Asymptotically Optimal Discrimination between Pure Quantum States

Transcription

1 Asymptotcally Optmal Dscrmnaton between Pure Quantum States Mchael Nussbaum 1, and Arleta Szko la 2 1 Department of Mathematcs, Cornell Unversty, Ithaca NY, USA 2 Max Planck Insttute for Mathematcs n the Scences, Lepzg, Germany Abstract. We consder the decson problem between a fnte number of states of a fnte quantum system, when an arbtrarly large number of copes of the system s avalable for measurements. We provde an upper bound on the exponental rate of decay of the averaged probablty of rejectng the true state. It represents a generalzed quantum Chernoff dstance of a fnte set of states. As our man result we prove that the bound s sharp n the case of pure states. Keywords: multple quantum state dscrmnaton, generalzed quantum Chernoff dstance, quantum hypothess testng, error exponents. 1 Introducton In varous branches of quantum theory such as quantum nformaton processng, quantum communcaton theory or quantum statstcs one of the basc problems s to determne the state of a gven quantum system. In the smplest case there s a fnte set of states specfyng the possble preparaton of the quantum system. In the Bayesan approach of quantum statstcs, the lkelhood of the dfferent states s determned by an a pror probablty dstrbuton. One makes a decson n favor of one of the states followng a specfed rule based on the outcomes of a generalzed measurement -called a quantum test. In the bnary case optmal tests,.e. tests mnmzng the averaged probablty of rejectng the true state, are known to be gven by Holevo-Helstrom projectons [5], [4]. These generalze the classcal lkelhood rato tests. Here we consder the scenaro where there s an arbtrarly large fnte number n of copes of the quantum system avalable for performng a measurement. The correspondng state s then descrbed by an n-fold tensor product of one of the assocated densty operators. There are two man goals: frstly, to construct a sequence of quantum tests n n whch maxmze the asymptotc exponental rate of decay of the averaged probablty of rejectng the true state. The second goal s to determne the correspondng optmal error exponent. It has been shown that n the bnary case asymptotcally optmal quantum tests, thus n partcular the Holevo-Helstrom tests, acheve an exponental rate of decay whch s equal to the quantum Chernoff bound, cf. [8], [1] and [2]. Surprsngly, the correspondng questons n the case of r>2 states Supported n part by NSF grant DMS W. van Dam et al. Eds.: TQC 2010, LNCS 6519, pp. 1 8, c Sprnger-Verlag Berln Hedelberg 2011

2 2 M. Nussbaum and A. Szko la have not yet receved a fnal answer, despte a number of efforts and numerous strong results obtaned n relaton to multple quantum state dscrmnaton, see [11], [7], [3], [10] and references theren. We defne a generalzed quantum Chernoff dstance of a fnte set of states as the mnmum of the bnary quantum Chernoff dstances over all possble pars of dfferent states. The bnary quantum Chernoff dstance has been ntroduced n the context of bnary quantum hypothess testng n [8]. Relyng on [8] we prove that the generalzed quantum Chernoff dstance specfes a bound on the achevable asymptotc error exponents n multple quantum state dscrmnaton. Ths s n lne wth results obtaned n the context of classcal multple hypothess testng, cf. [9]. As our man result we prove that n the specal case of pure quantum states ths bound, ndeed, s achevable and hence specfes the optmal asymptotc error exponent. The correspondng asymptotcally optmal quantum tests rely on a Gram-Schmdt orthonormalzaton procedure of the assocated state vectors. Smlar quantum tests were already consdered by Holevo n [6] n the context of quantum mnmal error decson problems. However, the queston of the correspondng asymptotc error exponent s not addressed n [6]. 2 Notatons and the Man Results Let S be a fnte quantum system and H be the assocated complex Hlbert space wth dm H = d<. Further denote by A the algebra of observables of S,.e. A s the algebra of lnear operators on H. Foreachn N denote by A n the algebra of lnear operators on the n-fold tensor product Hlbert space H n.it represents the algebra of observables of a compound quantum system S n wth ts n unt systems beng of the same type S. For each n N the set of densty operators n A n corresponds one-to-one to the state space SA n ofa n. Recall that a densty operator s defned to be a self-adjont, postve lnear operator of trace 1. Let r N and Σ be a set of densty operators ρ SA, = 1,...,r, representng the possble states of the quantum system S. Assume that for each n N there s a compound quantum system S n beng an n-fold copy of S. Ths means, n partcular, that the correspondng quantum state s n Σ n := {ρ n } r =1,.e. t s unquely determned by the ndex {1,...,r}. Further, let E n = {E n } r =1 be a postve operator valued measure POVM n A n,.e. each E n, =1,...,r, s a self-adjont element of A n wth E n 0 and r =1 En = 1. ThePOVMsE n descrbe quantum tests for dscrmnaton between the r states from Σ n,orsmplyquantum tests for Σ n, by dentfyng the measurement outcome correspondng to E n, =1,...,r, wth the densty operator ρ n, respectvely. If ρ happens to descrbe the true state of S, and correspondngly ρ n determnes the state of S n, then the assocated ndvdual success probablty s gven by Succ E n :=tr[ρ n E n ].

3 Asymptotcally Optmal Dscrmnaton between Pure Quantum States 3 The nddvdual error probablty refers to the stuaton when the densty operator ρ s dscarded as possble preparaton of S; t s gven by the formula Err E n :=tr[ρ n 1 E n ]. Assumng 0 <p < 1, =1,...,r,wth r =1 p = 1 to be the a pror dstrbuton of the r quantum states from Σ the averaged error probablty s defned by ErrE n = r =1 p tr [ρ n 1 E n ]. If the lmt lm n 1 n log ErrEn exsts, we refer to t as the asymptotc error exponent. Otherwse we have to consder the correspondng lm sup and lm nf expressons. For two densty operators ρ 1 and ρ 2 the quantum Chernoff dstance s defned by ξ QCB ρ 1,ρ 2 := log nf 0 s 1 tr [ρ1 s 1 ρ s 2 ]. 1 It specfes the optmal achevable asymptotc error exponent n dscrmnatng between ρ 1 and ρ 2, compare [8], [1], [2]. Quantum tests wth mnmal averaged error probablty for a par of dfferent densty operators ρ 1 and ρ 2 on the same Hlbert space H are well-known to be gven by the respectve Holevo-Helstrom projectors Π 1 := supp ρ 1 ρ 2 +, Π 2 := supp ρ 2 ρ 1 + = 1 Π 1. Here supp a denotes the support projector of a self-adjont operator a, whle a + means ts postve part,.e. a + = a + a/2 for a := a a 1/2, see [5], [4]. As mentoned n the ntroducton, the Holevo-Helstrom projectors generalze the lkelhood rato tests for two probablty dstrbutons. Ths can be verfed by lettng ρ 1 and ρ 2 be two commutng densty matrces, cf. [8]. For a set Σ = {ρ } r =1 of densty operators on H, wherer>2, we ntroduce the generalzed quantum Chernoff dstance ξ QCB Σ :=mn{ξ QCB ρ,ρ j : 1 <j r}. 2 Ths s n full analogy to the defnton of the generalzed Chernoff dstance n classcal multple hypothess testng, where the densty operators are replaced by probablty dstrbutons on a fnte sample space, cf. [9]. Our frst theorem s an mplcaton of Theorem 2.2 n [8]. Theorem 1. Let r N and Σ = {ρ } r =1 be a set of parwse dfferent densty operators on H wth correspondng a pror probablty dstrbuton {p } r =1.For any sequence E n, n N, ofquantumtestsforσ n, respectvely, t holds lm sup 1 n n log ErrEn ξ QCB Σ, where ξ QCB Σ s the generalzed quantum Chernoff dstance defned by 2.

4 4 M. Nussbaum and A. Szko la It turns out that the generalzed quantum Chernoff dstance s achevable as an asymptotc error exponent n the case of pure states. Ths s the statement of our man theorem below. Theorem 2. Let r N and Σ = {ρ } r =1 be a set of parwse dfferent pure states of a quantum system S. Then there exsts a sequence {E n } n N of quantum tests for Σ n, respectvely, wth lm 1 n n log ErrEn =ξ QCB Σ,.e. the generalzed quantum Chernoff dstance s an achevable asymptotc error exponent n multple pure state dscrmnaton. 3 Generalzed Quantum Chernoff Bound n Multple Quantum State Dscrmnaton In ths secton we gve a proof of Theorem 1 statng that the generalzed quantum Chernoff dstance specfes a bound on the asymptotcally achevable error exponents n multple quantum state dscrmnaton. It reles on ts bnary verson presented n Theorem 2.2 n [8]. Proof Theorem 1. Fx any two ndces 1 < j r. For n N let A n,b n A n be two postve operators such that A n + B n = 1 E n E n j. Then the postve operators Ẽ n n := E + A n and Ẽj n n := E j + B n represent a POVM Ẽn n A n, whch we consder a quantum test for the par {ρ n }. For the ndvdual error probabltes of the modfed quantum test Ẽ n we obtan the upper bounds,ρ n j Err Ẽn =tr[ρ n 1 Ẽ n ] tr [ρ n 1 E n ] = Err E n, and smlarly Err j Ẽn Err j E n. It follows a lower bound on the average error probablty wth respect to the orgnal tests {E n } r =1 : r ErrE n = p k Err k E n p Err E n +p j Err j E n k=1 p Err Ẽn +p j Err j Ẽn p mn Err Ẽn +Err j Ẽn where p mn := mn{p : 1 r}. The above bound mples lm sup n 1 n log ErrEn lm sup 1 n n log p mn + lm sup 1 Err n n log Ẽn +Err j Ẽn = lm sup 1 n n log 1 2 ξ QCB ρ,ρ j., Err Ẽn +Err j Ẽn

5 Asymptotcally Optmal Dscrmnaton between Pure Quantum States 5 Here the last nequalty s by Theorem 2.2 n [8], whch represents the statement of our Theorem 1 n ts bnary verson correspondng to the specal case r = 2. Snce the par of ndces, j was choosen arbtrary, the statement of the theorem follows. 4 Asymptotcally Optmal Pure State Dscrmnaton In ths secton we provde a constructve proof for Theorem 2. Roughly speakng, our quantum tests, whch can be shown to acheve an asymptotc error exponent equal to the generalzed quantum Chernoff dstance of Σ, are obtaned from a Gram-Schmdt orthonormalzaton procedure of the unt vectors assocated to the pure states n Σ. Proof Theorem 2. Observe that n vew of Theorem 1 t s suffcent to construct quantum tests for whch we can verfy lm nf n 1 n log ErrEn ξ QCB Σ. For each 1 r let v be a unt vector n H such that v v = ρ. 1. We assume that the set V Σ :={v } r =1 s lnearly ndependent and start wth the case n = 1, where no tensor products are ncluded. We defne for each k =1,...,r,ad k matrxψ k Ψ k := v 1,...,v k, 3.e. the columns of Ψ k are equal to the state vectors v,1 k. We refer to the k k-matrx Ψ k Ψ k =: Γ k as a Gram matrx of {v 1,...,v k }. By the assumpton of lnear ndependence of the set V Σ foreachk {1,...,r} the operator P k := Ψ k Ψ k Ψ k 1 Ψ k = Ψ k Γ 1 k Ψ k, represents an orthogonal projector onto a k-dmensonal subspace of H, whch s spanned by the k state vectors v 1,...,v k.further,wesetp 0 = 0 and defne for 1 k r E k := P k P k 1. The E k represent one-dmensonal orthogonal projectors, whch are mutually 1 orthogonal. Wth e k := E k v k E kv k we can wrte E k = e k e k,andtheset {e k } r k=1 represents a Gram-Schmdt orthonormalzaton of the lnearly ndependent set V Σ of unt vectors v k, k =1,...,r.

6 6 M. Nussbaum and A. Szko la Observe that by constructon r =1 E 1. IfE 0 := 1 r =1 E 0,we redefne E 1 to be E 1 + E 0, such that r =1 E = 1 s satsfed. By dentfyng E, =1,...,r,wthρ, respectvely, we obtan a quantum test E 1 = {E } r =1 for Σ. For 1 r the correspondng ndvdual success probablty reads Succ E 1 =tr[ρ E ]=tr[ v v E ]= v P P 1 v. 4 Snce the P s are constructed as orthogonal projectors onto span{v 1,...,v } t holds v v P and as a consequence v P v = 1. Then from the relaton Err E 1 =1 Succ E 1 weobtan Err E 1 = v P 1 v = v Ψ Γ 1 1 Ψ 1 v 1 λ mn Γ 1 v Ψ 1 Ψ 1 v = 1 λ mn Γ 1 Ψ 1v 2, 5 where λ mn denotes the mnmal egenvalue of a self-adjont matrx. By defnton 3 of Ψ we have Ψ 1 v 1 2 = v j v 2, =2,...,r. 6 j=1 Insertng expresson 6 nto 5 we obtan the upper bound Err E 1 1 j=1 v j v 2 λ mn Γ 1. 7 Recall that the densty operators ρ, =1,...,r, are expected to appear wth probablty p, respectvely. Then the averaged error probablty can be estmated from above as follows ErrE 1 = r p Err E 1 =1 r Err E 1 =1 r 1 =2 j=1 v j v 2 λ mn Γ 1, 8 where n the second lne we have appled Let n>1. Notce that stll assumng that V Σ sasetofr lnearly ndependent unt vectors, the same remans true for V Σ n consstng of the n-fold tensor product state vectors v n, =1,...,r. Hence we can adopt the constructon of the quantum test E 1 for Σ as t stands for the tensor product

7 Asymptotcally Optmal Dscrmnaton between Pure Quantum States 7 case. In partcular, we defne Ψ j,n,1 j r, analogously to 3 as the d n j- matrx Ψ j,n := v1 n,...,vj n, respectvely. Then the correspondng averaged error probablty Err E n can be upper bounded smlarly to 8: r 1 Err E n v n j v n 2 r 1 λ mn Γ 1,n = vj v 2 n λ mn Γ 1,n, =2 j=1 =2 j=1 where Γ 1,n := Ψ 1,n Ψ 1,n. Observe that each Gram matrx Γ j,n = Ψj,n Ψ j,n, j =1,...,r,sasquare matrx of fxed dmenson j, respectvely. Further, note that the dagonal entres γ j,n kk, k =1,...,j of Γ j,n are gven by v n k v n k, respectvely, and hence are all equal to 1. Snce for k l t holds v k v l < 1, the off-dagonal entres = v n k 1 j r γ j,n k,l v n l = v k v l n tend to 0 as n goes to nfnty. It follows for every Γ j,n I j as n, where I j denotes the dentty matrx of dmenson j.bycontnutyofthemnmal egenvalue ths mples We conclude Err E n λ mn Γ j,n 1 as n. r 1 vj v 2 n 1 + o1. =2 j=1 As n tends to nfnty the largest term domnates. As a consequence we have 1 n log Err En max{log v j v 2 : 1 j< r} + o1 = mn{ξ QCB ρ,ρ j, 1 j< r} + o1 = ξ QCB Σ+o1, 9 wherenthesecondlnewehaveusedthefactthatnthecaseoftwodfferent pure states on H, sayρ = v v and σ = w w, the correspondng bnary quantum Chernoff dstance ξ QCB ρ, σ takes the smple form log v w 2,cf.[8]. The last dentty s by defnton 2 of the generalzed quantum Chernoff dstance. The proof s complete under the assumpton of lnear ndependence of the set of egenvectors of Σ. 3. Fnally, notce that even f V Σ s not lnearly ndependent, the set V Σ N consstng of N-fold tensor product vectors becomes lnearly ndependent for N large enough. Then, for every n N we can adopt the constructon of quantum tests E n for Σ n as presented n parts 1 and 2 of the proof, and the asymptotc relaton 9 remans vald.

8 8 M. Nussbaum and A. Szko la Acknowledgments. The work of M. N. has been supported n part by NSF Grant DMS A. S. wshes to thank the research groups of Prof. Jost and Nhat Ay at the MPI MS for ther nterest and helpful dscussons. References 1. Audenaert, K.M.R., Casamgla, J., Munoz-Tapa, R., Bagan, E., Masanes, L.l., Acn, A., Verstraete, F.: Dscrmnatng States: The Quantum Chernoff Bound. Phys. Rev. Lett. 98, Audenaert, K.M.R., Nussbaum, M., Szko la, A., Verstraete, F.: Asymptotc Error Rates n Quantum Hypothess Testng. Commun. Math. Phys. 279, Barett, S., Croke, S.: On the condtons for dscrmnaton between quantum states wth mnmum error. J. Phys. A: Math. Theor Helstrom, C.W.: Quantum Detecton and Estmaton Theory. Academc Press, New York Holevo, A.: Investgatons n the general theory of statstcal decsons. Trudy Mat. Inst. Steklov 124 n Russan Englsh translaton n Proc. Steklov Inst. of Math. 3. Amer. Math. Soc., Provdence Kholevo, A.: On asymptotcally optmal hypothess testng n quantum statstcs. Theor. Probab. Appl. 23, Köng, R., Renner, R., Schaffner, C.: The operatonal meanng of mn- and maxentropy. IEEE Trans. Inf. Th Nussbaum, M., Szko la, A.: The Chernoff lower bound for symmetrc quantum hypothess testng. Ann. Stat. 372, Salkhov, N.P.: On one generalsaton of Chernov s dstance. Theory Probab. Appl. 432, Tyson, J.: Two-sded estmates of mnmum-error dstngushablty of mxed quantum states va generalzed Holevo-Curlander bounds. J. Math. Phys. 50, Yuen, H.P., Kennedy, R.S., Lax, M.: Optmum testng of Multple Hypotheses n Quantum Detecton Theory. IEEE Trans. Inform. Thoery IT-212,