OPTIMIZATION OF QUANTUM UNIVERSAL DETECTORS

Transcription

1 OPTIMIZATION OF QUANTUM UNIVERSAL DETECTORS G. M. D ARIANO, P. PERINOTTI, M. F. SACCHI QUIT Group, Unità INFM an Dipartimento i Fisica A. Volta, Università i Pavia, via A. Bassi 6, I Pavia, Italy The expectation value O of an arbitrary operator O can be obtaine via a universal measuring apparatus that is inepenent of O, by changing only the ataprocessing of the outcomes. Such a universal etector performs a joint measurement on the system an on a suitable ancilla prepare in a fixe state, an is equivalent to a positive operator value measure (POVM) for the system that is informationally complete. The ata processing functions generally are not unique, an we pose the problem of their optimization, proviing some examples for covariant POVM s, in particular for SU() covariance group. Universality an programmability are crucial features in quantum technology, for communication, processing, an storage of information. Different tasks shoul be achieve by a basic set of evices, that woul allow to perform ifferent kins of quantum information processing, such as in quantum computation 1,2, teleportation 3,4, entanglement etection 5, an entanglement istillation 6. In particular, a universal etector 7 achieves the estimation of the ensemble average of an arbitrary operator by changing only the ata processing of the outcomes. In some way it is analogous to a quantum tomographic apparatus 8 : however, the latter woul typically require a quorum of observables corresponing to a set of evices or to a single tunable evice whereas a universal etector woul measure only a single fixe observable on an extene Hilbert space that inclues a suitable ancilla. Universal etectors can be characterize via a necessary an sufficient conition given in terms of frames of operators (i. e. spanning sets of operators), an can be achieve via Bell measurements, which are escribe by projectors on maximally entangle states 7. Entanglement, however, is not an essential ingreient, an there are universal etectors which are escribe by separable POVM s as well 9. When attention is restricte to the system Hilbert space only, universal etectors are equivalent to informationally complete (shortly info-complete ) POVM s 10, which are frames mae of positive operators. Info-complete POVM s are necessarily not-orthogonal, whence universal etectors have a more physical counterpart, in terms of an observable an an apparatus ancilla. When using a universal etector the ensemble average of an arbitrary operator is by choosing the appropriate ata processing function of the measurement outcomes. As we will see, the ata processing functions are generally not unique, an are relate to the concept of ual operator frame. In the following, after reviewing the main results on universal etectors, we pose the problem of optimization of ata-processing functions, with particular focus 86

2 on the case of covariant POVM s, an in particular for the SU() covariance group. Let us introuce the concept of universal etector, or, more abstractly, of universal POVM. We consier a quantum system in a Hilbert space H, couple to an ancilla with Hilbert space K. A POVM {Π i }, Π i 0 an i Π i = I H I K on the Hilbert space H K is universal for the system iff there exists a state of the ancilla ν such that for any operator O one has Tr[ρO] = i f i (ν, O)Tr[(ρ ν)π i ], (1) where f i (ν, O) parametrize by ν an O is a suitable function of the outcome i of the measurement, an we will refer to it as the ata processing function. The etector will be calle universal when it is escribe by a universal POVM. In orer to give a necessary an sufficient conition for universality, we nee to introuce some notation, an the concept of frame of operators. We will use the following symbols for bipartite pure states in H K A = imh n=1 imk m=1 A nm n m, (2) where n an m are fixe orthonormal bases for H an K, respectively. Equation (2) exploits the isomorphism 11 between the Hilbert space of the Hilbert-Schmit operators A, B from K to H, with scalar prouct A, B = Tr[A B], an the Hilbert space of bipartite vectors A, B H K, with A B A, B. It is easy to show the following ientities A B C = ACB τ, Tr K [ A B ] = AB, Tr H [ A B ] = A τ B, (3) where τ an enote transposition an complex conjugation with respect to the fixe bases. A frame 12 for operators say A from K to H is just a set of operators spanning a norme linear space of operators, i. e. there are constants 0 < a b < such that for all operators A one has a A 2 i A, Ξ i 2 b A 2. Here, for simplicity, we will consier the (Hilbert) space of Hilbert-Schmit operators from K to H, an use the equivalent vector notation introuce in Eq. (2). Frames of operators have been alreay use isguise as spanning sets of operators 13 in the context of quantum tomography. For {Ξ i } an operator frame there exists another frame {Θ i } calle ual frame proviing operator expansions in the form A = i Tr[Θ i A]Ξ i. (4) 87

3 The completeness relation of the frame an its ual reas ψ Ξ i φ ϕ Θ i η = ψ η ϕ φ, (5) i for any φ, ϕ H an ψ, η K. For continuous sets, the sums in Eqs. (4) an (5) are replace by integrals. Given a frame {Ξ i }, generally the ual set is not unique. However, all uals {Θ i } of a given frame can be obtaine via the linear relation 14 Θ i = F 1 Ξ i + Y i j Ξ j F 1 Ξ i Y j, (6) where Y i are arbitrary, an the positive an invertible operator F writes F = i Ξ i Ξ i. (7) The operator F is calle frame operator in frame theory 12, whereas the set of operators corresponing to the vectors F 1 Ξ i through the above isomorphism is calle canonical ual frame. As we will show immeiately, the ual frame provies the ata processing function, whence Eq. (6) allows a useful flexibility in the ata-processing, with the possibility of optimizing the statistical error in the estimation by minimization over the free operators Y i. Let us now consier a universal POVM on H K. The elements {Π i } can be iagonalize as follows r i Π i = Ψ (i) j Ψ(i) j, (8) j=1 where the vectors Ψ (i) j have norm equal to the j-th eigenvalue of Π i, an r i is the rank of Π i. From the normalization conition i Π i = I H I K, it follows that the set of operators {Ψ (i) j } from K to H must be an operator frame itself. The characterization of universal POVM s is then given by the conition that there exists a state ν K such that the following operators r i Ξ i [ν] Ψ (i) j ντ Ψ (i) j (9) j=1 are a frame for operators on H. In fact, using Eq. (8), Eq. (1) rewrites Tr[ρO] = r i f i (ν, O)Tr ρ Ψ (i) j ντ Ψ (i) j, (10) i an this is true inepenently of ρ iff j=1 O = i f i (ν, O)Ξ i [ν]. (11) 88

4 From linearity one has f i (ν, O) = Tr[Θ i [ν]o], (12) where Θ i [ν] is a ual frame of Ξ i [ν]. Hence, after fining a ual frame for Ξ i [ν], the ata processing function is easily evaluate via Eq. (12). When restricting our attention just on the system Hilbert space, notice that from Eqs. (9) an (11) a universal etector correspons to a system POVM whose elements make a frame of positive operators. Then, from Eqs. (11) an (12) it follows that such POVM is informationally complete 10, namely it allows evaluation of the expectation of an arbitrary system operator. Since the number of elements of an operator frame for H cannot be smaller than (im H) 2, an info-complete POVM is necessarily not orthogonal. Viceversa, it is simple to prove that an arbitrary frame for operators in H mae of positive operators {K i } allows to construct an info-complete POVM. In fact, since the operator S i K i is invertible, the set { K i = S 1/2 K i S 1/2 } satisfies the completeness relation i K i = I H. The irect construction of info-complete POVM s is not trivial, since it involves the searching of positive operator frames. A way to construct universal POVM s is suggeste by grouptheoretic techniques 7. For example, one can consier projectors on maximally entangle states, namely a Bell POVM on H H. In the notation of Eq. (2), a Bell POVM has elements of the form Π i = α i U i U i, (13) where is the imension of H, α i are suitable positive constants an [ U i ] are unitaries. When the POVM is orthogonal, one has α i = 1 an Tr U i U j = δ ij. Particular cases of Bell POVM s are those in which U i are a unitary irreucible representation (UIR) of some group G. As an example, consier a projective UIR of an abelian group, which therefore satisfies the relation U α U β U α = eic(α,β) U β. (14) In this case the Bell POVM is orthogonal, with number of elements equal to the carinality of the group 2. One can show 7 that for any ancilla state ν such that Tr[U α ντ ] 0 for all α, the set of Ξ α [ν] = 1 U αν τ U α is an operator frame. By ientifying U 1 I, a possible choice of the ancilla state is ν = 1 I + 1 ( 2 U α. (15) 1) α>1 The ual frame in this case is unique, an is given by Θ α [ν] = 1 2 β=1 U β Tr[U β ν ] e ic(β,α). (16) Corresponingly, accoring to Eq. (12), also the ata processing function is unique. 89

5 There are universal Bell POVM s also from non-abelian groups. An interesting example is provie by the group SU(). In this case the universality of the corresponing Bell POVM is prove by showing that the set of Ξ α [ν] = U α ν τ U α is an operator frame. Let us start by evaluating the frame operator, which is given through Eq. (7) by F = α (U α Uα ) ντ ν τ (U α Uτ α ) = 1 I I + Tr[(ντ ) 2 ] (I 1 ) I I, (17) where we use Shur s lemma to compute the integral. It can be notice that F is expresse in iagonal form with eigenvalues 1 an Tr[(ντ ) 2 ] 1 2 1, thus it is invertible for any ν τ unless Tr[(ν τ ) 2 ] = 1, corresponing to the state ν = I/. The expression for the inverse of the frame operator is easily evaluate F 1 = 1 I I (I Tr[(ν τ ) 2 1 ) ] 1 I I. (18) The canonical ual set Θ 0 α [ν] for Ξ α[ν] is is obtaine by efinition as follows an one has Θ 0 α[ν] = F 1 U α ν τ U α, (19) Θ 0 α[ν] = a U α ν τ U α + b I, (20) where a = 2 1 Tr[(ν τ ) 2 ] 1 an b = Tr[(ντ ) 2 ] Tr[(ν τ ) 2 ] 1. Accoring to Eq. (12) the processing function corresponing to the canonical ual frame is then f α (ν, O) = a Tr[U α ν τ U α O] + btr[o]. (21) The knowlege of the canonical ual frame allows to parameterize all the alternate uals as in Eq. (6) by the arbitrary operators Y α, greatly simplifying the task of optimizing the statistical error in the estimate of a given operator. Such noise can be generally efine in terms of the eigenvalues of the covariance matrix ( ) (Ref) C = 2 Ref 2 RefImf Ref Imf RefImf Ref Imf (Imf) 2 Imf 2, (22) where g = α g α (ν, O)Tr[ρΞ α [ν]]. (23) The noise clearly epens on the state on which the estimate is one. For a state-inepenent efinition of noise one coul use either the maximum noise or the average noise over all (pure or mixe) states. If one consiers averages of Hermitian operators, the imaginary parts of the processing functions can be 90

6 iscare, an this is equivalent to consier Ref only. The noise can thus be evaluate by the customary variance (Ref) 2 Ref 2. As an example, we now evaluate the optimal ual frame for the estimation of Hermitian operators, restricting our attention on covariant ual frames, i. e. of the form Θ α [ν] = U α ξu α. (24) It can be prove that such a set is a ual frame of Ξ α [ν] iff Tr[ξ] = 1, Tr[ν τ ξ] =. (25) Since we are consiering Hermitian operators, the processing function can be written Ref α (ν, O) + iimf α (ν, O) = Tr[U α (Reξ)U αo] + itr[u α (Imξ)U αo], (26) where Reξ = 1 2 (ξ+ξ ), an Imξ = 1 2i (ξ ξ ). As state before, we can restrict attention on Ref α (ν, O), an thus we nee to consier only the self-ajoint case ξ Reξ. Our optimization consists in minimizing the average variance over all pure states, namely ( ξ O 2 ) = 1 β α ψ 0 U β Θ α[ν]u β ψ 0 Tr[Ξ α[ν]o] 2 1 β ψ 0 U β OU β ψ 0 2, (27) where the pure states have been parametrize as U β ψ 0, for a fixe arbitrary ψ 0 H an U β SU(). We will compare Eq. (27) with the variance of the ieal measurement of O average over all pure states, namely ( obs O 2 ) = 1 α ψ 0 U αo 2 U α ψ 0 1 α ψ 0 U αou α ψ 0 2. (28) Equations (27) an (28) can be evaluate using the following ientities α U α AU α = Tr[A]I α U 2 α 2 AUα = Tr[P SA]P S Tr[P AA]P A Tr[EB B] = Tr[B 2 ], (29) where P S = 1 2 (I + E) an P A = 1 2 (I E) are the projections on the totally symmetric an antisymmetric subspaces of H 2, an E enotes the swap operator E φ ψ = ψ φ. The results are ( obs O 2 ) = (Tr[O 2 ] 1 Tr[O]2 ) (30) ( ξ O 2 ) = Tr[ξ2 ] 1 ( obs O 2 ). (31) 1 91

7 The optimization can be achieve by minimizing the coefficient Tr[ξ2 ] 1 1 with the constraints Tr[ξ] = 1 an Tr[ν τ ξ] =. By the metho of Lagrange multipliers, one can write the variational equation ( ) δ ξ ξ 1 λ ξ ν τ µ ξ I = 0, (32) δ ξ 1 which leas to the following result ξ opt = 2 1 Tr[(ν τ ) 2 ] 1 ντ Tr[(ντ ) 2 ] Tr[(ν τ ) 2 ] 1 I, (33) namely, the optimal covariant ual frame is the canonical one. The optimization can be finally complete by looking for the least noisy ancilla state. By calculating Tr[ξ 2 opt] an substituting in Eq. (31) one obtains ( opt O 2 ) = p ( obs O 2 ), (34) p 1 where p Tr[(ν τ ) 2 ]. A simple ifferentiation of the expression in Eq. (34) with respect to p shows that the best choice correspons to p = 1, namely the minimal ae noise is achieve by an arbitrary pure ancilla state. In this case the expression is simplifie an is equal to Acknowlegments ( opt O 2 ) = ( + 2)( obs O 2 ). (35) This work has been cosponsore by EEC through the ATESIT project IST an by MIUR through Cofinanziamento-2002 P. P. an M. F. Sacchi also acknowlege support from INFM through the project PRA CLON, an G. M. D. also acknowleges partial support from MURI program Grant No. DAAD References 1. Introuction to Quantum Computation an Information, e. by H.-K. Lo, S. Popescu, an T. Spiller (Worl Scientific, Singapore, 1998). 2. M. A. Nielsen an I. L. Chuang, Quantum Information an Quantum Computation (Cambrige Univ. Press, Cambrige, 2000). 3. C. H. Bennett, G. Brassar, C. Crepeau, R. Jozsa, A. Peres, an W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). 4. S. L. Braunstein an H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998). 5. J. M. G. Sancho an S. F. Huelga, Phys. Rev. A 61, (2000); O. Guhne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, an A. Sanpera, Phys. Rev. A 66, (2002). 6. C. H. Bennett, G. Brassar, S. Popescu, B. Schumacher, J. A. Smolin, an W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996). 92

8 7. G. M. D Ariano, P. Perinotti, an M. F. Sacchi, quant-ph/ G. M. D Ariano, M. G. A. Paris, an M. F. Sacchi, quant-ph/ G. M. D Ariano, Phys. Lett. A 300, 1 (2002). 10. P. Busch, M. Grabowski, P. Lahti, Operational Quantum Physics, Lecture Notes in Physics 31 (Springer, Berlin 1995) 11. G. M. D Ariano, P. Lo Presti, an M. F. Sacchi, Phys. Lett. A 272, 32 (2000). 12. P. G. Casazza, Taiw. J. Math. 4, 129 (2000). 13. G. M. D Ariano, L. Maccone, an M. G. A. Paris, J. Phys. A 34, 93 (2001). 14. S. Li, Numer. Funct. Anal. an Optimiz 16, 1181 (1995). 93