Maximally entangled mixed states with fixed marginals Giuseppe Baio SUPA & University of Strathclyde, Glasgow, UK 51 Symposium of Mathematical Physics, Toruń, Poland 17 th June 2019 17 th June 2019 Giuseppe Baio 51 SMP Toruń 1 / 22
My research activity @ Strathclyde Computational Nonlinear and Quantum Optics Cold Atoms, Nanophotonics, Quantum Information and Many-Body Physics, Structured Light etc. http://cnqo.phys.strath.ac.uk ColOpt ITN Collective effects and optomechanics in ultra cold matter https://www.colopt.eu/ 17 th June 2019 Giuseppe Baio 51 SMP Toruń 2 / 22
My research activity @ Strathclyde Computational Nonlinear and Quantum Optics Cold Atoms, Nanophotonics, Quantum Information and Many-Body Physics, Structured Light etc. http://cnqo.phys.strath.ac.uk ColOpt ITN Collective effects and optomechanics in ultra cold matter https://www.colopt.eu/ Talk based on recent paper: Phys. Rev. A 99, 062312 (2019) Joint work with: D. Chruściński, G. Sarbicki (Toruń, Poland), P. Horodecki (Gdańsk, Poland), A. Messina (Palermo, Italy) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 2 / 22
Outline Maximally entangled mixed states (MEMS) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 3 / 22
Outline Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information 17 th June 2019 Giuseppe Baio 51 SMP Toruń 3 / 22
Outline Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case 17 th June 2019 Giuseppe Baio 51 SMP Toruń 3 / 22
Outline Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case Higher dimensions: Two qutrit case and quasidistillation 17 th June 2019 Giuseppe Baio 51 SMP Toruń 3 / 22
Outline Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case Higher dimensions: Two qutrit case and quasidistillation Future directions 17 th June 2019 Giuseppe Baio 51 SMP Toruń 3 / 22
Preliminaries: Mixed bipartite entanglement and measures von Neumann entropy for pure states | Ψ AB �� Ψ AB | : E (Ψ AB ) = S ( ρ A ) = − Tr( ρ A log ρ A ) (1) 1 M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 4 / 22
Preliminaries: Mixed bipartite entanglement and measures von Neumann entropy for pure states | Ψ AB �� Ψ AB | : E (Ψ AB ) = S ( ρ A ) = − Tr( ρ A log ρ A ) (1) For mixed states, i.e. Tr( ρ 2 AB ) < 1 : convex roof construction, e.g.: � EOF( ρ AB ) = min p k E (Ψ k ) (2) p k , Ψ k k 1 M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 4 / 22
Preliminaries: Mixed bipartite entanglement and measures von Neumann entropy for pure states | Ψ AB �� Ψ AB | : E (Ψ AB ) = S ( ρ A ) = − Tr( ρ A log ρ A ) (1) For mixed states, i.e. Tr( ρ 2 AB ) < 1 : convex roof construction, e.g.: � EOF( ρ AB ) = min p k E (Ψ k ) (2) p k , Ψ k k Several tools adopted: concurrence and negativity 1 : N ( ρ AB ) ≡ 1 2 ( � ρ τ AB � 1 − 1) (3) Partial transpose: ρ τ AB = ( I ⊗ τ ) ρ AB 1 M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 4 / 22
Maximally entangled mixed states (MEMS) Relation between entanglement and purity Tr( ρ 2 AB ) : 2 2 W. J. Munro et al. , Phys. Rev. A 64, 030302(R) (2001). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 5 / 22
Maximally entangled mixed states (MEMS) Relation between entanglement and purity Tr( ρ 2 AB ) : 2 2 W. J. Munro et al. , Phys. Rev. A 64, 030302(R) (2001). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 5 / 22
Maximally entangled mixed states (MEMS) Relation between entanglement and purity Tr( ρ 2 AB ) : 2 Werner 2 W. J. Munro et al. , Phys. Rev. A 64, 030302(R) (2001). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 5 / 22
Maximally entangled mixed states (MEMS) Relation between entanglement and purity Tr( ρ 2 AB ) : 2 Werner MEMS : states ρ ∗ such that any measure E ( ρ ∗ ) ≥ E ( Uρ ∗ U † ) , ∀ U . 2 W. J. Munro et al. , Phys. Rev. A 64, 030302(R) (2001). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 5 / 22
Maximally entangled mixed states (MEMS) Two qubit MEMS found solving the spectral constrained analogue 3 3 F. Verstraete et al. , Phys. Rev. A 64, 012316 (2001). 4 T. C. Wei et al. , Phys. Rev. A 67, 022110 (2003). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 6 / 22
Maximally entangled mixed states (MEMS) Two qubit MEMS found solving the spectral constrained analogue 3 Theorem (Verstraete, 2001) Given a state ρ = ΦΛΦ † , the unitary maximising EOF and negativity is: 0 0 0 1 √ √ 1 / 2 0 1 / 2 0 √ √ D φ Φ † U = ( U 1 ⊗ U 2 ) (4) 1 / 2 0 − 1 / 2 0 0 1 0 0 MEMS depend on the entanglement measure considered 4 3 F. Verstraete et al. , Phys. Rev. A 64, 012316 (2001). 4 T. C. Wei et al. , Phys. Rev. A 67, 022110 (2003). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 6 / 22
Maximally entangled mixed states (MEMS) Two qubit MEMS found solving the spectral constrained analogue 3 Theorem (Verstraete, 2001) Given a state ρ = ΦΛΦ † , the unitary maximising EOF and negativity is: 0 0 0 1 √ √ 1 / 2 0 1 / 2 0 √ √ D φ Φ † U = ( U 1 ⊗ U 2 ) (4) 1 / 2 0 − 1 / 2 0 0 1 0 0 MEMS depend on the entanglement measure considered 4 � 2 For negativity: ρ MEMS = 1 − r 4 I 2 ⊗ I 2 + rP + P + 2 = 1 2 , i,j =1 | ii �� jj | 2 3 F. Verstraete et al. , Phys. Rev. A 64, 012316 (2001). 4 T. C. Wei et al. , Phys. Rev. A 67, 022110 (2003). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 6 / 22
Fixing marginals: reconstructing states from local info Entanglement characterization: 5 5 G. Adesso et al. , Phys. Rev. A 68, 062318 (2003) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 7 / 22
Fixing marginals: reconstructing states from local info Entanglement characterization: 5 → Maximally entangled marginally mixed states ( MEMMS ), i.e. MEMS with respect to local purities 5 G. Adesso et al. , Phys. Rev. A 68, 062318 (2003) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 7 / 22
Fixing marginals: reconstructing states from local info Entanglement characterization: 5 → Maximally entangled marginally mixed states ( MEMMS ), i.e. MEMS with respect to local purities What is the upper bound E max ( ρ ) on bipartite entanglement when only marginals are known? MEMS with respect to fixed marginals Given ρ A , ρ B , find E max ( ρ ) : Tr B ( ρ ) = ρ A , Tr A ( ρ ) = ρ B 5 G. Adesso et al. , Phys. Rev. A 68, 062318 (2003) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 7 / 22
Fixing marginals: reconstructing states from local info Entanglement characterization: 5 → Maximally entangled marginally mixed states ( MEMMS ), i.e. MEMS with respect to local purities What is the upper bound E max ( ρ ) on bipartite entanglement when only marginals are known? MEMS with respect to fixed marginals Given ρ A , ρ B , find E max ( ρ ) : Tr B ( ρ ) = ρ A , Tr A ( ρ ) = ρ B Characterizing states from local measurements : Quantum marginal constraints (Klyachko), Quantum tomography etc. 5 G. Adesso et al. , Phys. Rev. A 68, 062318 (2003) 17 th June 2019 Giuseppe Baio 51 SMP Toruń 7 / 22
What is known: Two qubits Let ρ A = diag { 1 − λ A , λ A } , ρ B = diag { 1 − λ B , λ B } be two qubit states. 0 , 1 � � λ A , λ B ∈ , λ A ≥ λ B 2 17 th June 2019 Giuseppe Baio 51 SMP Toruń 8 / 22
What is known: Two qubits Let ρ A = diag { 1 − λ A , λ A } , ρ B = diag { 1 − λ B , λ B } be two qubit states. 0 , 1 � � λ A , λ B ∈ , λ A ≥ λ B 2 C ( ρ A , ρ B ) set of two-qubit states with fixed marginals: ǫ ∆ 12 ∆ 13 ∆ 14 − ǫ ∆ 23 − ∆ 13 ρ AB = ρ A ⊗ ρ B + (5) − ǫ − ∆ 12 (c.c) ǫ 17 th June 2019 Giuseppe Baio 51 SMP Toruń 8 / 22
What is known: Two qubits Let ρ A = diag { 1 − λ A , λ A } , ρ B = diag { 1 − λ B , λ B } be two qubit states. 0 , 1 � � λ A , λ B ∈ , λ A ≥ λ B 2 C ( ρ A , ρ B ) set of two-qubit states with fixed marginals 6 : ǫ · · ∆ 14 − ǫ ∆ 23 · ρ AB = ρ A ⊗ ρ B + (5) − ǫ · (c.c) ǫ 6 F. Verstraete et al. , Phys. Rev. A 64, 012316 (2001). 17 th June 2019 Giuseppe Baio 51 SMP Toruń 8 / 22
What is known: Two qubits Let ρ A = diag { 1 − λ A , λ A } , ρ B = diag { 1 − λ B , λ B } be two qubit states. � 0 , 1 � λ A , λ B ∈ , λ A ≥ λ B 2 C ( ρ A , ρ B ) set of two-qubit states with fixed marginals: � 1 − λ A · · (1 − λ A ) λ B · 0 · · ρ AB = ˜ (5) · · λ A − λ B · � (1 − λ A ) λ B · · λ B � � ( λ A − λ B ) 2 + 4 λ B (1 − λ A ) ρ AB ) = 1 � Maximal neg: N (˜ λ A − λ B − 2 17 th June 2019 Giuseppe Baio 51 SMP Toruń 8 / 22
Recommend
More recommend
