A bound on the mutual information, and properties of entropy reduction, for quantum channels with inefficient measurements

0.16 MB·English

by  Kurt Jacobs

A bound on the mutual information, and properties of entropy reduction, for
quantum channels with inefficient measurements
Kurt Jacobs1
1Centre for Quantum Computer Technology, Centre for Quantum Dynamics,
School of Science, Griffith University, Nathan 4111, Brisbane, Australia
The Holevoboundis a boundon themutualinformation for a given quantumencoding. In 1996
Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev.
Lett. 76, 3452 (1996)] derived a bound which reduces to the Holevo bound for complete measure-
ments, but which is tighter for incomplete measurements. The most general quantum operations
may be both incomplete and inefficient. Here we show that the bound derived by SWW can be
further extended to obtain one which is yet again tighter for inefficient measurements. This allows
us in addition to obtain a generalization of a bound derived by Hall, and to show that the average
6
reduction in the von Neumann entropy during a quantum operation is concave in the initial state,
0
for all quantum operations. This is a quantum version of the concavity of the mutual information.
0
We also show that both this average entropy reduction and the mutual information for pure state
2
ensembles,areSchur-concaveforunitarilycovariantmeasurements;thatis,forthesemeasurements,

n information gain increases with initial uncertainty.
a
J
PACSnumbers: 03.67.-a,03.65.Ta,89.70.+c,02.50.Tt

4
2
I. INTRODUCTION furtherreduced. In1996Schumacher,Westmorelandand

2 Woottersshowedthatwhenthereceiversmeasurementis
v
incomplete,itispossibletotakethisintoaccountandde-
6 The celebrated Holevo bound, conjectured by Gor-
rive a more stringent bound on the information. If the
0 don [1] and Levitin [2] and proved by Holevo in 1973 [3]
receiver’s measurement is the POVM described by the
0 gives a bound on the information which may be trans-
2 mitted from A to B (strictly, the mutual information, operators {Aj} (with jA†jAj = 1), so that the mea-
1 M, between A and B) when A encodes information in a surement outcomes are labeled by the index j, then the
4 quantum system using a set of states {ρ }, chosen with SWW bound is [6] P
0 i
probabilities{P(i)},andBmakesasubsequentmeasure-
/
h ment upon the system. The Holevo bound is M(I:J)≤χ− P(j)χj, (2)
p
j
- X
t M(I:J)≤χ≡S(ρ)− P(i)S(ρ ), (1)
n i where P(j) is the probability of outcome j [7], and χ
a Xi is the Holevo quantity for the ensemble that the systemj
u
q where ρ = iP(i)ρi (and which we will refer to as the remainsin(fromthe pointofview ofthe receiver),given
: ensemble state). We write the mutual information as outcome j. This bound can be at least partially under-
v M(I:J) toPsignify that it is the mutual information be- stood by noting that if the system still remains in some
i
X tweentherandomvariablesI andJ,whosevaluesiandj ensemble of possible states after the measurement, then
r label respectively the encoding used by A, and outcome future measurements can potentially extract further in-
a of the measurement made by B. More recent proofs of formation about the encoding, and so the information
the Holevo bound may be found in Refs. [4, 5, 6] The obtainedbythefirstmeasurementmustthereforebe less
bound is achieved if and only if the encoding states, ρ , than the maximum possible by at least by this amount.
i
commute with each other, and the receiver, B, makes a What the SWW bound tells us is that the bound on the
von Neumann measurement in the basis in which they information is reduced not only by the amount of infor-
are diagonal. (A von Neumann measurement is one that mation which could be further extracted after outcome
projects the system onto one of a complete set of mu- j has been obtained, but by the Holevo bound on this
tually orthogonal states. In this case the set of states information, χj.
is chosen to be the basis in which the coding states are If the initial state of the system is ρi, then after
diagonal.) With this choice of coding states and mea- outcome j the final state of the system is given by
surement the channel is classical, in that it can be im- ρ˜j|i = AjρiA†j/Tr[A†jAjρi]. Thus the states which make
plemented with a classical system. The Holevo bound up the final ensemble that remains after outcome j, are
takes into account that the sender may only be able to {ρ˜ }, and the probability of each state in the ensemble
j|i
sendmixedstates,andthismixingreducestheamountof is P(i|j) = P(j|i)P(i)/P(j), with P(j|i) = Tr[A†A ρ ].
j j i
informationthat canbe transmitted. However,if the re- The Holevo quantity for ensemble j is thus
ceiverisnotabletoperformmeasurementswhichalways
project the system to a pure state (so called complete χ =S(ρ˜ )− P(i|j)S(ρ˜ ), (3)
j j j|i
measurements), then in general the information will be
i
X
2
whereρ˜ =A ρA†/Tr[A†A ρ]. Ifatleastoneofthemea- welearnaboutthefinalstateofthesystem. Equivalently,
j j j j j
surement operators A are higher than rank 1, then the itcanbe saidtomeasurethe degreeof“state-reduction”
j
measurement is incomplete. If the measurement is com- which the measurement induces.
plete,thenforeachj allthefinalstatesρ˜ areidentical, While it is the mutual informationwhich is important
j|i
χ is zero and the SWW bound reduces to the Holevo for communication, the reduction in the von Neumann
j
bound. entropyisimportantforfeedbackcontrol. Feedbackcon-
The most general kind of measurement can also be trol is the process of performing a sequence of measure-
inefficient. A measurement is described as inefficient if mentsonasystem,andapplyingunitaryoperationsafter
the observer does not have full information regarding each measurement in order control the evolution of the
which of the outcomes actually occurred. The name in- system. Such a procedure is useful for controlling sys-
efficient comes from that fact that the need to consider tems which are drivenby noise. If the ability to perform
such measurements first arose in the study of inefficient unitary operations is unlimited, then the von Neumann
photo-detectors. [8] An inefficient measurement may be entropy provides a measure of the level of control which
described by labeling the measurement operators with canbeachieved: ifthesystemhasmaximalentropythen
two indices, so that we have A . The receiver has com- theunitaryoperationshavenoeffectonthe systemstate
kj
plete information about one of the indices, j, but no in- whatsoever; conversely, if the state is pure then the sys-
formation about the other, k. [9] As a result the final tem can be controlled precisely - that is, any pure state
state for each j (given the value of i) is now can be prepared. Thus the entropy measures the extent
towhichapurestate,orpureevolutioncanbeobtained,
ρ′ = P(k|j) AkjρiA†kj . (4) andthusthelevelofpredictabilitywhichcanbeachieved
j|i Tr[A† A ρ ] overthe future behaviorofthe system[10]. Theprimary
Xk kj kj i role of measurement in feedback control is therefore to
reduce the entropy of the system. As such the average
Sinceinefficiencyrepresentsalossofinformation,wewish
reductioninvonNeumannentropyprovidesarankingof
toaskwhetheritispossibletotakethisintoaccountand
the effectiveness of different measurements for feedback
obtain a more stringent bound on the mutual informa-
control,otherthingsbeingequal. Furtherdetailsregard-
tion. IfwemerelyapplytheSWWboundtothemeasure-
ingquantumfeedbackcontrolandvonNeumannentropy
mentA ,thentheboundinvolvesthe Holevoquantities
kj
can be found in reference. [11]
of the ensembles that remain when both the values of k
The entropy reduction is also relevant to the transfor-
andj areknown(thefinalensemblesthatresultfromthe
mation of pure-state entanglement, since the von Neu-
efficient measurement). That is
mannentropymeasuresthe entanglementofpure states.
Asaresultthisquantitygivestheamountbywhichpure-
M(I:J)≤χ− P(k,j)χ . (5)
kj
state entanglement is broken by a local measurement.
kj
X We give two corollaries of the general information
One therefore wishes to know whether it is possible to bound derived in the first part which involve h∆S(ρ)i.
derive a bound which instead involves the Holevo quan- The first is a generalization of a bound derived by
tities of the ensembles that remain after the inefficient Hall [12, 13] to inefficient measurements. Hall’s bound
measurement is made, that is, for the receiver who only states that for efficient measurements the mutual infor-
has access to j. mation is bounded by h∆S(ρ)i. We show that for ineffi-
In the first part of this paper we answer this question cient measurements this becomes
in the affirmative - for an inefficient measurement where
M(I:J)≤h∆S(ρ)i− P(i)h∆S(ρ )i, (7)
the known outcomes are labeled by j, the bound given i
byEq.(2)remains true,wherenowtheχj aretheHolevo Xi
quantitiesfortheensembleofstatesρ′ whichresultfrom
j|i where h∆S(ρi)i is the average entropy reduction which
the inefficient measurement. would have resulted if the initial state had been ρ , and
i
In the second part of the paper, we consider the aver- as above ρ= P(i)ρ .
i i
age reductionin the vonNeumann entropy induced by a The second is the fundamental property that, for all
measurement: quantum operPations, the average reduction in von Neu-
mann entropy is concave in the initial state ρ. That is
h∆S(ρ)i≡S(ρ)− P(j)S(ρ′). (6)
j
i
X
h∆S(ρ)i≥ P(i)h∆S(ρ )i. (8)
Here ρ′ is the state thatresults fromoutcomej, given i
j i
thattheinitialstateisρ. SincethevonNeumannentropy X
is a measure of how much we know about the state of Finally,inthethirdpartofthispaper,weusetheabove
the system, this is the difference between what we knew result to show that for measurements which are uniform
aboutthesystemstatebeforewemadethemeasurement, in their sensitivity across state-space (that is, measure-
andwhatweknow(onaverage)aboutthesystemstateat ments which are unitarily covariant), the amount which
the end of the measurement;it thus measures how much one learns about the final state always increases with
3
the initial uncertainty, where this uncertainty is charac- Proof. Webeginbycollectingvariouskeyfacts. Thefirst
terized by majorization. This is a quantum version of is that any efficient measurement on a system Q, de-
the much simpler classical result (which we also show) scribed by N = N N operators, A , (j = 1,...,N
1 2 kj 1
that the mutual information always increases with the and k = 1,...,N ) can be obtained by bringing up an
2
initial uncertainty for classical measurements which are auxiliary system A of dimension N, performing a uni-
permutation symmetric. In addition we show that, for tary operation involving Q and A, and then making a
unitarily covariant measurements, the mutual informa- von Neumann measurement on A. [14, 15] If the initial
tionforpure-stateensemblesalsohasthis property. One state of Q is ρ(Q), then the final joint state of A and Q
can sum up these results by saying that the statement after the von Neumann measurement is
that information gain increases with initial uncertainty
can fail to hold only if the measurement is asymmetric σ(AQ) =|kjihkj|(A)⊗ Akjρ(Q)A†kj. (13)
in its sensitivity. P(k,j)
where|kjiisthestateofAselectedbythevonNeumann
II. AN INFORMATION BOUND FOR measurement. Thesecondfactisthatthestatewhichre-
GENERAL QUANTUM OPERATIONS sults fromdiscarding allinformationabout the measure-
ment outcomes k and j can be obtained by performing
We now show that the bound proved by SWW can be a unitary operation between A and another system E
generalized to obtain a more stringent bound for chan- which perfectly correlates the states |kji of A with or-
nelsinwhichthereceiversmeasurementisinefficient. To thogonal states of E, and then tracing out E. The final
show this it turns out that we can use the same method keyfactwerequireisaresultprovenbySWW[6],which
employed by SWW, but with the addition of an extra isthattheHolevoχquantityisnon-increasingunderpar-
quantum system which allows us to include the ineffi- tial trace. That is, if we have two quantum systems A
ciency of the measurement. and B, and an ensemble of states ρ(AB) with associated
i
probabilities P , then
Theorem 1. For a quantum channel in which the en- i
coding ensemble is ε = {P(i),ρi}, and the measurement χ(A) = S(ρ(A))− S(ρ(A))
performed by the receiver is described by operators A i
kj
i
( A† A =1), where the measurement is in general X
kj kj kj ≤ S(ρ(AB))− S(ρ(AB))=χ(AB), (14)
inefficient so that the receiver knows j but not k, then i
thPe mutual information, M(I:J), is bounded such that Xi
M(I:J)≤χ− P(j)χj, (9) where ρ(iA) =TrB[ρ(iAB)]. To provethis result SWW use
strong subadditivity. [16]
j
X We now encode informationin system Q using the en-
where P(j) is the overall probability for outcome j, χ = semble ε, and consider the joint system which consists
S(ρ)− iP(i)S(ρi) is the Holevo quantity for the initial of the three systems Q, A, E and a forth system M,
ensemble and with dimension N . We now start with A, E and M in
P 1
purestates,sothattheHolevoquantityforthejointsys-
χ =S(σ )− P(i|j)S(σ ), (10)
j j j|i tem is χ(QAEM) = χ(Q). We then perform the required
is the Holevo quantity for tXhe ensemble, ε , that remains unitary operation between Q and A, and a unitary op-
j
(from thepoint of view of the receiver) oncethe measure- eration between A and E which perfectly correlates the
ment has been made, so that the receiver has learned the states |kji(A) of A with orthogonalstates of E. Unitary
outcome j, but not the value of k. Here the receiver’s operations do not change the Holevo quantity. Then we
overall final state is trace over E, so that we are left with the state
σ = kAkjρA†kj = P(i,k|j)σ , (11) |ψihψ|(M)⊗ P(k,j)|k,jihk,j|(A)⊗Akjρ(Q)A†kj. (15)
j P(j) kj|i P(k,j)
P Xik Xjk
where P(i,k|j) is the probability for both i and outcome After the two unitaries and the partial trace over E, the
k given j, and σkj|i is the final state that results given Holevoquantityfortheremainingsystems,whichwewill
the initial state ρi, and both outcomes j and k. The re- denotebyχ′(QAM),satisfiesχ′(QAM) ≤χ(QAEM) =χ(Q).
maining ensemble εj ={P(i|j),σj|i}, where We now perform one more unitary operation, this time
between M and A, so that we correlate the states of M,
kAkjρiA†kj which we denote by |jihj|(M) with the second index of
σ = P(k|j,i)σ = , (12)
j|i kj|i P(j|i) the states of A, giving
k P
X
and where P(k|j,i) is the probability for outcome k given |jihj|(M)⊗ P(k,j)|k,jihk,j|(A)⊗σ(Q) (16)
kj
j and the initial state ρ .
i j k
X X
4
where σ(Q) = A ρ(Q)A† /P(k,j) is the final state re- ρ, does increase, the average increase in the entropy for
kj kj kj
sulting from knowing both outcomes k and j, with no each of the coding states ρi is always more that this by
knowledge of the initial choice of i. Finally we trace out at least the mutual information.
A, leaving us with the state The secondresultthatweobtainfromEq.(20)is that,
becausethe mutualinformationisnonnegative,we have
σ(QM) = |jihj|(M)⊗ P(k,j)σ(Q) (17)
kj
Xj Xk h∆S(ρ)i≥ P(i)h∆S(ρi)i. (22)
i
After this final unitary, and the partial trace over A, X
the Holevo quantity for the remaining systems Q and That is, the reduction in the von Neumann entropy is
M, which we will denote by χ′′(QM), satisfies χ′′(QM) ≤ concave in the initial state. This parallels the fact that
χ′(QAM) ≤ χ(Q). We have gone through the above pro- themutualinformationisalsoconcaveintheinitialstate.
cess using the initial state ρ, but we could just as eas- The fact that this is true for inefficient measurements,
ily have started with any of the initial states, ρi, in the meansthatoncewehavemadeanefficientmeasurement,
ensemble, and we will denote the final states which we nomatterwhatinformationwethrowawayregardingthe
obtain using the initial state ρ as σ(QM). Calculating final outcomes (i.e. which outcomes we average over),
i i
χ′′(QM) we have h∆S(ρ)iisalwaysgreaterthantheaverageoftheentropy
reductionswhichwouldhavebeenobtainedthroughmea-
surement in each of the coding states, when we throw
χ′′(QM) = S(σ(QM))− P(i)S(σ(QM)) away the same information regarding the measurement
i
i results.
X
= H[J]− P(i)H[J|i]
i
X
IV. INFORMATION GATHERING AND
+ P(j) S(σ )− P(i|j)σ (18) STATE-SPACE SYMMETRY
j j|i
" #
j i
X X
= M(J :I)+ P(j)χ(Q) ≤χ(Q). (19) Inthissectionweshowthatmeasurementswhoseabil-
j ity to extract information is uniform over the avail-
j
X able state-space (that is, does not vary from point to
Rearranging this expression gives the desired result. pointinthestate-space)alwaysextractmoreinformation
(strictly, never extract less information) the less that is
known before the measurement is made. Thus, in this
III. PROPERTIES OF ENTROPY REDUCTION sense, one may regard “the more you know, the less you
get” as a fundamental property of measurement. We
will show that this is true both for the information ob-
Wenowrewritetheaboveinformationboundusingthe
tainedregardingthe finalstate(being h∆S(ρ)i), andthe
fact that P(i|j)P(j)=P(j|i)P(i). The result is
mutual information for a measurement on an ensemble
of pure states. We will consider here efficient measure-
M(I:J)≤h∆S(ρ)i− P(i)h∆S(ρ )i (20)
i ments only; no doubt inefficient measurements will also
i
X have this property, but only if the information which is
thrown away is also uniform with respect to the state-
where ρ = P ρ . Ozawa has shown that for efficient
i i i
space, and we do not wish to burden the treatment with
measurements h∆S(ρ)i is always positive[17] (for more
P this additional complication.
recentproofsofthisresultsee[18,19]). Forefficientmea-
surements Eq.(20) is therefore in general stronger than, To proceed we must make precise the notion that the
andgivesimmediately,Hall’sbound[12,13],whichstates sensitivity ofa measurementis uniformoverstate-space.
thatthemutualinformationisboundedbythereduction This is captured by stating that such a measurement
in the von Neumann entropy. The inequality in Eq.(20) should be invariant under reversible transformations of
isthenageneralizationofHall’sboundtoinefficientmea- the state-space. For classical measurements (which are
surements. Since the mutual information is always pos- simplyquantummeasurementsinwhichalloperatorsand
itive, but for inefficient measurements the reduction in density matrices commute [20]) this means that the set
the von Neumann entropy can be negative (that is the ofmeasurementoperatorsisinvariantunderallpermuta-
entropy of the quantum state can increase as a result of tionsoftheclassicalstates: wewillrefertotheseascom-
the measurement), the relation pletely symmetric measurements. Note that in this clas-
sical case, this is equivalent to saying that the measure-
M(I:J)≤h∆S(ρ)i (21) mentdistinguishesallstatesfromallotherstatesequally
well. The quantum generalization of this is invariance
is not necessarily satisfied for such measurements. How- under all unitary transformations. Such measurements
ever,Eq.(20)tellsusthatiftheentropyoftheintialstate, are referred to as being unitarily covariant. [21, 22]
5
Wemustalsoquantifywhatwemeanbytheobserver’s cause of the unitary covariance of the measurement, we
lack of knowledge, or uncertainty, before the measure- see from the form of h∆S(ρ)i that it is invariantunder a
mentismade. Thisiscapturedbythesimpleandelegant unitary transformationofρ. As a result,it only depends
concept of majorization. [23, 24] If two sets of probabil- upon the eigenvalues of ρ. Since the permutations are
ities p ≡ {P } and q ≡ {Q } satisfy the set of relations a subgroup of the unitaries, it is also a symmetric func-
i i
tion of its arguments (the eigenvalues), and thus Schur-
concave.
k k
P ≥ Q , ∀k, (23) Wewishfinallytoshowthatthemutualinformationis
i i
also Schur-concave in ρ for unitarily covariant measure-
i=1 i=1
X X ments on ensembles of pure states. This requires a little
whereitisunderstoodthattheelementsofbothsetshave
more work. First we need to show that once we have
beenplacedindecreasingorder(e.g.,P >P ,∀i),then
i i+1 fixed a set of encoding states, the mutual information is
p is said to majorize q, and this is written q ≺ p. While
concave in the vector of the ensemble probabilities P(i).
atfirstEq.(23)looksalittlecomplicated,afewmoments
This is straightforward if we first note that the mutual
considerationreveals that it captures precisely what one
information, because it is, in fact, symmetric between i
means by uncertainty - if p majorizes q, then p is more
and j, can be written in the reverse form
sharplypeakedthanq,andconsequentlydescribesastate
of knowledge containing less uncertainty. What is more, M(I:J) = H[P(j)]− P(i)H[P(j|i)], (24)
majorization implies an ordering with Shannon entropy
j
X
H[·]. Thatis,if pmajorizesq, thenH[p]≤H[q].[23,24]
Since,forafixedmeasurement,themutualinformationis
Inasense,majorizationisamorebasicnotionofuncer-
afunctionoftheensembleprobabilitieswewillwriteitas
taintythanentropyinthatitcapturesthatconceptalone
M({P(i)}). Denotingthepurestatesintheencodingen-
– the Shannon entropy on the other hand characterizes
semble as ρ =|ψ ihψ |, and choosing the ensemble state
the more specific notion of information. To characterize i i i
ρ = P σ , where the σ are built from the encoding
the uncertainty of a density matrix, we can apply ma- k k k k
states so that σ = P |ψ ihψ |, then
jorization to the vector consisting of its eigenvalues. If ρ k i i|k i i
P
and σ are density matrices, then we will write σ ≺ ρ if M({P(i)}) P
ρ’s eigenvalues majorize σ’s. Various applications have
= H[ P(k)P(j|k)]− P(i|k)P(k)H[P(j|i)]
beenfoundformajorizationinquantuminformationthe-
ory. [18, 19, 25, 26, 27, 28] Xk Xi Xk
Wethusdesiretoshowthatformeasurementswiththe ≥ P(k)H[P(j|k)]− P(k) P(i|k)H[P(j|i)]
specified symmetry, h∆S(σ)i≥h∆S(ρ)i whenever σ ≺ρ
k k i
X X X
(and similarly for the mutual information). Functions
= P(k)M({P(i|k)}), (25)
with this property (of which the von Neumann entropy,
k
S(ρ), is one example) are referred to as being Schur- X
concave. To show that a function is Schur-concave, it is being the desired concavity relation. The inequality in
sufficient to show that it is concave, and symmetric in the third line is merely a result of the concavity of the
its arguments [23, 24], which in our case are the eigen- Shannon entropy. Note that while we have written the
values of the density matrix ρ (if our functions did not measurement’s outcomes explicitly as being discrete in
depend only onthe eigenvaluesof ρ, then they couldnot the about derivation, the result also follows if they are
be Schur-concave, since the majorization condition only a continuum (as in the case of UC measurements) by
involves these eigenvalues). replacing the relevant sums with integrals.
The desired result for classical completely symmetric Now we need to note some further points about UC
measurementsisnowimmediate. Intheclassicalcasethe measurements: A UC measurement may be generated
mutualinformationistheuniquemeasureofinformation by taking all unitary transformations of any single oper-
gain, and M(I : J) = h∆S(ρ)i. The mutual informa- ator A, and dividing them by a common normalization
tion is concave in the initial classical probability vector factor. The resulting measurement operators are thus
P = (P1,...,Pn) (being the vector of the eigenvalues of AU ∝ UAU†, where U ranges over all unitaries. The
ρinourquantumformalism),asisindeedimpliedbythe normalizationfortheAU comesfrom UA†AU†dµ(U)=
concavity of h∆S(ρ)i. Since all operators commute with Tr[A†A]I where dµ(U) is the (unitarily invariant) Haar
R
thedensitymatrix,h∆S(ρ)iisonlyafunctionofthe{P }. measure [22, 29] over unitaries.
i
From the form of h∆S(ρ)i we see that a permutation of ItisnothardtoshowthatallUCmeasurementscanbe
the elements of P is equivalent to a permutationapplied obtainedbymixingdifferentUCmeasurements,eachgen-
tothemeasurementoperators,andsincetheseareinvari- eratedby adifferent operator. (Mixing a setofmeasure-
antundersuchanoperation,h∆S(ρ)i,andthusM(I:J), ments means assigning to each a probability, and then
is a symmetric function of its arguments. Thus M(I:J) making one measurement from the set at random based
is Schur-concave. on these probabilities [30]).
TheSchur-concavityofh∆S(ρ)iforunitarilycovariant Next, we need to show that for all UC measurements
(UC) quantum measurements is just as immediate. Be- the mutual information depends only on the eigenvalues
6
of the ensemble density matrix, and we state this as the V. CONCLUSION
following lemma.
Lemma 1. The mutual information for a UC measure- In using a quantum channel, if there are limitations
onthe completeness(or alternativelythe strength,inthe
ment on a pure-state ensemble, ε={P(i),|ψ i} depends
i
terminology of [19]) or efficiency of the measurements
on the ensemble only through the eigenvalues of the den-
that the receiver can perform, then it is possible to give
sity matrix ρ= P(i)|ψ ihψ |.
i i i a bound on the mutual information which is stronger
Proof. WefirstsPhowthisforUCmeasurementsgenerated than the Holevo bound. Further, this bound has a very
from a single operator. Writing the mutual information simple form in terms of the Holevo χ quantity, and the
in the reverse form one has χquantitiesofthe ensembles,oneofwhichremainsafter
the measurement is made.
M(I:J)) = H[P(U)]− P H[P(U|i)], (26) This bound also allows us to obtain a relationship be-
i
i tween the mutual information and the average von Neu-
X
mannentropyreductioninduced by ameasurement,and
whereU isthecontinuumindexforthemeasurementop-
encompassesthe factthat this vonNeumannentropyre-
erators (and thus the measurementoutcomes) which are
duction is concave in the initial state.
A =UAU† for some appropriately normalized A. Nat-
U Fromthe concavityofthe mutualinformationandthe
urally all this means is that P(U|i) is a function of U,
von Neumann entropy reduction, it follows that these
whereU rangesoverallunitaries. Sincethemeasurement
quantities are Schur-concave (the former naturally for
isunitarilycovariant,H[P(U|i)]isthesameforallinitial
pure-state ensembles) for completely symmetric classi-
states |ψ i, and therefore the second term is the same
i cal measurements, and for unitarily covariant quantum
for all initial ensembles. Thus M depends only on the
measurements. Thus the possibility that either of these
first term H[P(U)] = H[Tr[UA†AU†ρ]], which depends
kinds of information gain decreases with increasing ini-
only on ρ, and is invariant under all unitary transforma-
tial uncertainty is associated with the asymmetry of the
tions of ρ. Thus M depends only on the eigenvalues of
measurement in question.
ρ. Since the mutual information for a mixture of mea-
surements is merely a function of the respective mutual
informations for each measurement (in particular it is a
Acknowledgments
linear combination of them), the result holds for all UC
measurements.
The author would like to thank Gerard Jungman,
Since M depends only on ρ, in establishing the Schur Howard Barnum, Howard Wiseman, Terry Rudolph and
concavity of M with respect to ρ, we need only consider Michael Hall for helpful discussions. The author is also
one ensemble for each ρ. We therefore choose the eigen- grateful both to Vlatko Vedral for hospitality during a
ensemble{λ ,|φ i},whereλ and|φ iaretheeigenvalues visit to ImperialCollege, andLucien Hardy for hospital-
i i i i
andeigenvectorsofρrespectively. Weknowthatthemu- ity during a visit to the Perimeter Institute where some
tualinformationisconcaveinthevectorofinitialensem- of the initial stages of this work was carried out. This
ble probabilities, and for the ensemble we have chosen, work was supported by the Australian Research Coun-
the initial probabilities are the eigenvalues of ρ. As a re- cil and the State of Queensland. Note added: After
sultthe mutualinformationisconcaveinthe eigenvalues submitting this manuscript, which was first posted as
ofρ. SinceM isinvariantunderunitarytransformations, eprint quant-ph/0412006, it was brought to my atten-
and since unitary transformations include permutations tion that the work presented here overlaps with concur-
asasubgroup,itisalsoasymmetricfunctionoftheeigen- rent work by Barchielli and Lupeiri (quant-ph/0409019
values. Thus M is Schur-concave. and quant-ph/0412116).
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