Steering and Entropic Uncertainty Relations

Transcription

1 Steering and Entropic Uncertainty Relations Ana C. S. Costa, Roope Uola, Otfried Gühne Department of Physics, University of Siegen - Germany September 5, 2017 na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

2 Summary 1 Steering LHS model Entropic steering criteria: Shannon entropy Entropic steering criteria: Tsallis entropy 2 Applications Werner states Isotropic states General two-qubit states One-way steerable states 3 Final remarks na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

3 Steering LHS model Local hidden state model Assume that Alice and Bob share a quantum state ρ AB. Alice makes measurements on her system and claims that with these measurements she can steer the state inside Bob s laboratory. Local hidden state model: p(a, b A, B) = λ p(λ)p(a A, λ)tr B [B b ρ B λ ]. September 5, / 15

4 Steering Entropic steering criteria: Shannon entropy Entropic steering criteria Shannon entropy: S(A) = k a k log(a k ), Relative entropy: D(A B) = k a k log ( ak b k ) If A 1, A 2 are independent distributions, with A(x, y) = A 1 (x)a 2 (y) - and similar for B 1, B 2, D(A B) = D(A 1 B 1 ) + D(A 2 B 2 ). September 5, / 15

5 Steering Entropic steering criteria: Shannon entropy Entropic steering criteria Consider the quantity: F (A, B) = D(A B A 1) 1. F (A, B) = ( ) pij p ij log = S(A, B) S(A) log(n). p i 1/N i,j Now, for a product distribution, p(a, b A, B) = λ p(λ)p(a A, λ)tr B[B b ρ B λ ], F (A, B) [ ] p(λ) D(p(a A, λ) p(a A, λ)) + D(p q (b B, λ) 1/N) λ = λ p(λ)s(b λ) log(n). 1 AB = joint measurement distribution (p ij = Tr[(A i B j )ρ]), A = marginal distribution (p i = Tr[(A i 1)ρ]), 1 = equal distribution: q j = 1/N j. na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

6 Steering Entropic steering criteria: Shannon entropy Entropic steering criteria Consider a set of measurements A k B k : [ ] S(B k A k ) log(n) ( ) p(λ)s(b k λ) log(n). k k λ If B k obey some entropic uncertainty relation k S(B k) CB S, we have for all unsteerable states: k = 2: S(B 1 ) + S(B 2 ) log(ω), k [ ] S(B k A k ) CB S. Ω min i,j ( 1 B i 1 Bj 2 2 ). The same criteria obtained by Walborn et. al. [PRL 106, (2011)]. September 5, / 15

7 Steering Entropic steering criteria: Tsallis entropy Entropic steering criteria: Tsallis entropy Tsallis entropy: S q (A) = 1 q 1 [ 1 i aq i Tsallis relative entropy: D q (A B) = 1 1 q ]. [ 1 i ] a q i. b q 1 i Product distributions: D(A B) = D(A 1 B 1 ) + D(A 2 B 2 ) + (q 1)D(A 1 B 1 )D(A 2 B 2 ) k [ ] S q (B k A k ) + (1 q)c(a k, B k ) CB T. In terms of probabilities, 1 q 1 k 2 1 ij (p (k) ij ) q (p (k) i ) q 1 CB T. 2 C(A, B) = i pq i (ln q(p i )) 2 i,j pq ij ln q(p i ) ln q (p ij ), and ln q (x) x1 q 1 1 q is the q-logarithm. September 5, / 15

8 Applications Werner states Application: Two-qubit Werner states ρ W = (1 p) 1 + p ψ ψ, where ψ = 1 ( ) S2(ρW) p S3(ρW) q q = 2,3 Ana C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty ofp Siegen -Relations Germany) September 5, / 15

9 Applications Isotropic states Application: Isotropic states Isotropic states: ρ iso = p φ + d φ+ d + 1 p d 2 1, where φ + = 1 d 1 i i. d Consider a set of mutually unbiased bases in an arbitrary dimension. p i = 1/d for all i, p ii = (1 + (d 1)p)/d 2 (occurring d times), p ij = (1 p)/d 2 (i j occurring d(d 1) times). These probabilities are the same for all measurements. [ m 1 1 ] q 1 d q ((1 + (d 1)p)q + (d 1)(1 p) q ) where CB T = m ln ( md q d+m 1) for q (0; 2] 3. 3 A. E. Rastegin, Eur. Phys. J. D 67, 269 (2013). i=0 C T B, na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

10 Applications Isotropic states Application: Isotropic states For a complete set of MUBs (if it exists) and q = 2, the violation of the generalized entropic steering criteria occurs for p > 1 d + 1. p crit d Entropic steering (q = 2) Entropic steering (q 1) Calculated via SDP [PRA 96, (2017)] Infinite meas. [PRL 98, (2007)] September 5, / 15

11 Applications General two-qubit states Application: General two-qubit states ρ = 1 4 ( 1 AB + a σ A 1 B + 1 B b σ B + 3 c i σ A i i=1 σ B i ) Entropic steering for three measurements (q = 2): { 3 [ 1 a S3 e 2 (ρ) = max 0, 1 i bi 2 ci 2 ] } + 2a i b i c i 2(1 ai 2). i=1 Linear steering quantifier for three measurements ( n k=1 A k B k 3, where A i = û i σ, B i = ˆv i σ): { } c S3(ρ) l 2 = max 0, 1 + c2 2 + c A. C. S. Costa and R. M. Angelo, PRA 93, 2, (R) (2016). na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

12 Applications General two-qubit states Application: General two-qubit states September 5, / 15

13 Applications One-way steerable states Application: One-way steerable states ρ AB = p ψ(θ) ψ(θ) + (1 p)1/2 ρ θ B, [ ] where ψ(θ) = cos(θ) 00 + sin(θ) 11 and ρ θ B = Tr A ψ(θ) ψ(θ). For three measurement settings, this state is one-way steerable 5 for 1 θ [0, π/4] if 3 < p 1/ sin 2 (2θ). Generalized entropic steering criteria and q = 2: this state is one-way steerable for sin 2 (2θ) 2 cos(2θ) 1 < p sin 2 (2θ) pcrit θ 5 Y. Xiao et. al, Phys. Rev. Lett. 118, (2017). na C. S. Costa, Roope Uola, Otfried Gühne (Department Steering of Physics, and Entropic University Uncertainty of Siegen -Relations Germany) September 5, / 15

14 Final remarks Final remarks Generalized entropic steering criteria based on Tsallis entropy (paper in preparation). Applications: isotropic states, general two-qubit states, one-way steerable states. Future work: Extension to multipartite systems; entropic uncertainty relations in the presence of quantum memory; bound entangled states. September 5, / 15

15 Final remarks Thank you for your attention! Muito obrigada! September 5, / 15