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1 ŠܹÈÐ Ò ¹ÁÒ Ø ØÙØ für Mathematk n den Naturwssenschaften Lepzg Asymptotcally optmal dscrmnaton between multple pure quantum states by Mchael Nussbaum, and Arleta Szkola Preprnt no.:

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3 Asymptotcally optmal dscrmnaton between multple pure quantum states Mchael Nussbaum 1, Arleta Szko la 2 1 Department of Mathematcs, Cornell Unversty Ithaca NY, 14853, USA e-mal: nussbaum@math.cornell.edu 2 Max Planck Insttute for Mathematcs n the Scences Inselstrasse 22, Lepzg, Germany e-mal: szkola@ms.mpg.de January 11, 2010 Abstract We consder the decson problem between a fnte number of states of a fnte quantum system, when an arbtrary large number of copes of the system s avalable for measurements. We provde an upper bound on the asymptotc exponental 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. 1 Introducton In dfferent 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 occurence of the dstnct states s determned by an a pror probablty dstrbuton. One makes a decson n favor of one of the states accordng to a specfed rule based on the outcomes of a generalzed measurement -called quantum test. In the bnary case optmal tests,.e. tests Supported n part by NSF Grant DMS

4 mnmzng the averaged probablty of rejectng the true state, are known to be gven by Holevo-Helstrom projectons, [5], [4]. They generalze the classcal lkelhood rato tests. Here we want to consder the scenaro where there s an arbtrary large fnte number n of copes of the quantum system avalable for performng a measurement. Then the correspondng state s 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) decay of the averaged probablty of rejectng the true state. Secondly, 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 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 have not yet receved a fnal answer, although many efforts has been made and numerous strong results has been obtaned related 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 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 has been already consdered by Holevo n [6] n the context of quantum mnmal error decson problems. However, the queston of the correspondng asymptotc error exponent has not been addressed theren. 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. For each n 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 S(A (n) ) of A (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 S(A), = 1,...,r, representng the 2

5 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 E(n) = 1. The POVMs E (n) descrbe quantum tests for dscrmnaton between the r states from Σ n, or smply quantum tests for Σ n, by dentfyng the measurement outcome correspondng to E (n), = 1,...,r, wth the densty operator ρ n, respectvely. If ρ occurs to descrbe the true state of S, and correspondngly ρ n determnes the state of S n, then the assocated ndvdual succes probablty s gven by Succ (E (n) ) := tr [ρ n E (n) ]. (1) The nddvdual error probablty refers to the stuaton when the densty operator ρ s dscarded as possble preparaton of S and s gven by the formula Err (E (n) ) := tr [ρ n (1 E (n) )]. (2) 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 Err(E (n) ) = r =1 p tr [ρ n (1 E (n) )]. (3) If the lmt lm n 1 n log Err(E(n) ) exsts, we refer to t as the asymptotc error exponent. Otherwse we have to consder the correspondng lmsup and lmnf 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 tr [ρ1 s1 ρ s 2]. (4) 0 s 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 3

6 dstrbutons. Ths can be verfed by settng for ρ 1 and ρ 2 two commutng densty matrces, cf. [8]. For a set Σ = {ρ } r =1 of densty operators on H, where r > 2, we ntroduce the generalzed quantum Chernoff dstance ξ QCB (Σ) := mn{ξ QCB (ρ, ρ j ) : 1 < j r}. (5) Ths s n full analogy to the defnton of 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, of quantum tests for Σ n, respectvely, t holds lm sup 1 n n log Err(E(n) ) ξ QCB (Σ), (6) where ξ QCB (Σ) s the generalzed quantum Chernoff dstance defned by (5). 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 Err(E(n) ) = 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) := E (n) + A (n) and Ẽj (n) (n) := E j + B (n) represent a POVM Ẽ(n) n A (n), whch we 4

7 consder as a quantum test for the par {ρ n, ρ n j }. For the ndvdual error probabltes of the modfed quantum test Ẽ(n) we obtan the upper bounds Err (Ẽ(n) ) = tr [ρ n (1 Ẽ (n) )] tr [ρ n (1 E (n) )] = Err (E (n) ), and smlarly Err j (Ẽ(n) ) Err j (E (n) ). It follows a lower bound on the average error probablty wth respect to the orgnal tests {E (n) } r =1 : Err(E (n) ) = r p k Err k (E (n) ) k=1 where p mn := mn{p : 1 r}. The above bound mples lm sup 1 n n log Err(E(n) ) ( ) p Err (E (n) ) + p j Err j (E (n) ) ( ) p Err (Ẽ(n) ) + p j Err j (Ẽ(n) ) ) p mn (Err (Ẽ(n) ) + Err j (Ẽ(n) ), lmsup 1 n n log p mn + lmsup 1 (Err n n log (Ẽ(n) ) + Err j (Ẽ(n) ) = lm sup 1 n n log 1 2 ξ QCB (ρ, ρ j ). ) ( ) Err (Ẽ(n) ) + Err j (Ẽ(n) ) 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 n 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 Err(E(n) ) ξ QCB (Σ). 5

8 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, a (d k) matrx Ψ k Ψ k := (v 1,...,v k ), (7).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 (8) as a Gram matrx of {v 1,...,v k }. By the assumpton of lnear ndependence of the set V (Σ) for each k {1,...,r} the operator P k := Ψ k (Ψ k Ψ k) 1 Ψ k = Ψ kγ 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, we set P 0 = 0 and defne for 1 k r E k := P k P k 1. (9) The E k represent one-dmensonal orthogonal projectors, whch are mutually orthogonal. 1 Wth e k := E k v k E kv k we can wrte E k = e k e k, and the set {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. Observe that by constructon r =1 E 1. If E 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, wth ρ, 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. (10) 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) ) we obtan Err (E (1) ) = v P 1 v (11) = v Ψ (Γ 1 ) 1 Ψ 1 v (12) 1 λ mn (Γ 1 ) v Ψ 1 Ψ 1 v = 1 λ mn (Γ 1 ) Ψ 1v 2, (13) 6

9 where λ mn ( ) denotes the mnmal egenvalue of a self-adjont matrx. By defnton (7) of Ψ we have 1 Ψ 1v 2 = v j v 2, = 2,...,r. (14) Insertng expresson (14) nto (13) we obtan the upper bound j=1 Err (E (1) ) 1 j=1 v j v 2 λ mn (Γ 1 ). (15) Recall that the densty operators ρ, = 1,...,r, are expected to appear wth probablty p, respectvely. Let p max := max{p : 1 r}. Then the averaged error probablty can be estmated from above as follows Err(E (1) ) = r p Err (E (1) ) =1 r p max Err (E (1) ) =1 1 r p max =2 j=1 v j v 2 λ mn (Γ 1 ), (16) where n the second lne we have appled (15). 2. Let n > 1. Notce that stll assumng that V (Σ) s a set of r 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 case. In partcular, we defne Ψ j,n, 1 j r, analogously to (7) as the (d n j)-matrx ( ) Ψ j,n := v1 n,...,v n j, respectvely. Then the correspondng averaged error probablty Err (E (n) ) can be upper bounded smlarly to (16): r 1 Err (E (n) ) p max =2 j=1 vj n v n 2 λ mn (Γ 1,n ) = p max r 1 =2 j=1 ( vj v 2) n λ mn (Γ 1,n ), where Γ 1,n := Ψ 1,n Ψ 1,n. Observe that each Gram matrx Γ j,n = Ψ j,n Ψ j,n, j = 1,...,r, s a square 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 7

10 v k v l < 1, the off-dagonal entres γ (j,n) k,l nfnty. It follows for every 1 j r = v n k Γ j,n I j as n, v n l = v k v l n tend to 0 as n goes to where I j denotes the dentty matrx of dmenson j. By contnuty of the mnmal egenvalue ths mples λ mn (Γ j,n ) 1 as n. (17) We conclude r 1 Err (E (n) ( ) p max vj v 2) n (1 + o(n)). (18) =2 j=1 As n tends to nfnty the largest term domnates. As a consequence we have 1 n log Err (E(n) ) max{log v j v 2 : 1 j < r} + o(n) = mn{ξ QCB (ρ, ρ j ), 1 j < r} + o(n) = ξ QCB (Σ) + o(n), (19) where n the second lne we have used the fact that n the case of two dfferent pure states on H, say ρ = 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 (5) of generalzed quantum Chernoff dstance. The proof s completed 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 (19) remans vald. Acknowledgments. The second author wshes to thank the research groups of Prof. Jost and Nhat Ay at the MPI MS for ther nterest and helpful dscussons. References [1] K.M.R. Audenaert, J. Casamgla, R. Munoz-Tapa, E. Bagan, Ll. Masanes, A. Acn and F. Verstraete, Dscrmnatng States: The Quantum Chernoff Bound, Phys. Rev. Lett. 98, (2007) [2] K.M.R. Audenaert, M. Nussbaum, A. Szko la and F. Verstraete, Asymptotc Error Rates n Quantum Hypothess Testng, Commun. Math. Phys. 279, (2008) 8

11 [3] S. Barett and S. Croke, On the condtons for dscrmnaton between quantum states wth mnmum error, J. Phys. A: Math. Theor. 42 (2009) [4] C.W. Helstrom, Quantum Detecton and Estmaton Theory, Academc Press, New York (1976) [5] A. Holevo, 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, RI. (1978)] [6] A. Kholevo, On asymptotcally optmal hypothess testng n quantum statstcs, Theor. Probab. Appl. 23, (1978) [7] R. Köng, R. Renner, and C. Schaffner, The operatonal meanng of mon- and maxentropy, arxv: [8] M. Nussbaum and A. Szko la, The Chernoff lower bound for symmetrc quantum hypothess testng, Ann. Stat. Vol. 37, No. 2, (2009) [9] N.P Salkhov, On one generalsaton of Chernov s dstance, Theory Probab. Appl. Vol. 43, No. 2, (1997) [10] J. Tyson, Two-sded estmates of mnmum-error dstngushablty of mxed quantum states va generalzed Holevo-Curlander bounds, J. Math. Phys. 50, (2009) [11] H.P. Yuen, R.S. Kennedy, and M. Lax, Optmum testng of Multple Hypotheses n Quantum Detecton Theory, IEEE Trans. Inform. Thoery Vol. IT-21, No. 2, (1975) 9