Entanglement Measures and Monotones

Transcription

1 Entanglement Measures and Monotones PHYS Southern Illinois University March 30, 2017 PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

2 Quantifying Entanglement The next objective will be to quantify the amount of entanglement in a given state. We want to be able to say how much entanglement a state possesses. Again we will adopt an operational philosophy and measure entanglement in terms of LOCC. To begin, let us focus on two extremes: No Entanglement and Maximal Entanglement. As discussed above, every separable state possesses zero entanglement. What about for maximal entanglement? PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

3 Maximal Entanglement Recall the Teleportation Fidelity: Figure: The teleportation fidelity as a function of κ when performing the standard teleportation protocol T 0 using the shared state cos κ 00 + sin κ 11. The fidelity is maximized by Φ + = 1/2( We define Φ + as a maximally entangled two-qubit state. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

4 Bipartite Entanglement Measures Let D(H A H B ) denote the set of density matrices for systems A and B. A bipartite entanglement measure µ is any non-negative function defined on D(H A H B ) for all finite-dimensional Hilbert spaces H A and H B that satisfies the following three properties: 1. Vanishing on Separable States µ(ρ) = 0 if ρ is separable. 2. Normalized by Φ + µ ( Φ + Φ + ) = 1. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

5 Bipartite Entanglement Measures 3. Monotonically decreasing under LOCC (Golden Rule of Entanglement) If {A λ B λ } are Kraus operators corresponding to any LOCC operation, µ does not increase on average across all outcomes: µ(ρ) p λ µ(ρ λ ), λ µ does not increase if outcome is discarded: µ(ρ) µ( p λ ρ λ ) λ where ρ λ = 1 p λ (A λ B λ )ρ(a λ B λ ) and p λ = tr[(a λ A λ B λ B λ)ρ]. Any function µ just satisfying 3. is called an entanglement monotone. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

6 Bipartite Entanglement Monotones Remark If µ is an entanglement monotone, then µ(ρ) = µ(u V ρu V ) for all ρ and all local unitaries U and V. Remark If µ is an entanglement monotone, then µ(ρ) = constant for all separable states ρ. Remark (really important!) µ is an entanglement monotone if and only if it is a monotone for a single local measurement by either Alice or Bob. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

7 Pure State Entanglement Monotones LOCC protocols not discarding information transforms a pure state Ψ according to Ψ Ψ λ := 1 pλ A λ B λ Ψ, where {A λ B λ } are Kraus operators for the protocol. The LOCC transformation Ψ {p λ, Ψ λ } is called a pure state entanglement transformation. A pure state entanglement monotone is a function µ defined on pure states that is monotonic under all pure state entanglement transformations: µ( Ψ ) p λ µ( Ψ λ ). λ PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

8 Pure State Entanglement Monotones Construction of Monotones For a d d bipartite system, let f : L(C d ) R be any function with the following two properties: 1 Unitary invariance: f (σ) = f (UσU ) for all σ L(C d ) and all unitaries U; f only depends on eigenvalues for hermitian operators σ! 2 Concavity: f (λσ 1 + (1 λ)σ 2 ) λf (σ 1 ) + (1 λ)f (σ 2 ) for all σ 1, σ 2 L(C d ) and 0 λ 1. Theorem: µ( Ψ AB ) := f (tr B ( Ψ Ψ )) is a pure state monotone. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

9 Pure State Entanglement Monotones Proof: Remark For a pure state Ψ AB, the reduced states tr A ( Ψ Ψ ) and tr B ( Ψ Ψ ) have the same eigenvalues. Since f only depends on the eigenvalues, we therefore have f (tr B ( Ψ Ψ )) = f (tr A ( Ψ Ψ )). PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

10 A Recipe for Bipartite Entanglement Monotones Every pure state entanglement monotone can used to build a general (mixed-state) entanglement monotone using a technique called convex roof extensions. Let µ be a given pure state entanglement monotone. Then define the function µ on an arbitrary bipartite mixed state ρ by µ(ρ) = min q i µ( ϕ i ) q i, ϕ i where the minimization is taken over all pure state decompositions of ρ = i q i ϕ i ϕ i. i Theorem: µ is an entanglement monotone. PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11

11 A Recipe for Bipartite Entanglement Monotones Proof: PHYS Southern Illinois University Entanglement Measures and Monotones March 30, / 11