Optimal state reconstructions

Transcription

1 Quantum observations talk 3 Vladimír Bužek Research Center for Quantum Information, Bratislava, Slovakia 4 September 0 Sharif University of Technology, Tehran Optimal state reconstructions Reconstructions from individual clicks

2 Incomplete observations set of observables measured meanvalues Solution E.T.Jaynes How to find ρˆ? MaxEnt principle max S The MaxEnt principle is the most conservative assignment in the sense that it does not permit one to draw any conclusions not warranted by the data. E.T.Jaynes, Phys. Rev. 80, 7 (957); E.T.Jaynes, Am. J. Phys. 3, 66 (963). MaxEnt reconstruction of qubit Pure state of a spin -/ particle = cos + e i sin 0 density operator ( ) = ( Î + n x ˆ x + n y ˆ y + n z ˆ z ) ˆ = Î + n. MaxEnt reconstruction

3 Reconstruction of Qubits Pure state of a qubit = cos + e i sin 0 = n density operator ( ) = ( Î + n x ˆ x + n y ˆ y + n z ˆ z ) ˆ = Î + n. ) exact meanvalues infinite ensembles ) What is the best a posteriori estimation of a quantum state when a measurement is performed on a finite (arbitrary small) number of elements of the ensemble? S.Massar & S.Popescu: Phys. Rev. Lett. 74, 59 (995) C.W.Helstrom: Quantum Detection and Estimation Theory (Academic, NY, 976) A.S.Holevo: Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 98) Quantum Clickology Measurement: conditional distribution on a discrete state space of the aparatus A: Ô observables with eigenvalues λ i ( ) a priori distribution p 0 ˆ on the state space Ω of the system: joint probability distribution p( Ô, ˆ i ) = Tr( ˆP ˆ i,ô ) p( Ô, ; ˆ i ) = p( Ô, ˆ i )p 0 ( ˆ ) System Apparatus Measurement 3

4 Quantum Bayesian Inference Bayesian inversion from distribution on A to distribution on Ω ( ) = p Ô, i ˆ p ˆ Ô, i ( )p 0 ( ˆ ) p( Ô, ; ˆ i )d Reconstructed density operator given the result λ i d Ω invariant integration measure ˆ est = ˆ (,)p ( ˆ Ô, i )d K.R.W. Jones, Ann. Phys. (N.Y.) 07, 40 (99) V.Bužek, R.Derka, G.Adam, and P.L.Knight, Annals of Physics (N.Y.), 66, 454 (998) Example: Single qubit case Prior knowledge: state is pure p 0 ( ˆ ) = const. Density operator ( ) Projectors on eigenvectors of the apparatus ˆ = Î + sin cosˆ x + sin sinˆ y + cos ˆ z Invariant measure Distribution on d = sin d d# 4 Ô z = ( Î + ˆ z ); = ± z Ω p( ˆ Ô, = + i ) = + cos ( ) ˆ est = ˆ (,)p ( ˆ Ô, )d = Î + 3 ˆ % ( z # $ &' x y 4

5 Figure of merit: Mean fidelity Fidelity of the guess when ± m is measured + m ; F + = + m n = ( + n. m) / m ; F = m n = ( n. m) / Mean fidelity (averaged over all input states) F = d n # + n. m n m + n. m n m $ & = % 3 S.Massar & S.Popescu: Phys. Rev. Lett. 74, 59 (995) C.W.Helstrom: Quantum Detection and Estimation Theory (Academic, NY, 976) A.S.Holevo: Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 98) EXAMPLE 5

6 TWO QUBITS independent measurements of two qubits: Ô ± = ± m ± m ; = ± Ô ± = ± k ± k ; = ± adaptive measurements of two qubits: Ô ± = ± m ± m Ô ± = ± k ± m ± k ± m simultaneous measurements of two qubits: ˆP j = j j ; ˆPj = Î j Optimal two-qubit measurement simultaneous measurements of two qubits: ˆP j = j j ; ˆPj = Î j n = 0,0, ( ) n = 8 3,0, % # $ 3& ' n 3 = 3,, % # $ 3 3& ' n 4 = 3,, % # $ 3 3& ' j = 3 n j ; n j + s Four vectors point to the four vertices of the tetrahedron Mean fidelity of estimation F = 3 4 S.Massar & S.Popescu: Phys. Rev. Lett. 74, 59 (995) 6

7 Generalized quantum measurements r Positive operators Ô r not projectors; Ô r = Mean fidelity F via the cost function F = * r ############### N times ############### $ % dx Tr Ôr U ( x ) ) ˆ 0 U ( x) U ( x) ˆ 0 U ( x) Tr U x ( #$ # $ &' ( ) ˆ 0 U x ( )U r ˆ 0 U % r &' C.W.Holstrom, Quantum detection and estimation theory (Academic Press, New York, 976) A.S.Holevo, Probabilistic and statistical aspects of quantum theory (North Holland, Amsterdam, 98) M.G.A.Parisi and J.Rehacek, Quantum estimations, (Springer, Berlin, 004) Optimal quantum measurements k-dimensional representation of the group G corresponding to chosen quantum object ( ) U x x Abstract compact Lie Group G reflecting symmetries of chosen estimation problem N-fold direct product of the representation U transforming N copies of given object; dim U N U N ( x ) ( ) = $ N + k # $ k % &' Acting on the space of single quantum object 0 ( ) 0 U ( x) r r U x Single quantum object in a given state Density matrix of the reference point Here the POVM Ôr is applied to each corresponds a guess Ô r R { } r= The cost between guess and actual state is defined N copies of quantum object all prepared in the same state U N 0 x ( ) 0 U N x ( ) Acting the state space of N identically transformed quantum objects 7

8 Optimal estimation of direction average fidelity of estimation F = N + N + = + N $ # N + & % F = F = 3 = + $ # 3% & S.Massar and S.Popescu, Phys. Rev. Lett. 74, 59 (995) A.Latorre, P.Pascual, and R.Tarrach, Phys. Rev. Lett. 8, 35 (998) R.Gill and S.Massar, Phys. Rev. A 6, 043 (000) R.Derka, V.Bužek, and A.K.Ekert, Phys. Rev. Lett 80, 57 (998) V.Bužek, R.Derka, and S.Massar, Phys. Rev. Lett 8, 07 (999) M.Hayashi, Asymptotic Theory of Quantum Statistical Inference (Academic Press, NY, 005) F = Optimal reconstructions of qubits average fidelity of estimation F = N + N + = + N $ # N + & % F = 3 = + $ # 3% & Construction of optimal (& finitedimensional) POVM s maximize the fidelity F POVM via von Neumann projectors Naimark theorem Optimal decoding of information Estimated density operator on average ˆ est = s ˆ + s Î; s = F = N N + Optimal preparation of quantum systems Recycling of q-information S.Massar and S.Popescu, Phys. Rev. Lett. 74, 59 (995) A.Latorre, P.Pascual, and R.Tarrach, Phys. Rev. Lett. 8, 35 (998) R.Gill and S.Massar, Phys. Rev. A 6, 043 (000) M.Hayashi, Asymptotic Theory of Quantum Statistical Inference (Academic Press, NY, 005) 8

9 Information gain vs disturbance Trade-off: information gain vs disturbance F out 3 F est F est How much information is left in the system after it has been optimally (max extraction of information & minimal disturbance) measured? Sequence of observers No classical communication Observers use the same measurement device though with no knowledge of previous results Optimal observation of single qubit Non-trivial estimation for everybody? K. Banaszek: Phys. Rev. Lett. 87, 366 (00) Recycling of quantum information Classical systems can be observed infinitely many times An observe can in principle specify completely the state of the system Observer performs measurements without disturbing system Many independent observers can measure the system All observers (ideally) obtain same results conclusions. Quantum information is fragile Info encoded into q-system is disturbed by measurement From one copy of q-system only limited info can be obtained In the case of qubits via the optimal measurement just bit N identical qubits (ensembles) are needed to gain more info But even these ensembles are disturbed. Usually it is assumed that once measured q-systems are of no use. Is this true? 9

10 Recycling of quantum information Formulation of the problem Encoding of (classical) information n =U n o = n = Sequential observation with no classical communication Everybody has the same apparatus (POVM) M k The st observer optimizes his measurement in order to maximize his average fidelity F The st observer optimizes the after-measurement state (given his knowledge) in order to help to maximize the fidelity of estimation of the nd observer - This whole procedure can be represented as a map [] Via this procedure a state is prepared for the nd observer; who does exactly the same as the first one, etc. 0

11 Recycling single-qubit Optimal measurement (preparation): projectors on eigenvectors of the apparatus Ô k,+ = + m k + m k ; = + Ô k, = m k m k ; = Initial state n =U n o = n = Mean-fidelity for subsequent observers F = + $ # 3 k %& Recycling of q-information Trade-off: information gain vs disturbance How much information is left in the system after it has been measured? Sequence of observers No classical communication Observers use the same measurement device (up to its orientation ) though with no knowledge of previous results F = + $ # 3 k %& Optimal observation of single qubit Non-trivial estimation by everybody?

12 Q-channel & mutual information k k... % #$ &' H k = p(n,n dn dn k p(n,n k )log k ) p(n k )p(n) H = log e H k = log e + log p (k) + + p (k) (3 k )log # $ p + (k) p (k) % &' p (k) ± = ± % # $ 3 k &' N parallel qubits Encoding direction in 3-d space into N parallel qubits n N Optimal measurement covariant measurement due to Holevo F k = + N k $ $ # N + % & # %& Different encodings?

13 Robustness of quantum information Sequence of observers: k Size of ensemble: N F = + N k $ $ # N + % & # %& ' k N Robustness against observations Trade-off gain vs. disturbance Non-trivial estimation by everybody Sending q-information via q-channel Alice sends info to Bob about direction encoded in qubit n Given the preparator (information source) and the quantum channel there must exist an optimal encoding-decoding procedure that maximizes the knowledge of Bob about n n 3

14 Coding of direction: one qubit Two preparators: original one n Ô ± = ± m ± m ; = ± or the flipped preparator n Ô = m m ; = Average fidelity is the same optimal coding given the resources F = 3 Coding of direction: two qubits One preparator generates two copies; independent measurements Ô ± = ± m ± m ; = ± Ô ± = ± k ± k ; = ± Two preparators each generate one qubit; independent measurements Ô ± = ± m ± m ; = ± Ô = k k ; = Average fidelity is the same, though not optimal. 4

15 Coding of direction: Two qubits One preparator: simultaneous measurements of two qubits: ˆP j = j j ; ˆPj = Î j n = 0,0, ( ) n = 8 3,0, % # $ 3& ' n 3 = 3,, % # $ 3 3& ' n 4 = 3,, % # $ 3 3& ' S.Massar & S.Popescu: Phys. Rev. Lett. 74, 59 (995) j = 3 3 F = = n j ; n j + s Four vectors point to the four vertices of the tetrahedron Mean fidelity of estimation Gisin & Popescu: anti-parallel qubits Two preparators & simultaneous measurements of two qubits: ˆP j = j j ; ˆPj = Î j n = 0,0, ( ) n = 8 3,0, % # $ 3& ' n 3 = 3,, % # $ 3 3& ' n 4 = 3,, % # $ 3 3& ' j = n j ; n j Mean fidelity of estimation F = =.095; = 0.9 N.Gisin & S.Popescu: Phys. Rev. Lett. 83, 43 (999) S.Massar: Phys. Rev. A 6, 0400R (000) E.Bagan, M.Baig, A.Brey, & R.Munoz-Tapia: Phys. Rev. Lett. 85, 530 (000) $ k#j ( ) n k ; n k 5

16 Optimal encoding of direction The optimal encoding of a direction into N qubits in the original preparation as well as after kth measurement mk k ( ) =U m ( k ) A A U mk The effects of the optimal measurement ( ); A = A j j,0 ( ) M r read: N / j=0 M k ( mk ) =U m ( k ) B B U mk N / ( ); B = j + j,0 j=0 E. Bagan et al., Phys. Rev. A 63, (00) Optimal decoding of direction The optimal decoding of direction into N qubits in the original preparation as well as after kth measurement F k = + x N /+ # ( ) k $ %& Where x N /+ P N /+ x is the largest zero of Legendre polynomial ( ) 6

17 Classical vs. quantum Sequence of observers: k Size of ensemble: N Where F k = + N k $ $ # N + % & # %& F k + k % % 0 # $ N &' # $ &' is the first zero of the Bessel function 0.4 J0 ( x) Sequence of observers: k Size of ensemble is large: N Large N limit F k k N F k k 0 N Robustness against observations Trade-off gain vs. disturbance Non-trivial estimation by everybody Entangled states are much better for storing (classical) information in a sense that it can be read by many independent observers property of classicality 7

18 Fair sharing of information Information about direction is encoded into N parallel spins K observers sequentially measure spins and estimate direction All fidelities of estimations have to be same: n N F = F =... = F K Weak optimal measurements M k () = ( k )I + k M () The measurement strength-parameter k Fair sharing of information II. The measurement strength-parameter k+ = Weak optimal measurements Large N limit (N=M.K) ( N + ) N + ( ) k ( N + ) + ( N + )( k ) + 4 ( k )( + N k ) F N,K ( ) = + (K,N ) N # F N,K $ # k ( ) + N $ N + ( K ) # $ N + % & K = % &' + M % # $ M + & ' 8

19 Generalization of GP by Bagan et al. BBBM have shown that if one is allowed to perform only space rotations on qubits then the best fidelity one can achieve is the one obtained in the case of two anti-parallel spins. N n N GP- BBBM F GPBBBM.4 N F n N N E.Bagan, M.Baig, A.Brey, & R.Munoz-Tapia: Phys. Rev. Lett. 85, 530 (000) Can we do better? Assume just one single-qubit preparator Nevertheless can we encode information in two qubits better than GP? n 9

20 Quantum compression: Three qubits ( ) 3 n 3 = 0 + = 3 3; ; + 3 3; + 3 3;3 % $ # ' & dimension of the Hilbert space of 3 qubits = 8 dimension of symmetric subspace = 4 basis vectors of symmetric subspace N;k can we map symmetric subspace of 3 qubits on -qubit HS? Compression of quantum information just one single-qubit preparator n Nevertheless can we encode information in two qubits better than GP? 3 Magic box ( n ) 3 0

21 Position coding 3; ; ; ; n 3 0 ( ) mapping of symmetric subspace of 3 qubits on -qubit HS? does there exist a state-independent unitary transformation? Efficient two-qubit encoding ( n ) = ( ) 3 position encoding ( n ) 3

22 Mean fidelity: two-qubit decoding ( n ) 3 position decoding 3 F = 4 5 = 0.8 > F GP = > F = 0.75 Generalization to N qubits n = 0 + n N N = N k k P # 0 ( N k ) k & ) ) $% '( k=0 N * = ) N k k, N k=0 + k P - /. N;k transformation on symmetric subspace 0 N 0 N N;k 0 ( N log ( N + )) (k) (k) is a binary representation of number of ones k

23 Position encoding of symmetric states first encode the symmetric state into the position of Second encode the position of into the binary number Defined for every symmetric states Then specify transformation for non-symmetric states transformation on symmetric subspace is efficient quadratic in N N F = N + Comparison of fidelities N n N GP- BBBM compression N.Gisin & S.Popescu: Phys. Rev. Lett. 83, 43 (999) S.Massar: Phys. Rev. A 6, 0400R (000) E.Bagan, M.Baig, A.Brey, & R.Munoz-Tapia: Phys. Rev. Lett. 85, 530 (000) 3

24 Asymptotics Position encoding of information about direction would allow for exponential asymptotics of the fidelity F n N N F GPBBBM N.4 F = N N + = N Summary quantum Bayesian inference from finite ensembles optimal measurements of finite ensembles recycling of quantum information coding and decoding of quantum information simulation of non-physical maps unambiguous measurements 4