum compression, some code words will be shorter
thanothers. Butthismayresultincodewordswhichare
Step 2. Optional: The output code is cut off (projected)
in a superposition of different lengths — how can those
togetadeterminate-lengthcode(whichintroduces
code words be transmitted without disturbance?
some loss).
This problem is one of severalreasons why it was pre-
Step 3. Finally,the codeistransmittedoversomequantum
channel.
Actually, Schumacher and Westmoreland [7] give a
∗Electronic address: [email protected]; also at Institute
method of this form for compression of quantum infor-
ofMathematics, BerlinInstituteofTechnology (TUBerlin).
†Electronicaddress: [email protected] mation using a prefix-free quantum code. In fact, Step
‡Electronicaddress: [email protected] 1 in their setting is lossless — it is a unitary and thus
2
reversible operationthat minimizes the output’s average Our main result is Theorem VI.4 on page 10. It
•
length. states the optimal rate for prefix-free compression
Thentheyshowthataprojectiontothefirstn (S+δ) of the unknown output of a single quantum infor-
·
qubits does notdisturb the message verymuch, where S mation source, given the task to minimize the ex-
is the source’s entropy — this is Step 2 in the scheme pected base length.
above,whichintroducessome(small)loss. Astheresult- To state the theorem, we define “monotone en-
ing code word consists of a classically known, determi- tropy” and “sequential projections” and discuss
nate number of qubits, it is clearhow to transmitit over some properties that simplify the computation of
a channel (Step 3). their actual numerical values.
In this paper, we describe how Step 1 can be carried
out losslessly if the task is to minimize the output’s base
lengthinsteadofthe averagelength(both lengthnotions III. PREVIOUS WORK
willbediscussedindetailbelow,cf. DefinitionIII.2),and
we compute the best possible compression rate in terms The aim of lossless quantum compression is to com-
ofanentropy-likequantity(TheoremVI.4onpage10)for press the unknown output ψ of an ensemble =
j
the case of a single quantum information source. We do (p , ψ ) of quantum stat|es iusing a variable-leEngth
i i
this using prefix-free quantum bit strings such that code q{uant|umi }code so that the original state ψ can always
j
wordscanbe concatenatedinthecaseofseveralsources. be retrieved exactly and without error. W| hien the ψ ’s
i
| i
For this reason, we advance the theory of prefix-free areorthogonal,thisisequivalenttolosslessclassicalcom-
quantum strings, by generalizing the definition and re- pression. Thechallengeisthereforetoencode whenthe
E
sultsofSchumacherandWestmorelandinanaturalway. ψ ’sarenon-orthogonalandthecodewordsmighthave
i
| i
Moreover, we explain that Step 3 is unproblematic if indeterminate lengths.
the channel in question is always open, even if Step 2 Inthissection,wefirstgiveadefinitionofquantumbit
is dropped. All in all, this gives a lossless method of strings that consistof a superposition of classicalstrings
compression and transmission of quantum information. of different lengths. Then we describe previous work on
Thepricewepayforitisthatthereisnowaytoseewhen howtousesuchquantumstringsforcompression,andthe
the transmission is finished. Yet, the benefit is that the difficulties that arise in such models. Finally, we outline
probability of transmission errors can be reduced. a physical interpretation of these indeterminate-length
quantum strings.
II. SYNOPSIS
A. Indeterminate-Length Quantum Bit Strings
This paper is organized as follows:
The strategy of classical variable-length compression
is to assign short code words C(x) to frequent events
InSectionIII,wegiveabriefdescriptionofprevious
• x (e.g. to frequent symbols in a text in some natural
workon losslessquantum compression. In particu-
language), while rare events are assigned the remaining
lar,wereviewthe argumentswhyandinwhatway
long code words. Trying a similar approach in quantum
lossless quantum compression of unknown states
information theory naturally produces code words that
seems to be impossible. We define indeterminate-
aresuperpositions of classical strings of different lengths.
length quantum bit strings (as used by several au-
For example, suppose we have two letters A and B,
thors before) and give a physical interpretation.
andaclassicalcodeC oftheformC(A)=0andC(B)=
In Section IV, we give a communication model 11. If, as a first naive try, we extend this map unitarily
• whichdescribesasituationwherelosslessquantum to quantum states spanned by A and B , we get for
| i | i
compressionispossibleanduseful. Inshort,weex- example
plainthemodelofan“always-openchannel”where A + B 0 + 11
C | i | i = | i | i
neither Alice nor Bob know when the transmission
√2 √2
hasfinished,butbothpartiesbenefitfromcompres- (cid:18) (cid:19)
which does not have a determinate length, since it is
sion by reducing transmission errors.
in a superposition of lengths 1 and 2. It is called an
indeterminate-length quantum bit string. Such strings
Section V contains a review of some results on
• can formally be defined as follows:
prefix-free quantum bit strings, generalizing work
by Schumacher and Westmoreland [7]. We also Definition III.1 (Quantum Bit String) A quantum
give new results which have useful interpretations state ψ is a quantumbit string (or qubit string) if it is
| i
intheframeworkofourcompressionscheme. More- an element of the Fock space (or string space)
over,weprovethattheconcatenationofprefix-free
indeterminate-length quantum bit strings can in 0,1 ∗ := ∞ C2 ⊗n,
principle be implemented physically. H{ }
n=0
M(cid:0) (cid:1)
3
that is, if it can be expressed as a superposition of clas- receiver, then it becomes impossible to synchronise the
sical bit strings of the form different computational paths (taking different numbers
of time steps) that are performed on the strings.
ψ = αs s The second difficulty is that if a mixture of
| i | i
s∈X{0,1}∗ indeterminate-length strings is transmitted at a fixed
speed, then the recipient can never be sure when a mes-
with α C and α 2 =1.
s ∈ s 0,1 ∗| s| sage has arrived and the strings can be decompressed.
∈{ }
The third difficulty is that if the data compression is
For convenience,Pwe will sometimes drop the normal-
performed by a read/write head (like a Turing ma-
ization condition. Moreover, it sometimes makes sense
chine), then after the data compression, the head loca-
to call normal mixed states, i.e. density operators, on
tion of the sender is entangled with the “lengths” of the
0,1 ∗ qubit strings, too. The reason is that the pre-
fiHx{es }of pure qubit strings can be mixed, which will be indeterminate-length string which represents the com-
pressed data.
explained in detail below in Section V.
Koashi and Nobuyuki [6] argued that it is impossible
Bostr¨omandFelbinger[4]definedtwowaystoquantify
to faithfully encode a mixture of non-orthogonal quan-
the lengths of indeterminate-length strings.
tumstatesiftheparticularoutputstatesofthequantum
Definition III.2 (Length of Qubit Strings [4]) information source are not known. They modelled loss-
ThebaselengthLofanindeterminate-lengthstring ψ = less data compression as taking place in a register of N
| i
s 0,1 ∗αs|si is the length of the longest part of its su- qubits. A compressed state in the register would be an
perp∈o{siti}on unknownindeterminate-lengthquantumstringwithbase
P
lengthL,inwhichcase,onlytheremainingN Lqubits
−
wouldbeusablebyotherapplicationswithoutdisturbing
L(ψ)=L α s := maxℓ(s),
s the compressed state. However the base length L is not
s∈X{0,1}∗ | i αs6=0 an observable, thus the other applications cannot deter-
minehowmanyqubitsareavailable. Thustheremaining
or if the maximum does not exist. This can also be
∞ N L qubits are not available for other applications to
written as L(ψ) = max ℓ(s) sψ = 0 . The average −
length ℓ¯of an indetermin{ate-le|nhgt|hiqu6 ant}um bit string is use, unless there is some a priori knowledge about L for
some reason.
the expectation value of the length
Schumacher and Westmoreland [7] showed that loss-
less quantum compression cannot be carried out by
ℓ¯(ψ)=ℓ¯ α s = α 2ℓ(s) a unitary operation in a simple model of communica-
s s
| i | | tion. Theyenvisagedthatindeterminate-lengthquantum
s∈X{0,1}∗ s∈X{0,1}∗ strings would be padded with zeros to create determi-
which mayas well beinfinite. Itcan bewritten as ℓ¯(ψ)= nate length strings (we explain this in more detail below
ψ Λψ , where Λ is the length operator, defined by linear in Subsection IIIC). They modelled the data compres-
h | | i
extension of sion as taking place between two parties Alice and Bob
in which Alice sends Bob only the original strings (with
Λs =ℓ(s)s (s 0,1 ∗). the zero-padding removed) leaving Alice with a number
| i | i ∈{ }
of zeros depending on the length of the string she sent.
Formally, Λ is an unbounded self-adjoint operator, de-
Ifshesends Bobanindeterminate-lengthstring,then af-
fined on a dense subspace of H{0,1}∗. terthe transmission,Alice andBobareentangledby the
Ifthelengthofaquantumstringisobserved,thenℓ¯gives number of zeros that are left on Alice’s register.
theexpectedlengththatisobservedandLgivesthemax- Bostr¨om and Felbinger [4] argued that it is not useful
imum length that can be observed. However, given an to consider quantum generalizations of classical prefix-
unknown indeterminate-length string ψ , neither its av- free codes: classical prefix-free strings carry their own
eragelengthnoritsbaselengthcanbe|mieasuredwithout length information, but the length information in an
disturbing it. indeterminate-lengthquantumstringcannotbeobserved
without disturbing the string. Their solution to this
problem is to use a classical side channel to inform the
B. Can Indeterminate-Length Quantum Strings be receiver where to separate the code words.
Used for Coding? Ahlswede and Cai [3] followed the same idea by send-
ing the length information over a classical side channel.
Various papers [3, 4, 5, 6, 7] have described problems Compared to Ref. [4], they improved the compression
in using indeterminate-length strings for lossless quan- rate by giving a more efficient way to use the side chan-
tum data compression. Braunstein et al. [5] pointed out nel,andtheycharacterizedtheoptimalcompressionrate
threedifficultiesofdatacompressionwithindeterminate- in this setting. We describe both approaches in more
length strings. The first is that if the indeterminate- detail below in Subsection IIID.
length strings are unknown to both the sender and the However, in both cases, the use of the classical side
4
channelrequiresthatthesender(Alice)knowstheoutput The strings ψ and ϕ are prefix-free if
| i | i
of the quantum information source (at least partially),
and thus the length of the compressed code word. This ψ ρ1...n ψ =0.
h | | i
is in contrast to the situation examined in this paper.
This definition assumes that the two strings have deter-
minate length. Two of the authors defined prefix-free
C. Schumacher and Westmoreland’s Prefix-Free strings more generally such that they can be supported
Average Length Compression onsubspaceswhicharespannedbyindeterminate-length
quantum bit strings [8]. We give a review of this more
general definition in Section V, which in fact contains
Schumacher and Westmoreland [7] investigated the
Definition III.5 as a theorem (Lemma V.5).
general properties of indeterminate-length strings. An
Given many copies n of a quantum information
indeterminate-length string can be padded with zeros ⊗
E
source , SchumacherandWestmorelandfurthershowed
suchthat it consistsof a determinate number ofqubits.
E
how to use prefix-free quantum bit strings for lossless
Definition III.3 (Zero-extended form) compression (this corresponds to “Step 1” of the com-
If|ψi= ℓ(s) lmaxαs|siisaquantumstringinaregister pression process as described in the Introduction) us-
of l qubits,≤then its zero-extended form is ing appropriate unitary operations. The indeterminate-
max
P
lengthoutputisthenprojectedontothefirstn(S( )+δ)
ψ = α s0 lmax ℓ(s) . qubits(“Step2”),whereSisvonNeumannentropyE. This
zef s ⊗ −
| i | i
projection(orpartialtrace)introducesonlyasmallerror
ℓ(s)X≤lmax which vanishes in the asymptotic case n .
This string has a determinate length of lmax qubits. This can be seen as follows: Let ρ be→the∞density op-
erator corresponding to , with spectral decomposition
Given a sequence of N strings, it is useful to be able E
to “concatenate”them so that the strings are packedto-
ρ= p i i.
getheratthebeginningofthestringandthezero-padding i| ih |
allliesattheendofthesequence. SchumacherandWest- Xi
moreland [7] call this the “condensation operation”. Then can be compressed by encoding each i as a
E | i
prefix-freestringoflength log(p ) withzero-padding.
Definition III.4 (Condensable strings [7]) ρ n canbe compressedin⌈th−e samei f⌉ashion, by encoding
A code is condensable if for every N, there exists a uni- ⊗
every factor individually and applying the condensation
tary operation U such that
operation to the resulting code words. Every string ψ
U(|ψz1efi⊗...⊗|ψzNefi)=(|ψ1i⊗...⊗|ψNi)zef ibnittrhaeritlyypcilcoaslestuobs2pancSe(ρo)fρa⊗snnhgarsowprsolbaarbgeil.itTyhhψus|ρ,|vψeic|taorri-
−
for all the code words ψi which are “length eigenvec- statesinthetypicalsubspaceofρareencodedinaclassi-
tors”, i.e. if each ψi c|onitains in its superposition only calmannerasstringsoflengtharbitrarilyclosetonS(ρ).
| i
classical words of some fixed length. Theimageoftheprojectiononthefirstn(S(ρ)+δ)qubits
thus contains this typical subspace. As with overwhelm-
For example, if ψ1 = 0 , ψ2 = 10 and ψ3 = 111 ,
| i | i | i | i | i | i ing probability,the outputis veryclose to this subspace,
then the condensation operation U is
it canafterwardsbe decodedwith high(but not perfect)
U(ψ1 ψ2 ψ3 ) = U(000 100 111 ) fidelity.
| zefi⊗| zefi⊗| zefi | i⊗| i⊗| i Hencethiscompressionschemeconsistsoftwopartsas
= 0 10 111 000
| i⊗| i⊗| i⊗| i alreadymentionedintheIntroduction: inafirststep,the
= (0 10 111 )zef. quantum message is compressed losslessly, in the sense
| i⊗| i⊗| i
that the output has minimal average length of about
Superpositions of classical prefix-free strings are con-
n S( ). In a second part, some “cut-off” takes place,
densable. More generally, Schumacher and Westmore- · E
introducing some small error, but preparing the output
land gave a definition of prefix-free quantum strings and
to be transmitted over conventional channels by trans-
showed that they are condensable. According to their
forming it to fixed length [9].
definition, two strings are prefix-free if when the addi-
Oneoftheresultsofthispaperistoshowhowthefirst
tional qubits in the longer string are traced out, the re-
step can be accomplished to minimize the expected base
sulting prefixes are orthogonal.
length (in the case of a single source), thus complement-
ing the work by Schumacher and Westmoreland.
Definition III.5 (Zero-padded prefix-freedom [7])
Suppose ψ and ϕ are quantum strings with
| i | i
n := L(ψ ) < L(ϕ ) and that they are in a regis-
| i | i D. Compression with Classical Side Channels
ter of l qubits. The first n qubits of ϕ may be in
max zef
| i
a mixed state, described by the density operator
Bostr¨om and Felbinger [4] gave a scheme for lossless
ρ1...n =Tr (ϕ ϕ ). quantum compression of known ensemble outputs using
n+1,...,lmax | zefih zef|
5
classicalside channels. If = p , ψ is the mixture to the average length of a string can be interpreted as its
i i
E { | i}
be compressed, then they assume that the value of i is energy.
known to the compressor, Alice. If she encodes using A Hilbert space H n can be realised by a sequence of
⊗
E
a unitary operation C, then she sends the base length of photons φ ... φ in which φ representsexactly
1 n i
| i⊗ ⊗| i | i
thecompressedstringtoBob,thedecompressor,through one photon with frequency ω . The value of the qubit
i
aclassicalsidechannel. ShethensendsL(C(ψ ))qubits φ is realised by the polarisation of its photon, either
i i
| i | i
of ψ ’s zero-extended form to Bob. Since the length of horizontal 0 orvertical 1 . The absence ofa photonat
i
| i | i | i
the encoded string is encoded classically, it is not neces- a particular frequency can be represented by # which
| i
sarytouseaprefix-freecodetoencodethequantumpart isorthogonalto 0 and 1 . Indeterminate-lengthstrings
| i | i
— thus C is unitary but not necessarily a condensation are obtained by allowing the number of photons to exist
operation. in superposition and ordering the photons by their fre-
Ahlswede and Cai [3] studied quantum data compres- quencies. The first # (which can be in a superposition
| i
sion with classical side channels in more detail. They of positions) is used to mark the end of the string.
found an expression for the number of qubits that are The frequency of each photon φ is chosen to be ap-
i
| i
sent through the quantum channel in Bostr¨om and Fel- proximately equal so that ω ω for some value ω. The
i
≈
binger’s lossless quantum compression scheme [4]. energy in a superposition of photons is the average en-
Moreover,they showedby using counterexamplesthat ergy required to either create or destroy that superposi-
theoptimalrateofcompressionRofaone-onecodecan- tion (~ω per photon of frequency ω where ~ is Planck’s
not be achieved by a greedy algorithm. However, the constant). Thus the energy of an indeterminate-length
main goal of Ahlswede and Cai was to find a more effi- stringofphotons φ isproportionaltoitsaveragelength
cient way to use the classical side channel than just to and is given by (a|ppiroximately) ~ωℓ¯(φ ).
| i
report the base lengths. They showed that the quantum On the other hand, the base length of φ represents
| i
part could be compressedfurther than in the scheme set the number of photons at different frequencies that are
out by Bostr¨om and Felbinger. used to describe φ . Thus the base length of φ is the
| i | i
The basis for their scheme is as follows. If is the size of the system required to carry the state φ .
E | i
mixture to be compressed,andif there existssome small
subspaceX suchthatseveralstates ψ lieexactlywithin
i
| i
X, then this fact can be reported through the classical IV. COMMUNICATION MODEL FOR
side channel. Thus the amount of quantum information LOSSLESS QUANTUM COMPRESSION
that must be sent through the quantum channel is re-
duced. They gave an expression for the optimal rate of Now we describe a model of a communication chan-
compressionintheirschemeofanensemble = pi, ψi nelwherelosslessquantumcompressionofunknownmix-
E { | i}
when the states ψi are linearly independent (but not tures is possible and useful.
| i
necessarily orthogonal). Themainargumentwhylosslessquantumcompression
Compression with classical side channels has been ofunknownstatesseemstobeimpossibleisthatitisim-
studied in more detail for lossy compression [10]. possibletodeterminehowmanyqubitstotransmitwhen
Hayashi and Keiji [11] investigated variable- (but not the message is in a superposition of different lengths. If
indeterminate-) length universal compression. Alicehasanunknownindeterminate-lengthqubitstring,
RallanandVedral[12]gaveanotherschemeforlossless how can she find out when the transmission is finished
quantum compression with classical side channels which and the channel can be closed? To avoid this problem,
does not use zero-extended forms. They envisaged that we look at always-open channels.
the compressed state would be represented by photons
— thus using a tertiary alphabet 0 , 1 , # , where Blaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
{| i | i | i} transmission cell
# denotes the absence of a photon and marks the end
| i
of the string. They assumed that Alice has n copies of 0
an ensemble which she would like to send to Bob. In
E
this scheme,Alice onlysendsBobthe valueofnthrough swap conditional swap
the classical channel. This scheme has a nice physical
interpretation.
1 0 1
0 0 0 0 0 0 0
0 0 1
noise
Alice Bob
Blaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
E. Physical Interpretation of Indeterminate-Length
Strings
FIG.1: Schematicof an always-open channelas described in
Section IV.
Bostr¨om and Felbinger [4] pointed out that variable-
lengthquantumstringscanberealisedinaquantumsys-
tem whose particle number is not conserved. Rallan and A model of an always-open channel [13, 14] is shown
Vedral [12] described in detail an example system where inFig.1. Suppose Alice wantsto send Boba singlecode
6
word of a quantum prefix code, i.e. an indeterminate- message’sbaselength),Bobstopstoaccessthetransmis-
lengthqubitstring ψ whichisanelementofaprefix-free sion cell. Thus, any noise that affects the cell from that
| i
subspace of 0,1 ∗ that Alice and Bob have agreed point on will not disturb the communication any more,
upon in adHvancHe.{(T}his single code word might itself be because the channel to Bob is effectively closed. Thus,
aconcatenationofseveralprefix-freecodewords.) Aswe minimizing the number of qubits to be transmitted re-
shallseelaterinTheoremVI.4,wemayassumethat is duces he probability of transmission errors, even though
H
spanned by the classical code words of a classical prefix neither Alice nor Bob know the number of transmitted
code as in Subsection IIIC above. qubits.
The main part of the channel is a transmission cell It is clear that the optimal compression method de-
which carries exactly one qubit. Initially, this qubit is pends on the kind of noise that the system is exposed
set to zero, and so are all the qubits in Bob’s memory. to. Obviously, in the case that each transmitted qubit
Moreover, Alice’s memory contains a zero-padded form is independently subject to the same kind of pertur-
of her message string, as described in Definition III.3. bation, then Schumacher and Westmoreland’s average
Now we describe the communication protocol — for length compression method optimally minimizes trans-
each step, we describe what Alice and Bob are doing in missionerrors. Butthereareotherconceivablescenarios:
the case of classical bits (i.e. in the case that the mes- forexample,wemighthaveseveralchannelsatoncethat
sage qubit string ψ is just a classical string s out of are subject to the same kind of noise,or time-dependent
| i | i
the classical prefix-free orthonormal basis of ), and we noisethatgrowswiththe numberofqubits. Inthis case,
H
assume that the resulting operation is linearly extended itisnotsoclearanymorewhatthebestmethodofcom-
to a unitary operation on the corresponding quantum pression is.
system. The unitarity of the operations at Alice’s and In this paper, we compute the best possible compres-
Bob’s side is then assuredby the reversibility of the cor- sion method and the rate for minimizing the code’s ex-
responding classical operations. pected base length. Although we do not currently know
Atstepiofthetransmission,Aliceswapsthei-thqubit of a natural noise model where the expected base length
of her padded message string with the content of the determinestheerrorprobability,itseemslikelythatthere
transmission cell. Afterwards, Bob checks if the i 1 areindeednaturalsituationswherethiskindofcompres-
−
qubits he has receivedpreviously forma validcode word sion is superior to average length compression — for ex-
or not. (Due to prefix-freedom, if the answer is “yes”, ample,modelslikethosementionedinthelastparagraph
then the transmission must be over — cf. also Defini- where “later” qubits are subject to larger errors than
tion III.5 and Lemma V.5). If the answer is “no”, he “earlier” ones.
swapsthe i-thqubitofhismemory(whichis justazero)
with the content of the transmission cell; otherwise, he
does notdo anything. Thatis, Bobapplies aconditional
V. PREFIX-FREE QUANTUM BIT STRINGS
swap,wherethe conditionis thatthe transmissionisnot
yet finished.
Schumacher and Westmoreland [7] defined prefix-free
Thisway,themessagequbitstringistransmittedqubit
quantum strings in terms of their zero-extended forms
by qubit. In the end, Alice ends up with a memory full
using the partial trace, see Definition III.5 above. In
of zeroes, while the transmission cell contains a zero as
Ref. [8], two of the authors have given another way to
well,andBob’smemorycarriesthe zero-paddedmessage
define prefix-free quantum strings which is more general
string. Hence the entanglement problem described by
and a more direct generalization of the classical defini-
Schumacher and Westmoreland [7] is avoided, and the
tion. It can be shown to contain the definition by Schu-
message qubit string is transmitted reversibly and uni-
macher and Westmoreland as a special case. In this sec-
tarily from Alice to Bob.
tion,we brieflyreview the definitionandbasicresultson
But what is the advantage of compression for such
prefix-free quantum strings.
a communication channel, if that channel can never be
The notion of the prefix of a classical string is closely
switched off by Alice or Bob? It cannot be used to save
relatedtotheconcatenationoperation . Thus,beforewe
transmission time (considered as a resource), because ◦
define prefix-free quantum strings, we first explain how
both parties never know if the transmission is already
finished or not (unless some predefined maximal trans- tisoacnoyncqauteannattuemqubaitntsutmringb,itasntdrinsgs. If0,|1ψi ∈isHa{c0,l1a}s∗-
mission time tmax has passed). However, quantum com- ∈ { }∗
sical bit strings, then we can define ψ s by linear
pression can have other advantages: for example, sup- | i ◦
extension of the classical concatenation: Expand ψ =
pose the transmission cell is subject to noise during the | i
α x , and define
transmission. That is, the transmission of every single x 0,1 ∗ x| i
∈{ }
qubit has an inherent error probability. In this case, Al-
P
ice can minimize transmissionerrors by compressing her ψ s := ψ s:= α x s .
x
| ◦ i | i◦ | ◦ i
quantum messages before sending them. x s
X∈
To be more exact, as soon as the code word has been
fullytransmitted(i.e. atatimestepcorrespondingtothe Moreover,if |ϕi= t 0,1 ∗βt|ti is another qubit string
∈{ }
P
7
with finite base length, then we set where P( ) denotes the orthogonal projector onto .
H H
Equality holds for the left three terms if and only if every
ψ ϕ := ψ ϕ := βt ψ t . ei is a length eigenvector.
| ◦ i | i◦| i | ◦ i | i
t∈X{0,1}∗
Prefix-free subspaces have a remarkable property: ev-
This concatenation operation on the quantum strings is
ery basis vector of length n can be distinguished with
related to the tensor product: If ψ is a length eigen-
| i certainty from every other (even longer) basis vector by
state (i.e. an eigenvectorof the length operatorΛ), then
measuring the first n qubits only. Unfortunately, this
ψ ϕ = ψ ϕ . However,if ψ isnotalengtheigen-
| i◦| i | i⊗| i | i is only true in general for orthonormal bases of length
state, then the concatenation operation is not always an
eigenvectors [8].
isometry and thus not always physically meaningful [8].
We can now define prefix-free sets of quantum strings
(dei.rge.ctpgreefinxer-farleizeastuiobnspoafctehseocfltahsseicsatrlidnegfisnpiaticoenH. A{0l,t1h}∗o)ugbhy wLheimchmcaonVsi.s5tsAenntiorretlhyonoforlmenaglthsyesitgeemnveMctors⊂isHp{r0e,fi1}x∗-
free if and only if for every ϕ , ψ M with ϕ = ψ ,
there are several a priori possible generalizations, they | i | i∈ | i6 | i
it holds
allturnouttobeequivalent(foraproofseeRef.[8]). To
state them, we use the symbol λ for the empty string of
length zero. ψ ϕℓ(ψ) ψ =0, (1)
h | | i
Definition V.1 (Prefix-Free Sets of Qubit Strings) where ϕn denotes the restriction of the quantum state
A set M 0,1 ∗ of qubit strings is called prefix-free, ϕ ϕ to the first n:=ℓ(ψ) qubits.
if one of t⊂heHfo{ur}following equivalent conditions holds: | ih |
This lemma shows that if the subspace contains an or-
(1) For every ϕ , ψ M and classical string s
| i | i ∈ ∈ thonormal basis of length eigenvectors, our definition of
0,1 λ , it holds ϕψ s =0.
∗
{ } \{ } h | ◦ i prefix-freedom is equivalent to the definition by Schu-
(2) For every ϕ , ψ M and qubit string χ λ , macher and Westmoreland [7].
| i | i ∈ | i ⊥ | i
it holds ϕψ χ =0. Wehaveonlycollectedthebasicfactsaboutprefix-free
h | ◦ i
quantumbitstringsthatarerelevantforlosslessquantum
(3) For every ϕ , ψ M and classical strings s,t
| i | i ∈ ∈ compression. For more details, we refer the reader to
0,1 with s=t, it holds ϕ tψ s =0.
{ }∗ 6 h ◦ | ◦ i Refs. [7] and [8].
(4) For every ϕ , ψ M and qubit strings χ , τ In general, the concatenation operation does not pre-
| i | i ∈ | i | i ∈
H{0,1}∗ with |χi⊥|τi, it holds hϕ◦τ|ψ◦χi=0. saenrviseomtheetrnyoarmndohfenvceectnoorst pfrhoymsicHal{ly0,1m}∗e,ani.ien.gfuitl.isHonwot-
TherelevantcaseforquantumcompressionisthatM is ever, we shall now prove that concatenation can be im-
itself a closedsubspace of string space 0,1 ∗. To prove plemented in principle on a quantum computer (i.e. it is
prefix-freedomofsuchasubspace,itisHsu{ffici}enttoprove anisometry)if onerestrictstoprefix-freeHilbertspaces:
this property for an arbitrary orthonormal basis [8]:
Lemma V.2 A subspace 0,1 ∗ is prefix-free if Theorem V.6 (Isometry of Concatenation)
and only if it has a prefix-fHree⊂ortHho{nor}mal basis. In this If ϕ1 , ϕ2 0,1 ∗ is a prefix-free set, and
case, every orthonormal basis of is prefix-free. ψ1{,|ψ2i | i}0,1⊂∗, tHhe{n }
H | i | i∈H{ }
Eprxefiaxm-fprelee:V.3 The following subspace H ⊂ H{0,1}∗ is hϕ1◦ψ1|ϕ2◦ψ2i=hϕ1|ϕ2ihψ1|ψ2i.
1 1 Consequently, if 0,1 ∗ is a closed prefix-free sub-
:=span (1 + 01 ), (10 010 ) space, then thereHe⊂xisHts{a}unique isometry U :
H (cid:26)√2 | i | i √2 | i−| i (cid:27) 0,1 ∗ 0,1 ∗ such that U ϕ ψ = ϕ◦ ψHfo⊗r
Itiseasily checkedthatcondition (1)from DefinitionV.1 eHv{ery}ϕ→ H{an}d ψ 0,1 ∗◦.| i⊗| i | ◦ i
| i∈H | i∈H{ }
above is satisfied for the two orthonormal basis vectors.
Notethatinthespecialcasethat isspannedbylength
Similarlyasintheclassicalcase,closedprefix-freesub- H
eigenvectors, the map U corresponds to the “simple
spaces obey a Kraft inequality [8]: condensation operation” a◦s defined by Schumacher and
Westmoreland [7].
Lemma V.4 (Quantum Kraft Inequality) Let
ei i I 0,1 ∗ be a prefix-free orthonormal system, Proof. It is easy to check that for every pair of qubit
{sp|anin}i∈ng⊂a cHlo{sed}subspace H⊂H{0,1}∗. Then, it holds sstϕringss|ϕ=1i,ϕ|ϕ2ϕi∈.HN{0o,w1}∗suapnpdosse∈th{a0t,1a}d∗d,iwtieonhaalvlyehΦϕ:1=◦
2 1 2
| ◦ i h | i
2−L(ei) ≤ 2−ℓ¯(ei) ≤Tr 2−ΛP(H) ≤1, a{r|ϕb1itir,a|ϕry2iq}ubisitastprrienfigx.-Efrxepeasnedt,inagndψ|ψ=i ∈ H{0,1}∗γis san,
Xi∈I Xi∈I (cid:0) (cid:1) | i s∈{0,1}∗ s| i
P
8
we have Let now s and t be elements of maximal length
max max
in S(ϕ) and S(ψ) respectively. Clearly, s t ϕ
hϕ1◦ψ|ϕ2◦ψi = γ¯sγthϕ1◦s|ϕ2◦ti ψ = αsβt, where the sum is over ahllmsax◦S(mϕa)x|and◦
s,t∈X{0,1}∗ t i S(ψ) such that s t = smax tmax. B∈ut because
∈ P ◦ ◦
( ) of the maximum length property of s and t , it
=∗ γ¯sγs ϕ1 sϕ2 s max max
h ◦ | ◦ i follows that ℓ(s)=ℓ(s ) and ℓ(t)=ℓ(t ), and thus
s∈X{0,1}∗ s = smax and t = tmaxm.axConsequently, smmaaxx tmax ϕ
= hϕ1|ϕ2i |γs|2 =hϕ1|ϕ2ihψ|ψi. ψi=αsmaxβtmax 6=0, and L(ϕ◦ψ)≥Lh(ϕ)+L◦(ψ). | (cid:3)◦
s∈X{0,1}∗ Weexplainthemeaningoftheseresultsforlosslessquan-
In( ), wehaveusedthe factthatΦ is prefix-free,andso tum data compression below after Definition VI.1, the
ϕ1∗sϕ2 t =0 if s=t. Finally, if ψ1 , ψ2 0,1 ∗ definition of a lossless quantum code.
hare ◦arb|itra◦ryiqubit str6ings, then choo|seian| arib∈itrHar{y o}r-
thonormal basis ei i N of 0,1 ∗ such that ψ1 =
λe withλ R,{a|ndi}ex∈pand Hψ{ a}s ψ = |α ie . VI. LOSSLESS QUANTUM DATA
It|f1oillows ∈ | 2i | 2i i∈N i| ii COMPRESSION
P
ϕ ψ ϕ ψ = α ϕ ψ ϕ e
1 1 2 2 i 1 1 2 i Our aim is to compute the best possible rate for com-
h ◦ | ◦ i h ◦ | ◦ i
i N pressing the unknown output of a single quantum infor-
X∈
( ) mation source, where the source is given by an ensem-
∗=∗ α ϕ ψ ϕ e
1 1 1 2 1
h ◦ | ◦ i ble = p , ψ of ingeneralnon-orthogonalquantum
i i i
= α λ ϕ ϕ e e E { | i}
1 h 1| 2ih 1| 1i states ψi with probabilities pi > 0. As motivated in
| i
=1 theIntroduction,wewanttominimizetheexpectedbase
= ψ ψ ϕ ϕ . length ofthe code, and we wantto use a prefix-freecode
h 1| 2ih 1||2{iz }
toallowconcatenationofcodewordsinthecaseofseveral
In( ),we haveagainusedthe factthatΦ isprefix-free,
sources.
∗∗
and consequently ϕ ψ ϕ e = 0 for i 2, since
1 1 2 i
ψ e . h ◦ | ◦ i ≥ (cid:3) Definition VI.1 (Lossless Quantum Code)
1 i
| i⊥| i Let = p , ψ bean ensemble of quantum states in a
We show now that the base length of a concatena- E { i | ii}i
Hilbert space, with :=span ψ ,..., ψ . A lossless
tion of two qubit strings is the sum of the individual H {| 1i | ni}
code C is an isometric linear map from into a closed
base lengths. Note that this is in general not true for H
average length ℓ¯: for example, if |ψi = √12(|1i+ |01i) preTfihxe-freexepescutbesdpbaacseeHle′n⊂gtHh {o0f,1c}o∗m. pression of C is
and ϕ = 1 (10 010 ) are two vectors from the
| i √2 | i − | i E(L(C( )))= p L(C(ψ )). (2)
prefix-freeHilbertspace inExampleV.3,andif χ := i i
E | i
1 (ψ + ϕ ),thenitiseaHsytocheckthat 19 =ℓ¯(χ|ϕi)> Xi
√2 | i | i 4 ◦
ℓ¯(χ)+ℓ¯(ϕ)=2+ 5. C is optimal if for any other code C′,
2
Lemma V.7 (Additivity of Base Length) E(L(C( )) E(L(C′( ))).
E ≤ E
Ilefn|gϕthi,s|,ψii.e.∈LH(ϕ{)0,<1}∗ araendquLb(iψt )st<rings, twheitnhLfi(nϕiteψb)as=e The expression(2) defines the compressionrate of the
∞ ∞ ◦ code as the expected base length of the encoding of the
L(ϕ)+L(ψ).
output of a single instance of the ensemble. What if we
Psroof. 0,1For evseϕry |=ϕi 0∈. HI{t0,f1o}l∗l,owdsefithnaetSL(ϕ(ϕ)) :== shtaavteesnacroepipersoEdu⊗cnedofianndeepnesnedmebnltelyE,ani.de.idseevnetricaallolyutdpiust-
∗
{ ∈ { } | h | i 6 }
max ℓ(s) s S(ϕ) . If we expand ϕ =: tributed according to ?
{ | ∈ } | i E
α s and ψ =: β t , then Supposewehavetwodifferentensembles = p , ψ
s 0,1 ∗ s| i | i t 0,1 ∗ t| i E { i | ii}
∈{ } ∈{ } and = q , ϕ whichhaveoptimalcodesC andC
j j
P ϕ ψ = Pα β s t . respeFctive{ly. |Asit}he codes are prefix-free, we Emay conF-
s t
| ◦ i | ◦ i
s,t catenatethemtoobtainacodeC C for . Theo-
X rem V.6 proves that this concatenEa◦tioFncanEb⊗e dFone uni-
It follows that
tarily,i.e. canbeimplementedinprincipleonaquantum
S(ϕ ψ) S(ϕ) S(ψ):= s t s S(ϕ), t S(ψ) , computer, and Lemma V.7 tells us that the base lengths
◦ ⊆ ◦ { ◦ | ∈ ∈ }
then just add up. Explicitly,
and thus
E(L(C C )) = p q L(C (ψ ) C (ϕ ))
L(ϕ ψ) = max ℓ(s) s S(ϕ ψ) E ◦ F i j E | ii ◦ F | ji
◦ { | ∈ ◦ ij
max ℓ(s) s S(ϕ) S(ψ) X
≤ { | ∈ ◦ } = p q L(C (ψ )+L(C (ϕ ))
= max ℓ(s t) s S(ϕ),t S(ψ) i j E | ii F | ii
{ ◦ | ∈ ∈ } ij
= max ℓ(s)+ max ℓ(t)=L(ϕ)+L(ψ). X (cid:0) (cid:1)
= E(L(C ))+E(L(C )).
s S(ϕ) t S(ψ)
∈ ∈ E F
9
ThusC C isacodefor withthesimpleproperty As is well-known, this best rate is given by the Shannon
thatitsEr◦ateFisjustthesumE⊗ofFthe ratesofthetwocodes. entropy H(p); thus, we would get back (up to possibly
However,it is not necessarily optimal any more. In fact, one bit) Shannon entropy. This implies
denoting the optimal compression rate of an ensemble
by R( ), Theorem VI.4 below will show that if e.g. E Hmon(p) H(p), (3)
E ≥
1 1 1 and justifies that we call Hmon an entropy. Note that
:= , , ,(0 , 1 , 2 ) H changesifwepermutetheentriesofp(whileShan-
E 3 3 3 | i | i | i mon
(cid:26)(cid:18) (cid:19) (cid:27) non entropy stays constant). If the elements of p are in
where 0 , 1 and 2 arethreearbitraryorthonormalvec- decreasingorder,thenmonotoneentropyequalsShannon
tors,th|ein|R(i )=|5i,whileR( )= 29 <2R( )= 30. entropy up to possibly one bit:
E 3 E⊗E 9 E 9
Hence concatenation of codes does not always produce
p p ... p H(p) H (p) H(p)+1.
optimal codes (although they are typically quite good), 1 ≥ 2 ≥ ≥ n ⇒ ≤ mon ≤
(4)
and our result will for example not give a simple expres-
usipopnefrobrotuhnedasRy(mp).totic rate limn→∞ n1R(E⊗n), only the oTthhiesrisheaansdi,lyifpwroevseedtbℓyi :i=nse⌈rlotignng⌉ℓifo:=r e⌈v−erlyogi,pwi⌉e. gOent tthhee
E universal upper bound
Yet, the result is nevertheless useful, in particular if
there is only one output of the source, or if there are
H (p) logn , (5)
severalsources 1 2 ... k whicharenotknownin mon ≤⌈ ⌉
E ⊗E ⊗ ⊗E
advance to the compressor. Then, the compression can if n denotes the number of elements in p.
be done sequentially, for one source after the other, and
Nowweexplainthenotionofasequentialprojection. It
thecodewordsareconcatenatedwhiletheratesjustadd
is a certain probability distribution which is constructed
up. As for the compressionrate,we get the useful upper
from in a sequential manner.
bound R k R( ) even if there is no translation E
invariance≤in thie=s1equEeince of sources. Definition VI.3 (Sequential Projection) Let =
This subadPditivity property of the optimal rate also {(pi,|ψii)}ni=1 be an ensemble of quantum states. EA se-
showsthatinthecaseofncopiesofonesource,blockcod- quentialprojectionp′ =(p′1,p′2,...,p′k) is any probability
ing with concatenation will produce the optimal asymp- distribution which can be constructed by the following al-
totic compression rate: write gorithm:
⊗n = ⊗n1 ⊗n2 ... ⊗nk • Choose an arbitary integer i1 ∈ {1,...,n}. Then,
E E ⊗E ⊗ ⊗E add up all the probabilities p that correspond to
j
with ki=1nk = n such that the sequence (nk)k N is vectors |ψji which are linearly dependent on (par-
incnrekaPsseinpga.raTtehleyn,,aunsdectohnecoaptteinmaatelctohdeecCodneksfotoregaecthab∈cloodcke allel to) |ψi1i to get the value p′1, i.e.
⊗
fEor ⊗n. The corresponding compression rate will be I1 :={j ∈{1,...,n} | |ψji∈span{|ψi1i}}
E
asymptotically optimal.
(in particular, i J), and p := p .
To state the optimal compression rate for single 1 ∈ ′1 j I1 j
∈
sources,weintroducethenotionofmonotoneentropyand Choose an arbitrary remainingPinteger i
2
ofasequentialprojectionofsomeensembleE ={pi,|ψii}. • {1,...,n}\I1. Add up all the probabilities pj tha∈t
correspond tovectors ψ which arelinearly depen-
j
| i
dent on ψ and the previously chosen vectors in
Definition VI.2 (Monotone Entropy) | i2i
I to get the value p , i.e.
Let p=(p ,p ,...,p ) be a probability vector. Then, we 1 ′2
1 2 n
define the monotone entropy Hmon(p) as I2 :={j ∈{1,...,n}\I1 | |ψji∈span({|ψi2i}∪I1)}
n n
and p := p .
Hmon(p) := min piℓi 2−ℓi 1, ′2 j∈I2 j
(Xi=1 (cid:12)(cid:12)(cid:12) Xi=1 ≤ • C1h,o.o.s.e,nanP(IarbitIra)r.y Ardedmauipnianlgl thinetepgreorbabii3litie∈s
(cid:12) 1 2
ℓ1 ≤ℓ2 ≤((cid:12))...≤ℓn, ℓi ∈N0. v{peaejcrtltyohradstetpco}eonrg\dreeetsntptho∪eonndva|tlψuoie3vipeac,tnodir.est.h|eψjpirewvihoiucshlyacrheolsienn-
∗ ′3
Note that the Kraft in|equality{zon the}right-handside I := j 1,...,n (I I )
3 1 2
implies that the values ℓ are code word lengths of a { ∈{ }\ ∪ |
prefix code. { i}i |ψji∈span({|ψi3i}∪I1∪I2)}
Suppose we removed ( ) from the definition. This
and p := p .
wouldmeanthatwelookf∗orthesmallestpossiblerateof ′3 j∈I3 j
any prefix code for the given probability distribution p. ... P
•
10
Iteratethesestepsuntiltherearenoremainingvec- Ourtheoremtells us that we cancompressthe ensem-
• tors in the ensemble. ble atleastasgoodasR H(p)+1=H 1,1,...,1 +
≤ 8 8 8
1 = 4, but we can do better than that. To compute the
As an example of a sequential projection, consider the (cid:0) (cid:1)
optimalcompressionrate,we haveto look atall possible
states from an ensemble p , ψ 4
{ i | ii}i=1 sequential projections.
We constructa sequentialprojectionp: first, we arbi-
0 + 1 ′
ψ = 0 , ψ = + = | i | i, trarily choose one of the vectors, say, ψ . As there is
| 1i | i | 2i | i √2 | 1i
no other vector which is linearly dependent on (parallel
|ψ3i = |1i, |ψ4i=|2i, to) |ψ1i, the first entry to p′ is p′1 :=p1 = 81.
As the second step, we choose one of the remaining
where 0 , 1 and 2 denote orthonormal basis vectors
froman| airb|itiraryH|ilibert space. Then,applyingthe def- vectors, say, |ψ2i. We have to check if there are any re-
mainingvectorsthatareinthespanof ψ and ψ (i.e.
inition above,and noting that ψ is in the span of ψ | 1i | 2i
| 3i | 1i linearlydependentonthosetwo),whichisbyassumption
and ψ , we get one possible sequential projection as
| 2i not the case. Thus, we get p′2 :=p2 = 18.
We go on by choosing the next vector arbitrarily, say,
p =p , p =p +p , p =p ,
′1 1 ′2 2 3 ′3 4 ψ . As there are no remaining vectors in the span of
3
where we have chosen i1 = 1, i2 = 2 and i3 = 4. Other ||ψ1ii, |ψ2i and |ψ3i, we also get p′3 :=p3 = 18.
choices of indices yield different sequential projections. Then we select another remaining vector, say, ψ4 .
| i
Thatis,toeveryensemble ,thereareseveralpossiblese- But now, all the remaining vector ψ5 , ψ6 , ψ7 and
quential projections of . BEy combinatorics,the number ψ8 , are in the linear span of ψ1 ,|ψ2i,|ψ3i a|ndiψ4 .
of sequential projectionEs to an ensemble of n elements T| huis, we have to add the corre|spoind|ingip|robiabilitie|s tio
is upper-bounded by n!. Each sequentialEprojection is a get p′4 :=p4+p5+p6+p7+p8 = 85. Thus,
probability vector with dim elements.
E 1 1 1 5
To get an idea how sequential projections are related p′ = , , , .
tobaselengthcompression,supposeacodeQcompresses 8 8 8 8
(cid:18) (cid:19)
the state 0 with base length l and the state + with
| i 1 | i In this example, repeating the process with different
base length l l , then, since 1 is on the span of
2 ≥ 1 | i choices of vectors will always result in the same proba-
0 and + , 1 will typically also be compressed to l .
|Suippose| 2i is| ciompressed to length l (which can safel2y bilitydistributionp′. Thus,inthiscase,thereisonlyone
| i 3 possible sequential projection of which is given above.
be achieved if l3 ≥l2), then the compression rate of E is We get the rate R by computingER = Hmon(p′). First,
E(L(Q(E)))=p1l1+(p2+p3)l2+p4l3 =p′1l1+p′2l2+p′3l3. wweithkntohwe hfreolpmo(f5a) tlihtatlteHcommonp(upt′e)r≤pr⌈ologgra4m⌉,=it2i.sIenasfyactto,
We can now state the optimal rate of compression in seethattheminimuminthedefinitionofHmon isindeed
terms of monotone entropy and sequential projection, attained at this value, that is,
which can both be calculated combinatorially.
R=H (p)=2.
mon ′
Theorem VI.4 (Optimal Compression Rate)
Let = p , ψ be an ensemble of quantum states in Now we prove this theorem.
i i
someEHilb{ert s|paic}e. Then the optimal base length lossless Proof. The proof consists of two parts: first, we
quantum prefix compression code C can be constructed show that a rate of R is achievable, then we show
suchthatitmapsintoaHilbertspace whichisspanned that this rate is optimal. We shall denote our en-
′
by an orthonormal basis of length eigHenvectors. The rate semble by E = {p(|ψii),|ψii}ni=1, and we write H :=
R of this optimal code is given by
=:pi
span ψ ,..., ψ . For sequential projections, we use
1 n
R=min H (p) p is a sequential projection of . {| i | i|}{z }
mon ′ ′ the nomenclature from Definition VI.3.
{ | E}
To see the achievability, let p = (p ,...,p ) be an
In particular, if the vectors ψ are linearly indepen- ′ ′1 ′d
dent, then H(p) R H({p|)i+i}1i, i.e. the rate is es- arbitrary sequential projection of E. Let (c1,...,cd) ⊂
sentially given by≤the S≤hannon entropy of ’s probabil- {0,1}∗beaprefixcodewithcodewordlengthsℓi :=ℓ(ci)
ity distribution. In any case, we have the Eupper bound which are minimizers in the definition of Hmon(p′) (such
a code exists due to the Kraft inequality). Let :=
R H(p)+1. H′
≤ span c1 ,..., cd 0,1 ∗. We will now construct
Beforewegiveaproof,weillustratethetheoremwithone a cod{e| (ia linea|r iis}om⊂etHri{c m}ap) C : ′. For i
H → H ∈
example. Suppose our ensemble consists of eight states 1,...,d , let Ψi be the set of vectors from that have
{ } E
ψ1 , ψ2 ,..., ψ8 from some Hilbert space, each with been chosenin step i of the constructionof p′, such that
p|roiba|biliity p1|= pi2 = ... = p8 = 18, such that the span Ψi ∩Ψj = 0 for i 6= j, ∪di=1Ψi = {|ψ1i,...,|ψni}, and
of those eight states has dimension four. Furthermore, p′i = ψ Ψip(|ψi).
suppose that any four of those states are linearly inde- We st|air∈t by specifying the action of C on the vectors
P
pendent. of Ψ . All the vectors in Ψ are equal up to some phase
1 1
