Quantum hypothesis testing for Gaussian states

Transcription

1 Quantum hypothesis testing for Gaussian states Wataru Kumagai (Tohoku University) Joint Work with Masahito Hayashi Bernoulli Society Satellite Meeting to the ISI World Statistics Congress 203 (203/9/4)

2 Contents Quantum System and Gaussian State Formulation of Quantum Hypothesis Testing Optimal Test in Min-Max Criterion Conclusion 2

3 Quantum System and Quantum State Quantum theory assumes that every physical object has a quantum state on a quantum system. Quantum system and quantum state Quantum system complex Hilbert space H, Quantum state linear operator ρ on H satisfying (i) positive definiteness 0 ρ (ii) normalization condition Trρ= { When dimh=d<, every quantum state can be represented as Udiag( p,..., p ) U. d Probability Distribution Unitary Transformation 3

4 Quantum System and Quantum State When physical objects are identically and independently prepared, the quantum state has a tensor product form. i.i.d. system and i.i.d. state n-i.i.d quantum system tensor product space H n, n-i.i.d quantum state quantum state of tensor product form ρ n on H n corresponds to i.i.d. random variable 4

5 Gaussian state Gaussian State Quantum state of optical laser with Gaussian noise e C N 2 N C d 2-dim Gaussian noise quantum state of laser { N C: 0 : mean parameter number parameter Mean photon number in the Gaussian state 5

6 Gaussian State Quantum LAN (Gut ă, Kahn, 2006) n Le Cam For quantum state model S c on 2-dim quantum system, S n c S G J c n Gauss, 0 0 ( ) i.i.d. quantum state Gaussian distribution+ Gaussian state Uniform Convergence (Quantum Le Cam Distance) Gaussian state = Quantum version of Gaussian distribution 6

7 Contents Quantum System and Gaussian State Formulation of Quantum Hypothesis Testing Optimal Test in Min-Max Criterion Conclusion 7

8 Quantum Hypothesis Testing A method to statistically decide whether a quantum state satisfies a hypothesis(=condition) of interest For an unknown quantum state H Decision method : vs. H 0 0 : in a model S, ( 0 ) test operator 0 T I ( two-valued measurement) Error probability Type I error prob.: Type II error prob.: ( ) Tr( ) T T ( ) Tr ( T ) T 0 8

9 Level of test Min-Max criterion Min-Max Criterion Test operator T with level α α T (θ) α (θ Θ 0 ). When quantum state model has a nuisance parameter H S,, : vs. H 0 0 : Min-Max criterion requires that the type II error probability of a test operator is as little as possible w.r.t. the nuisance parameter ( ) 0 Min-Max test T arg min sup T ': test T ', for 9

10 Correspondence Table Conventional Setting Sample space Probability distribution (i.i.d probability distribution) LAN Gaussian distribution Test function Quantum Setting Quantum system Quantum state (i.i.d. quantum state) Quantum LAN Gaussian state Test operator 0

11 Contents Quantum System and Gaussian State Formulation of Quantum Hypothesis Testing Optimal Test in Min-Max Criterion Conclusion

12 Min-Max Test Theorem (WK, MH 203) For { Gaussian model testing problem Min-Max test is given as follows. n C, N 0, H N N vs. H : N, : N 0 0 0,N n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 2

13 Min-Max Test Theorem (WK, MH 203) Invertible Transformation n For { Gaussian BS model n testing problem Min-Max test is given as follows. n C, N 0,, 0 H N N vs. H : N, : N 0 0 0,N n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 3

14 Min-Max Test Theorem (WK, MH 203) 2Discard the first component For { n Gaussian model n 0 testing problem Min-Max test is given as follows. n nc 0, 0, N H N vs. H : N, : N 0 0 0,N n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 4

15 Min-Max Test Theorem (WK, MH 203) 2Discard the first component Due to quantum Hunt-Stein theorem, { n Gaussian model N For the performance testing problem does not change Min-Max test is given as follows. C, N 0,, H N N vs. H : N, : N 0 0 0,N n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 5

16 Min-Max Test Theorem (WK, MH 203) For { n 3Sufficient Statistic Gaussian model Min-Max test is given as follows. n n C, N 0, ~ k kn H N N vs. H : N, 0 Geom,..., testing problem N : N 0 0 0,N n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 6

17 Min-Max Test Theorem (WK, MH 203) For { 3Sufficient Statistic from n Gaussian quantum model state to probability testing problem Min-Max test is given as follows. C, N 0, H N N vs. H : N, : N distribution,n n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 7

18 Min-Max Test Theorem (WK, MH 203) For { Gaussian model testing problem Min-Max test is given as follows. n C, N 0, OTF H N N vs. H : N, 4Optimal testing function k k i : N 0 0 0,..., n,n n n,n,n BS 0,N 0,N k kn OTF BS=Beam splitter, =Photon number measurement, OTF=Optimal testing function. i ( 0or) 8

19 Table of Min-Max Tests Gaussian distribution G(, v) (μ:mean, v: variance) v:known v:unknown μ R 0 vs. μ >R 0 N-P lemma + MLR (Bioequivalence problem) μ:known μ:unknown v V 0 vs. v>v 0 χ 2 -test χ 2 -test Gaussian state ( ) (ζ:mean: number) N:known N:unknown ζ R 0 vs. ζ >R 0 ( * if R 0 =0) ζ:known ζ:unknown N N 0 vs. N>N 0 * :optimal under unbiasedness condition 9

20 Table of Min-Max Tests Equivalence between Gaussian distributions v:known G(, v ) and G(, v ) 0 0 v:unknown μ 0 =μ vs. μ 0 μ (v 0 =v ) N-P lemma + MLR t-test μ 0, μ :known μ 0, μ :unknown v 0 =v vs. v 0 v F-test F-test Equivalence between Gaussian states N 0 :known ( ) and ( ) 0 0 N 0 :unknown ζ 0 =ζ vs. ζ 0 ζ (N 0 =N ) ζ 0, ζ :known * ζ 0, ζ :unknown N 0 =N vs. N 0 N * * * : optimal under unbiasedness condition 20

21 Contents Quantum System and Gaussian State Formulation of Quantum Hypothesis Testing Optimal Test in Min-Max Criterion Conclusion 2

22 Conclusion Quantum hypothesis testing was considered. Model of Gaussian states was treated. Gaussian state corresponds to Gaussian distribution. Several optimal tests in Min-max criterion (Min-Max test) ware derived. 22

23 Thank you References M. Gut ă., J. Kahn. Local asymptotic normality for qubit states, Phys. Rev. A, 73, (2006). J. Kahn, M. Gut ă, "Local Asymptotic Normality for Finite Dimensional Quantum Systems", Commun. Math. Phys. 289, (2009). N. A. Bogomolov, "Minimax measurements in a general statistical decision theory", Theor. Prob. Appl. Vol. 26, (982). M. Ozawa, On the noncommutative theory of statistical decision. Research Reports on Information Sciences (980). W. Kumagai, M. Hayashi, Quantum hypothesis testing for quantum Gaussian states: Quantum analogues of chi-square, t and F tests, Commun. Math. Phys., 38(2), (203). 23