Generalized Bell Inequality and Entanglement Witness

Transcription

1 Nonlocal Seminar 2005 Bratislava, April 29th 2005 Reinhold A. Bertlmann Generalized Bell Inequality and Entanglement Witness Institute for Theoretical Physics University of Vienna

2 Motivation Composite quantum system: bipartite multipartite in mixed state Quantum correlations? not reproducible by classical means nonlocal contextual features J.S.Bell Entanglement easy for pure states Bell Inequality entanglement criterion difficult for mixed states entanglement criterion separable states entangled states classical correlations quantum correlations given a composite mixed system quantum state ρ separable or entangled? Generalized Bell Inequality 1/17

3 Contents Composite quantum system: Alice & Bob in mixed state Usual Bell inequalities (BI) fail as entanglement criterion Generalized Bell inequality (GBI) entanglement witness Theorem: entangled state distance violation of GBI Example: 2 spins σ A and σ B Alice & Bob Geometry in spin space 2/17

4 States observables consider tensor product of Hilbert spaces H = H A H B Alice and Bob Observable: A hermitian operator, matrix State: ρ density matrix ρ, A elements of H scalar product expectation value of A (ρ A) = tr(ρa) A ψ norm A 2 = ( tra 2) 1 2 3/17

5 Separability Set of separable states defined by { S = ρ = p i ρ i A ρ i B i 0 pi 1, i } p i = 1 Entangled state if not separable ω S c complement of S S S c = H Separability criterion (2 2 and 2 3 dimensions) 1. Theorem: positive partial transposition Peres, Horodecki (½ A T B ) ρ 0 ρ separable 2. Theorem: reduction Horodecki ½ A ρ B ρ 0 ρ separable reduced density matrix ρ B = tr A ρ 4/17

6 Usual BI s consider usual Bell inequalities CHSH inequality A CHSH... CHSH operator (ρ A CHSH ) 2 with A CHSH = n σ A ( m m ) σ B + n σ A ( m+ m ) σ B ρ local ρ separ QM (ρ A CHSH ) = 2 2 ρ = ψ ψ Bell singlet with ψ = 1 2 ( A B A B ) 5/17

7 Werner states shift CHSH-operator (ρ 2 ½ A CHSH ) 0 and (ρ 2 ½ A CHSH ) < 0 inequality valid for all entangled states? YES for pure entangled states NO for mixed entangled states Example: Werner state ρ W = p ρ + (1 p) 1 4 ½ for 1 3 < p 1 2 satisfies CHSH Bell inequality! usual BI s not good to find entangled states 6/17

8 GBI entanglement witness Generalized Bell inequality GBI detects entanglement of state observable A such that (ρ A) 0 ρ S (ω A) < 0 for some ω S c given entangled state ω (always) some operator A satisfying GBI A A w entanglement witness A A t tangent functional if ρ 0 S such that (ρ 0 A) = 0 then ρ 0 S A t characterizes convex set S of separable states Entanglement measure defined by distance D(ω) E(ω) D(ω) = min ρ S ρ ω 2 2 distance D(ω) of entangled state ω S c to set S of separable states 7/17

9 GBI theorem consider GBI GBI (ρ A) (ω A) 0 ρ S maximal violation of GBI B(ω) = max [ min (ρ A) (ω A) ] A α 2 1 ρ S Theorem B(ω) = D(ω) min of D ρ 0 Bertlmann-Narnhofer-Thirring ω S c max of B A max A t A max = ρ 0 ω (ρ 0 ρ 0 ½ ω) ρ 0 ω 2 8/17

10 Illustration of GBI theorem D(ω) = B(ω) ω ρ 0 S H S A max maximal violation of GBI B(ω) is equal to distance D(ω) ω set of separable states S opt. oper. A max is tangent to set S 9/17

11 Example Alice & Bob 2 spins σ A and σ B Alice & Bob consider quantum states ω α = 1 4 ( ½ α σa σ B ) 1 3 α 1 possible range separable state with minimum distance difference ρ 0 ω α = 1 4 ρ 0 = 1 4 ( ½ 1 3 σ A σ B ) S mixed product states ( 1) α σa σ B for α 1 10/17

12 Entanglement witness Distance of ω to set S measure of entanglement D(ω α ) = ρ 0 ω α 2 = Entanglement witness 3 2 ( 1) α 3 σ A σ B 2 = 2 3 A max = ρ 0 ω α (ρ 0 ρ 0 ω α )½ = 1 ρ 0 ω α σ A σ B ) (½ Tangent functional { 1 (ρ 0 A max ) = tr 4 ( 1 ½ 3 σ ) 1 } A σ B σ A σ B ) (½ = 0 11/17

13 GBI Alice & Bob Separable state general ρ = 1 4 ( ½ + n σa ½ B + ½ A m σ B + n σ A m σ B ) GBI (ρ A max ) = 1 2 (1 + cosδ) 0 n m = cos δ δ = 0 0 δ = 180 ρ 0 { 1 ( ) 1 } (ω α A max ) = tr ½ α σa σ B σ A σ B ) (½ = (1 3α) < 0 for 1 3 < α 1 = 1 3 for α = 1 ω α=1 = ρ Bell singlet 12/17

14 GBI maximal violation maximal violation of GBI B(ω α ) = max [ min(ρ A) (ω α A) ] = (ω α A max ) A α 2 1 ρ S (ρ 0 A max ) = 0 A max = 3 2 (α 1 3 ) D(ω α) distance in H nature of states 1 α ω α S separable 1 3 < α 1 mixed entangled Werner states α = 1 pure & maximally entangled Bell singlet ρ 13/17

15 Geometry of pure entangled states 4 Bell states ψ ρ = 1 4 (½ σx A σ x B σ y A σy B σz A σ z B) P 0 φ ω = 1 4 (½ σx A σ x B + σ y A σy B + σz A σ z B) P 1 φ + ω + = 1 4 (½ + σx A σ x B σ y A σy B + σz A σ z B) P 2 ψ + ρ + = 1 4 (½ + σx A σ x B + σ y A σy B σz A σ z B) P 3 density matrix in (c 1, c 2, c 3 ) parameter space ω c = 1 4 ( ½ + 3 i=1 c i σa i σb i ) positivity = tetrahedron (P 0, P 1,P 2, P 3 ) partial transposition positivity separable Thm Peres Horodecki 14/17

16 P1 P2 P3 P0 intersection of tetrahedrons = separable states double-pyramide (convex set) 15/17

17 Summary GBI criterion for entanglement entanglement measure distance in H S D(ω) = B(ω) maximal violation of GBI usual BI criterion for nonlocality contextuality basis for quantum communication and information 16/17

18 Literature GBI R.A. Bertlmann, H. Narnhofer W. Thirring: Phys.Rev.A 66, (2002) M. Horodecki, P. Horodecki, R. Horodecki: in Quantum Information, eds. G. Alber et al., Springer Tracts in Modern Physics Vol. 173, p. 151 (2001) K.G.H. Vollbrecht, R.F. Werner: quant-ph/ B. Terhal: Phys.Lett.A 271, 319 (2000) and quant-ph/ D. Bruß: quant-ph/ M. Lewenstein, D. Bruß, J.I. Cirac, B. Kraus, M. Kuś, J. Samsonowicz, A. Sanpera, R. Tarrach: quant-ph/ M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki: Phys.Rev.A 62, (2000) and quant-ph/ /17