We present optimal and minimal measurements on identical copies of an unknown
state of a qubit when the quality of measuring strategies is quantified with
the gain of information (Kullback of probability distributions). We also show
that the maximal gain of information occurs, among isotropic priors, when the
state is known to be pure. Universality of optimal measurements follows from
our results: using the fidelity or the gain of information, two different
figures of merits, leads to exactly the same conclusions. We finally
investigate the optimal capacity of $N$ copies of an unknown state as a quantum
channel of information.Comment: Revtex, 5 pages, no figure

A. Hobson

A. Peres

A. Uhlmann

A.S. Holevo

C.A. Fuchs

E.B. Davies

G. Vidal

Guifré Vidal

J.I. Latorre

R. Derka

R. Josza

Rolf Tarrach

S. Kullback

S. Massar

English

arXiv

We present optimal and minimal measurements on identical copies of an unknown state of a quantum bit when the quality of measuring strategies is quantified with the gain of information (Kullback-or mutual information-of probability distributions). We also show that the maximal gain of information occurs, among isotropic priors, when the state is known to be pure. Universality of optimal measurements follows from our results: using the fidelity or the gain of information, two different figures of merits, leads to exactly the same conclusions for isotropic distributions. We finally investigate the optimal capacity of N copies of an unknown state as a quantum channel of information

Tarrach, R., 1948-

Vidal Bonafont, Guifré

Diposit Digital de la Universitat de Barcelona

RAPID COMMUNICATIONSPHYSICAL REVIEW A NOVEMBER 1999VOLUME 60, NUMBER 5Universality of optimal measurementsRolf Tarrach and Guifre´ Vidal*Departament d’Estructura i Constituents de la Mate`ria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain~Received 30 July 1999!We present optimal and minimal measurements on identical copies of an unknown state of a quantum bitwhen the quality of measuring strategies is quantified with the gain of information ~Kullback—or mutualinformation—of probability distributions!. We also show that the maximal gain of information occurs, amongisotropic priors, when the state is known to be pure. Universality of optimal measurements follows from ourresults: using the fidelity or the gain of information, two different figures of merits, leads to exactly the sameconclusions for isotropic distributions. We finally investigate the optimal capacity of N copies of an unknownstate as a quantum channel of information. @S1050-2947~99!51311-5#PACS number~s!: 03.67.2a, 03.65.BzConsider an unknown state of a two-level quantum sys-tem described by the density matrix r(bW ), bW being the Blochvector, b[ubW u<1. The preparation device provides N iden-tical copies of the system, so that the state at our disposal isr(bW ) ^ N. In the past few years the optimal measuring strat-egy, i.e., the most successful at revealing the identity of theunknown state, has been obtained, first for pure states @1–3#and then for mixed states @4#. Also the minimal ones amongthe optimal strategies, i.e., the ones with the smallest numberof outcomes, have been constructed, both for pure states @5#and mixed states @4#. In the processing of information con-tained in quantum states, knowing the most efficient readoutprocedures, i.e., the optimal and least resource consumingones, is of course of importance.In all these contributions the quality of the measuringstrategy, characterized by a resolution of the identity(iM i51, ~1!in terms of positive operators M i>0, has been quantified bythe fidelity @6#. In other words, when outcome i ~related toM i) happens, one guesses the unknown state to be r˜ i[r(pW i) and one quantifies the quality of the guess byFr~bW !,r~pW i![$Tr@Ar~bW !1/2r~pW i!r~bW !1/2#%2. ~2!One can arrive at Eq. ~2! from several different startingpoints. One of them is based on a measure of distinguishabil-ity of the probability distributions associated with r and r8by performing general positive operator valued measure-ments @as in Eq. ~1!# on them @7# and minimizing,F~r ,r8!5minS (j ATr@rM j#ATr@r8M j# D2. ~3!Another is based on the standard Hilbert-space scalarproduct of the two pure states, which, belonging to C 2^ C 2, lead to r and r8 when reduced @8#,F~r ,r8!5maxz^cuc8& z2, ~4!where maximization is performed over $uc&,uc’&%/r5Tra@ uc&^cu# , r85Tra@ uc’&^c’u# .*Electronic address: guifre@ecm.ub.esPRA 601050-2947/99/60~5!/3339~4!/$15.00These equivalent definitions of the fidelity, plus thefollowing properties that characterize it further, make it aunique quantification of the comparison of two general quan-tum states: ~i! 0<F~r,r8!5F~r8,r!<1. ~ii! F~r,r8!51,r5r8; F~r,r8!50,rr850. ~iii! F(UrU†,Ur8U†)5F(r ,r8), UU†5U†U51. ~iv! F(uc&^cu,r)5^curuc&.~v! F(r ^ s ,r8^ s8)5F(r ,r8)F(s ,s8). ~vi! F@r ,pr11(12p)r2#>pF(r ,r1)1(12p)F(r ,r2),0<p<1. In Refs.@2,3,5# the unknown state was known to be pure, b51, butno knowledge of the direction of the Bloch vector was as-sumed. In reference @4# the unknown state was a mixed statedrawn stochastically from a known isotropic distributionf (b), and although the best guess r˜ i depended on f (b), theoptimal measuring strategy, that is, the set $M i% of positiveoperators of the different outcomes, did not. For isotropicdistributions optimal measurements are thus distribution, i.e.,f (b), independent.However, proposing an outcome-dependent guess andevaluating its quality through the fidelity are only two of thecriteria that could have been used to define optimal measure-ments. A sound alternative, the one we shall investigate inthis work and probably the most sensible choice in the con-text of quantum information theory, consists of quantifyingthe quality of measuring strategies through the gain of infor-mation about the unknown state. In fact, information theoryalready supplies a universally accepted, unambiguousscheme for this purpose, which we shall follow. It is basedon the Bayes formula, which provides a conditional~outcome-dependent!, posterior distribution f c(bW ui) from the~here isotropic! prior distribution f (b), and on the Kullbackformula, which quantifies the gain of information acquiredwhen replacing f (b) with f c(bW ui).More specifically, if Pi(bW )[Tr@r(bW )M i# is the probabil-ity of outcome i when the unknown state is r(bW ) andPap~ i ![E d3b f ~b !Pi~bW !S E d3b f ~b !51 D ~5!is the a priori probability of outcome i, then the Bayes for-mula states that the posterior distribution f c(bW ui), the onethat collects our knowledge about the unknown state r(bW )after measuring when the initial knowledge is given by f (b),readsR3339 ©1999 The American Physical SocietyRAPID COMMUNICATIONSR3340 PRA 60ROLF TARRACH AND GUIFRE´ VIDALf c~bW ui !5 f ~b !Pi~bW !/Pap~ i ! . ~6!The gain of information about r(bW ), DI , is then given, inbits, by the Kullback formula of f c(bW ui) relative to f (b) @12#Ki@ f c / f #[E d3b f c~bW ui !log2@ f c~bW ui !/ f ~b !# . ~7!This expression, the only one satisfying a series of intuitivelyreasonable conditions @13#, is well-defined for continuousdistributions ~it has no dependence on the measure in thespace of quantum states!, and its average over possible out-comes,K¯ @ f c / f #[(iPap~ i !Ki@ f c / f # , ~8!is precisely the difference of the a priori and average a pos-teriori entropies H of the corresponding probability distribu-tions of states,H@ f #2H¯ @ f c#[2E d3b f ~b !log2 f ~b !1(iPap~ i !3E d3b f c~bW ui !log2 f c~bW ui !, ~9!as can be checked by considering Eqs. ~6!–~8! and that( iPi(bW )51 @9#. This quantification is therefore equivalent tothe one already used in previous works on quantum-stateestimation with discrete distributions ~see, e.g., Ref. @10#!.First, the question of which are the optimal measurementsaccording to this information theoretically based criterionwill be addressed. We will check explicitly for N51 andN52, and provide clues for any N, that optimal—and alsominimal—measuring strategies are universal, i.e., indepen-dent of whether the fidelity or the increase of information isused for their quantification @11#, and will compute the cor-responding optimal gain of information DI . Then we willmove to consider which is the isotropic prior f (b) for whichoptimal measurements extract most information, so that itcorresponds to the optimal ~isotropic! quantum channel ofinformation. After introducing a reversible compression pro-cedure we conclude that the optimal amount of extractableinformation is, as N→‘ , of one bit per effective quantum bit~qubit! isotropic distributions.In order to find an optimal measuring strategy, i.e., a setof operators M i as in Eq. ~1! maximizing the gain of infor-mation @Eq. ~8!#, the following theorem and subsequent cor-ollaries, valid for any number of copies N, will be very use-ful.Theorem. Let the positive operator M i>0 be such that itsprobability Pi(bW )5Tr@M ir(bW ) ^ N# can be written, for any bW ,as the sum of two contributions of the form Pi ,k(bW )[Tr@M i ,kr(bW ) ^ N# , k51,2, where the operators M i ,1 ,M i ,2are also positive ~and M i ,11M i ,2 is not necessarily equal toM i). Let us introduce corresponding prior probabilitiesPap(i ,k) and posterior distributions f c(bW ui ,k) as in Eqs. ~5!and ~6!. Then,Pap~ i !Ki@ f c / f #<(k512Pap~ i ,k !Ki ,k@ f c / f # . ~10!Proof. It follows from the inequality~x11x2!lnx11x2y11y2<x1 lnx1y11x2 lnx2y2, ~11!; x1 ,x2 ,y1 ,y2>0. jCorollary 1. An optimal measuring strategy with rank-1operators always exists ~cf. @14#!.Proof. Indeed, suppose ( iM i51 corresponds to an opti-mal measurement. Then, if M i5(kui ,k&^i ,ku is the spectraldecomposition of M i , it follows from the theorem that therank-1 positive operator valued measurement ( i ,kui ,k&^i ,ku51 is also optimal. jWe can already consider the case N51, that is, whenonly one copy of the unknown state is available. One canconvince oneself immediately that an optimal ~and also mini-mal! measurement is just a standard von Neumann measure-ment. In fact, any will do because of the isotropy of f (b).Suppose that we measure sz . Then, for bW5(b sin u cos f,b sin u sin f,b cos u), we havef c~bW u6 !5~16b cos u! f ~b ! ~12!and the gain of information isDI (1)5pE01db b2 f ~b ![~11b !2/b]log2~11b !2@~12b !2/b#log2~12b !2log2 e/2 . ~13!The function in square brackets in Eq. ~13! is monotoni-cally increasing, so that the distribution for which the abso-lute increase in knowledge is maximal isf m(1)~b !5~1/4p!d~b21 !, ~14!i.e., an isotropic distribution of pure states.It is interesting to point out that, if instead of using in Ref.@4# the mean average fidelity F¯ (1) we had used the meanaverage increase in fidelity,DF (1)[F¯ (1)2Fap(1), ~15!with the optimal guess r˜ 0[r(0) if no measurement wasperformed, so thatFap(1)5 12 1I1/25Fap(N) ~16!with ~cf. @4#!Ia[4pE01db b2 f ~b !@~12b2!/4#a ~17!(I051,Ia>4Ia11), we would have obtainedDF (1)5AI1/22 1 136 ~124I1!22I1/2 . ~18!It is then easily verified that the maximum value of DF (1)also corresponds to the distribution equation ~14!. Thus, forN51, quantifying with the fidelity or with the Kullback in-formation leads to the same ~for N51 somewhat obvious!optimal and minimal measuring strategy and to the samedistribution that maximizes DI (1) and DF (1). Is this also truefor N52?In order to answer this question we need to present asecond corollary. Notice first that with the following notation~borrowed from @4#! for the composite Hilbert space of Ncopies of the unknown state r(bW ),H (N)[HA ^ HB ^ . . . HN , ~19!RAPID COMMUNICATIONSPRA 60 R3341UNIVERSALITY OF OPTIMAL MEASUREMENTSfor the corresponding local spin operators,SW A[12sW ^ I ^ N21,SW B[12 I ^ sW ^ I ^ N22,~20!ASW N[12 I^ N21^ sW ,and for the partial and total spin operators,SW (a)[ (b5AaSW b , a5A ,B , . . . ,N , SW [SW (a5N) , ~21!the following spin invariances hold @4#:@SW (a)2,r ^ N#50, a5A , . . . ,N , ~22!and since@SW (a)2,SW (b)2 #50, ; a ,b , ~23!the total Hilbert space can be written as a direct sumH (N)5 % $s(a)%E $s(a)% , ~24!where E $s(a)% are the simultaneous eigenspaces of all the op-erators SW (a)2,;aÞA , with corresponding eigenvalues$sa(sa11)%, ordered with decreasing a ~see @4# for moredetails!. For instance, for N52 only SW (B)2 (s (B)) is relevant,i.e., E $s(a)%5Es(B), and the decomposition readsH (N52)5E1 % E0 , ~25!where E1 is the triplet or symmetric ~under exchange of cop-ies! subspace, with total spin s[s (B)51, whereas E0 is thesinglet or antisymmetric subspace, with total spin s50.Then, we have the following.Corollary 2. There always exists an optimal measuringstrategy consisting only of rank-1 operators of the formu$s (a)%&^$s (a)%u, where the not necessarily normalized vectoru$s (a)%& is an eigenvector of all partial and total spin opera-tors, i.e.,SW (b)2 u$s (a)%&5s (b)~s (b)11 !u$s (a)%&, ; b , ~26!and thus it belongs to the subspace E $s(a)% .Proof. Let ( iM i51 correspond to an optimal measure-ment with rank-1 operators M i5ui&^iu ~where the ui& do notneed to be orthogonal nor normalized! and let P$sa%5P$sa%2be a projector onto the whole subspace E $sa% . Then it fol-lows from Eq. ~24! that($sa%P$sa%1P$sa%5($sa%P$sa%51, ~27!so that if we replace 1 with ( iM i in the left-hand side of thisequation, we obtain a new measurement(i ,$sa%ui ,$sa%&^i ,$sa%u51, ui ,$sa%&[P$sa%ui&. ~28!Now, since Eq. ~22! implies that for each ui&,Tr@r~bW ! ^ Nui&^iu#5($sa%Tr@r~bW ! ^ Nui ,$sa%&^i ,$sa%u# , ~29!the theorem guarantees that the measurement of Eq. ~28! isalso optimal. j~Notice that exactly the same conclusion was alsoachieved, for any N, when the fidelity was used as a criterionfor optimality @4#, this being indicative of the universality weare considering here.!Thus, in order to find an optimal measuring strategy forN52 we can always choose the pure states on which themeasurement projects to be symmetric or antisymmetric un-der the exchange of the two qubits. Let us next computeDI (2) for the optimal strategy of Ref. @4#, that is, correspond-ing to a resolution of the identity of the form15us&^su134 (i514~ unˆ i&^nˆ iu! ^ 2, ~30!where us& is the ~normalized! singlet state, sW nˆ unˆ &5unˆ &(^nˆ unˆ &51) and the four unitary vectors nˆ i point to the fourdirections of the vertices of a regular tetrahedron. Onereadily obtainsf c~bW us!5 [~12b2!/4]f ~b !Pap~s!, Pap~s!5I1 , ~31!f c~bW unˆ !5 316 ~11bW nˆ !2@ f ~b !/Pap~nˆ !# ,Pap~nˆ !512 ~12I1!, ~32!so thatDI (2)5pE01db b2 f ~b !$[~11b !3/b#log2~11b !2@~12b !3/b#log2~12b !1~12b2!log2~12b2!%2~12I1!$2 log2 e/3 1log2@~12I1/3#%2I1 log2 I122. ~33!Can we do better, i.e., is there another resolution of the iden-tity that leads to a larger DI (2)? Let us prove that there isnone. Because of corollary 2, the whole question boils downto whether symmetric entangled states could do better thanthe symmetric product states unˆ i&unˆ i& used in Eq. ~30!. Con-sider therefore a general symmetric state of Schmidt decom-positionuc&5Apu1&u1&1A12pu2&u2&, pP@0,1# , ~34!where the isotropy of f (b) has been taken into account inchoosing the basis. One can readily obtain the average Kull-back information corresponding to this state,DIc(2)512E01db b2 f ~b !E02pdfE211dmh log2 h/@~12I1!/3# ,h[k1l cos 2f , l[2Ap~12p !b2~12m2!,k[11b2m21~2p21 !2bm , ~35!which after integration of f givesDIc(2)5p2 E01db b2 f ~b !E211dm$~11b2m2!3log2@3e/2~12I1!# 1k log2~k1Ak22l2!%.~36!RAPID COMMUNICATIONSR3342 PRA 60ROLF TARRACH AND GUIFRE´ VIDALThis is a function of p that we want to maximize. Onlyk log2(k1Ak22l2) depends on p. The part 2l2 is maximizedfor p50 and p51. The other part, too, as one can see easilyneglecting the term l2. Thus DIc(2) is maximized when uc& isa product state and the resolution of Eq. ~30! is indeed opti-mal.As we did for N51, it is interesting to recall, with thehelp of Ref. @4#, the average increase in fidelity for N52DF (2)5A~I1/22I3/2!21 116 ~124I1!21I3/22I1/2 . ~37!One can now check that both DI (2) and DF (2) are againmaximized for the distribution equation ~14!. For DI (2) thisfollows by observing that the part in square brackets in Eq.~33! is an increasing function of b and that the other part,which depends on I1, increases as I1 goes towards zero.We have thus checked for N51 and N52 that both thefidelity and the Kullback information lead to the same opti-mal measuring strategy and to the same, pure-state distribu-tion that maximizes their increases. We conjecture, while notforeseeing any feature that could jeopardize extending theproof to N.2, that the universality of optimal measurementsholds for any number N of copies of the unknown state @15#.Corollary 2 makes this conjecture very plausible. The preciseoptimal strategy is in fact determined to a great extent by theisotropy of the prior distribution, the symmetries of the stater(bW ) ^ N that allow us to choose each positive operator M i toact only on one of the subspaces E $s(a)% , and the fact thatboth the fidelity and the Kullback information favor strate-gies with outcomes i whose normalized probability of occur-rence Tr@r(bW )NM i#/Tr@M i# spans the largest possible rangeas a function of the direction of bW .Now, suppose we want to use the N qubits as a quantumchannel of classical information. Alice prepares N copies of agiven state r(bW ) ~the classical information being encoded inthe vector bW ) and sends them to Bob, who will perform acollective measurement in order to recover as much informa-tion about bW as possible. The previous results single out,using, when restricted to isotropic prior distributions, onlypure states (b51) to encode classical information as theoptimal method. We can then easily compute the optimalcapacity of this isotropic quantum channel for any N, to findthatDI (N)5log2~N11 !2@N/~N11 !#log2 e , ~38!which for large N gives log2 N/N bits carried per qubit. No-tice that this is a purely quantum channel, no additional flowof classical information being required at any stage. Its poorcapacity can be exponentially enhanced without spoiling thisfact if we take into account that a pure state f ^ N belongs tothe symmetric subspace S (N) of the whole Hilbert spaceH (N). Since the dimension of S (N) is N11, which corre-sponds to the dimension of a Hilbert space H (M ) of M[log2(N11) qubits, Alice can always compress, by meansof a state-independent, unitary ~and thus fully reversible!transformation, the state f ^ N to fit in M qubits, which willthen be transferred to Bob. In this case the capacity increasesup to 12O(1/log N) bits per qubit, which is asymptoticallythe classical one ~as expected, since for any two inequivalentstates f and f8, f ^ N, and f8^ N become orthogonal as N→‘), and which is consistent with the Levitin-Holevobound @16# for the classical capacity of a quantum channel.Summarizing, using the gain of information as a guide,we have constructed optimal and minimal measurements onN51,2 identical copies and have shown that for isotropicdistributions the maximal gain of information is achieved forpure states. Also the universality of optimal measurementshas been proven, since these measurements exactly coincidewith those obtained in previous work, where the fidelity wastaken as the figure of merit. We conjecture that also for N>3 the most informative measurements are the most faithfulones, and vice versa.G.V. acknowledges CIRIT Grant No. 1997FI-00068 PG.Financial support from CIRYT, Contract No. AEN98-0431;CIRIT, Contract No. 1998SGR-00026; and from the ESF-QIT program is also acknowledged. We thank Antonio Acinand Chris Fuchs for their comments.@1# A.S. Holevo, Probabilistic and Statistical Aspects of QuantumTheory ~North-Holland, Amsterdam, 1982!, Chap. IV.@2# S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 ~1995!.@3# R. Derka et al., Phys. Rev. Lett. 80, 1571 ~1998!.@4# G. Vidal et al., Phys. Rev. A 60, 126 ~1999!.@5# J.I. Latorre et al., Phys. Rev. Lett. 81, 1351 ~1998!.@6# A. Uhlmann, Rep. Math. Phys. 9, 273 ~1976!.@7# C.A. Fuchs and C.M. Caves, Open Syst. Inform. Dynam. 3,345 ~1995!; also e-print quant-ph/9604001.@8# R. Josza, J. Mod. Opt. 41, 2315 ~1994!.@9# Furthermore, and due to the symmetry in the i and bW distribu-tions of Bayes’ formula, Eq. ~6!, the expressions in Eqs. ~8!and ~9! are also equal to the corresponding expressions forwhich f and f c have been traded for Pap and P.@10# A. Peres and W.K. Wootters, Phys. Rev. Lett. 66, 1119 ~1991!.@11# Holevo @1# presented examples of the universal character ofoptimal measurements in the context of phase estimation, andrelated universality to convexity properties of the figures ofmerits ~as in our theorem!.@12# S. Kullback and R.A. Leibler, Ann. Math. Stat. 22, 78 ~1951!;S. Kullback, Information Theory and Statistics ~Wiley, NewYork, 1959!.@13# A. Hobson, J. Stat. Phys. 1, 383 ~1969!.@14# E.B. Davies, IEEE Trans. Inf. Theory IT-24, 596 ~1978!.@15# We have also been able to check, for an arbitrary number N ofcopies of the unknown state, that the optimal measurementsaccording to the fidelity ~as presented in @4#! are at least locallyoptimal ~that is, better than any other measurement that fol-lows from infinitesimally perturbing the former! for the Kull-back information.@16# L.B. Levitin ~unpublished!; A.S. Holevo, Probl. Inf. Transm.9, 177 ~1973!.

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Universality of optimal measurements

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Universality of optimal measurements

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