Quantum Fisher Information. Shunlong Luo Beijing, Aug , 2006

Transcription

1 Quantum Fisher Information Shunlong Luo Beijing, Aug , 2006

2 Outline 1. Classical Fisher Information 2. Quantum Fisher Information, Logarithm 3. Quantum Fisher Information, Square Root 4. Two Open Problems 5. Summary

3 1. Classical Fisher Information Fisher, 1922, 1925 For a parametric family of probabilities p θ, its Fisher information is defined as ( ) 2 I (p θ ) = θ logp θ(x) p θ (x)dx ( ) 2 = 4 pθ (x) dx. θ Logarithm and Square Root

4 Statistical Inference Data: n samples x 1, x 2,, x n p θ (x). Aim: Estimate the parameter θ. Cramér-Rao: Unbiased estimate θ θ 1 ni (p θ ). Maximum Likelihood: θ(x 1,, x n ) n( θ θ) N(0, 1/I (pθ )).

5 Fisher Information and Shannon Entropy For a probability density p, its Shannon entropy is S(p) = p(x)lnp(x)dx. de Bruijin identity: t S(p g t) = 1 t=0 2 I (p), where g t (x) = 1 2πt e x2 /2t.

6 Informational Isoperimetric Inequality Classical isoperimetric inequality: Area Length2. 4π Informational isoperimetric inequality: e 2H(p) 1 2πe I (p). Fisher Information: Length 2 Shannon entropy power: Area

7 Fisher Information and Uncertainty Relations 1927, Heisenberg, Uncertainty Principle 1925, Fisher, Fisher Information, Statistical Inference What would happen if Heisenberg had met Fisher to discuss quantum measurement?

8 Variance, Fisher Information, Cramér-Rao If ψ θ satisfies i ψ θ θ = Aψ θ, then ψθ Ã = 1 4 I ( ψ θ 2 ). A = Re( Aψθ ψ ): local value (weak value). θ Ã = A A: quantum fluctuation. 4( ψθ A) 2 I ( ψ θ 2 ) 1 ( ψθ θ) 2.

9 Husimi Phase Space Probability Let F be an analytic function on C such that ρ F (z) = F (z) 2 e z 2 is a probability on (C, π 1 dzd z). Define its Fisher information matrix as I(ρ F ) = 4 ( ρ F (z)) ρ F (z)π 1 dzd z.

10 Theorem. ( ) 1 + ReΛ ImΛ I(ρ F ) = 2. ImΛ 1 ReΛ Here Λ = AF, F (a ) 2 F, F, AF = (a F ) 2 F. In particular, we have the information identity I 11 (ρ F ) + I 22 (ρ F ) = 4.

11 2. Quantum Fisher Information, Logarithm Recall the classical Fisher information can be written as I (p θ ) = l 2 θ (x)p θ (x)dx with l θ determined by θ p θ = p θ l θ = l θ p θ = 1 2 (l θp θ + p θ l θ ).

12 Analogy between Classical and quantum From Classical to Quantum: Probability p θ Density operator ρ θ Integral Trace operation tr Score l θ Quantum score L θ

13 Generalizing to Quantum Scenario Helstrom, Holevo, 60 Quantum Fisher information Here L θ is defined by I (ρ θ ) = trρ θ L θ L θ. θ ρ θ = ρ θ L θ right θ ρ θ = L θ ρ θ left θ ρ θ = 1 2 (L θρ θ + ρ θ L θ ) symmetric

14 More generally, we may define ( 1 ) I µ (ρ θ ) = tr ρ t θl θ ρ 1 t θ L θ dµ(t), with L θ determined by θ ρ θ = ρ t θl θ ρ 1 t θ dµ(t).

15 Comparison Proposition. I (δ0 +δ 1 )/2(ρ θ ) I µ (ρ θ ) I δ1 (ρ θ ). I (δ0 +δ 1 )/2(ρ θ ): Quantum Fisher Information via symmetric logarithmic derivative I δ1 (ρ θ ): Quantum Fisher Information via right logarithmic derivative

16 Uncertainty Relation, Quantum Cramér-Rao If ρ θ satisfies the Landau-von Neumann equation i θ ρ θ = [H, ρ θ ], and M is an unbiased measurement of the parameter θ, then 4 ρθ H min µ I µ (ρ θ ) 1 ρθ M.

17 3. Quantum Fisher Information, Square Root Wigner, Yanase, 1963 Skew information I (ρ, H) = 1 2 tr[ ρ, H] 2. Quantum measurement

18 Basic Properties of Skew Information 1 I (ρ, H) ρ H := trρh 2 (trρh) 2. 2 Invariance: I (UρU, H) = I (ρ, H) if UH = HU. 3 Additivity I (ρ 1 ρ 2, H 1 +H 2 ) = I (ρ 1, H 1 )+I (ρ 2, H 2 ). 4 Convexity I (λ 1 ρ 1 +λ 2 ρ 2, H) λ 1 I (ρ 1, H)+λ 2 I (ρ 2, H).

19 Three Interpretations of Skew Information 1 As a measure of non-commutativity between ρ and H, 2 As a kind of quantum Fisher information, 3 As the quantum uncertainty of H in the state ρ.

20 Skew Information as Quantum Fisher Information Generalize classical Fisher information ( ) 2 p θ (x) I F (p θ ) := 4 dx θ to the quantum scenario, we may define ( ) 2 ρθ I F (ρ θ ) := 4tr θ as a kind of quantum Fisher information. Here ρ θ is a family of density operators.

21 In particular, if ρ θ satisfies the Landau-von Neumann equation then i ρ θ θ = [H, ρ θ], ρ 0 = ρ, I F (ρ θ ) = 4tr[ρ 1/2, H] 2 = 8I (ρ, H).

22 Skew Information as Quantum Uncertainty Decompose the variance ρ H formally as Uncertainty =Classical + Quantum ρ A = C(ρ, A) + I (ρ, A).

23 Example Composite system: H 1 H 2. Observables H 1 = i α i i 1 i 1, H 2 = i β i i 2 i 2. Consider states ρ = Ψ Ψ, with Ψ = 1 N σ = 1 N i i 1 i 2 i i, with i = i 1 i 2. i

24 ρ: Bell State(maximally entangled) σ: classical mixture Put A = H 1 H 2, then I (ρ, A) = 1 αi 2 βi 2 1 ( ) 2 α i β i N N C(ρ, A) = 0, and i i I (σ, A) = 0 C(σ, A) = 1 N αi 2 βi 2 1 ( ) 2, α i β i N i i

25 Comparing with von Neumann Entropy Problem: How to quantify the uncertainty of density operator ρ? 1927, von-neumann, quantum entropy: S(ρ) = trρlogρ, which quantifies the classical mixing uncertainty of ρ.

26 Simple Properties: 1. S(UρU ) = S(ρ). 2. S(λ 1 ρ 1 + λ 2 ρ 2 ) λ 1 S(ρ 1 ) + λ 2 S(ρ 2 ). 3. S(ρ 1 ρ 2 ) = S(ρ 1 ) + S(ρ 2 ). 4. S(ρ) S(tr 2 ρ) + S(tr 1 ρ).

27 A Quantum Uncertainty Measure {H j : j = 1, 2, n 2 }: orthogonal base for the real n 2 -dimensional space of observables on C n with the Hilbert-Schmidt product. For a state ρ on C n, define I (ρ) = j I (ρ, H j ), then I (ρ) = n (tr ρ) 2.

28 Heisenberg Uncertainty Relation for Mixed States Variance characterization ρ X ρ Y 1 4 trρ[x, Y ] 2. Informational characterization: Put U(ρ, X ) = ( ρ X ) 2 C 2 (ρ, X ). Then U(ρ, X )U(ρ, Y ) 1 4 trρ[x, Y ] 2.

29 Quantum Cramér-Rao Inequality If ρ θ satisfies the Landau-von Neumann equation with energy operator H, and M is an unbiased estimate of the parameter θ, then ρθ M I (ρ, H) 1 4.

30 Bell Type Inequality 1. For separable states I (ρ 1 ρ n, H H n ) n. Here H i are spin observables. 2. For general states I (ρ, H H n ) n 2. Chen ZQ, Physical Review A 71 (2005),

31 4. Two Open Problems For any α (0, 1), define the Wigner-Yanase-Dyson information as I α (ρ, H) = 1 2 tr[ρα, H][ρ 1 α, H].

32 Problem 1 Conjecture: I α (ρ, H) I α (ρ 1, H 1 ) + I α (ρ 2, H 2 ). Here H = H H 2, H 1 and H 2 are observables on the Hilbert spaces H 1 and H 2, respectively, ρ is a state on the tensor product space H 1 H 2, ρ 1 = tr H2 ρ is the partial trace of ρ over H 2, and ρ 2 = tr H1 ρ.

33 Problem 2 Let each operator E j commutes H, and they satisfy E j E j = 1. Conjecture: j j I α (E j ρe j, H) I α(ρ, H).

34 5. Summary Classical Fisher information: unique Classical inference, physical law, powerful tool for analysis, Quantum Fisher information: Infinite family quantum inference, quantum information, operator theory,

35 References 1 Physical Review A, 73, (2006), ; 72, (2005), ; 69 (2004), Theoret. Math. Phys., 143 (2005), IEEE Trans. Information Theory, 50 (2004), Proc. American Math. Soc., 132 (2004), Physical Review Letters, 91 (2003), Foundations of Physics, 32 (2002), International Journal of Theoretical Physics, 41 (2002), Journal of Statistical Physics, 102 (2001), Letters in Mathematical Physics, 53 (2000), 243.

36 Thank You!