Entanglement Measures and Monotones Pt. 2

Transcription

1 Entanglement Measures and Monotones Pt. 2 PHYS Southern Illinois University April 8, 2017 PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

2 Entanglement Monotones in Greater Detail A function µ : D(H A H B ) R is called an entanglement monotone if it does not increase on average during any LOCC operation. Every LOCC operation is described by product Kraus operators {A λ B λ } λ combined with some coarse-graining of the measurement outcomes λ (i.e. forgetting or discarding information). A coarse-graining is a partitioning of measurement outcomes λ into disjoint sets S k. Every LOCC operation then generates a transformation of the form: ρ 1 Pr{S k } λ S k (A λ B λ )ρ(a λ B λ ), This is the post-measurement state as described by an experimenter not knowing the exact outcome λ, but knowing that λ belongs to subset S k. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

3 Entanglement Monotones in Greater Detail Notice that the coarse-graining map generalizes all scenarios: If the experimenter is able to learn all of the measurement outcomes when they occur, then each S k contains only one outcome λ k and Pr{S k } = p λk : ρ 1 Pr{S k } λ S k (A λ B λ )ρ(a λ B λ ) = 1 p λk (A λk B λk )ρ(a λk B λk ). If the experimenter learns none of the measurement outcomes when they occur (or they are forgotten), then there is only one subset S k =: S containing all outcomes and Pr{S} = 1: ρ 1 λ B λ )ρ(a λ B λ ) Pr{S} λ S(A = λ (A λ B λ )ρ(a λ B λ ). PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

4 Entanglement Monotones in Greater Detail A function µ : D(H A H B ) R is an entanglement monotone if µ(ρ) k Pr{S k }µ(ρ k ), for ρ k = 1 Pr{S k } λ S k (A λ B λ )ρ(a λ B λ ), where {A λ B λ } λ is a set of Kraus operators generated by any LOCC protocol, and {S k } k is any collection of coarse-graining sets for the outcomes λ. Two extremes : { Learn All Outcomes: µ(ρ) λ p λµ(ρ λ ) Learn No Outcomes: µ(ρ) µ( λ p λρ λ ) PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

5 Entanglement Monotones in Greater Detail In this course we will only consider entanglement monotones that are convex. That is, for any set of density operators {ρ i } i and any probability distribution {p i } i : ( ) Convexity: p i µ(ρ i ) µ p i ρ i. i For a convex monotone, its average cannot be increased by coarse-graining: implies: {p λ, ρ λ } coarse-graining {Pr{S k }, p λ µ(ρ λ ) λ k Pr{S k }µ i 1 Pr{S k } 1 Pr{S k } λ S k p λ ρ λ } p λ ρ λ. λ S k PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

6 Entanglement Monotones in Greater Detail Therefore, to prove that a convex function µ is an entanglement monotone, we just need to show that µ(ρ) ( ) 1 p λ µ (A λ B λ )ρ(a λ B λ ) p λ λ for all LOCC Kraus operators {A λ B λ } λ. Monotonicity under coarse-graining follows by convexity! Since the Kraus operators {A λ B λ } λ for an LOCC protocol are generated by sequences of local measurements, a convex function µ will be an entanglement monotone if and only if it is a monotone under each local measurement {A λ I} λ (for Alice) or {I B λ } λ (for Bob): µ(ρ) λ p λ µ ( ) Aλ I)ρ(A λ I) p λ, µ(ρ) λ p λ µ ( ) I Bλ )ρ(i B λ ) p λ. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

7 Review: A Recipe for Entanglement Monotones LOCC protocols not discarding information (no coarse-graining) 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 monotone is a function µ defined on pure states that is monotonic under all pure state entanglement transformations: µ( Ψ ) λ p λ µ( Ψ λ ). PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

8 Review: A Recipe for Entanglement Monotones Constructing Pure State Monotones 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 (tr B ( Ψ Ψ )) = f (tr A ( Ψ Ψ )) A function of just the Schmidt coefficients! 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 Pt. 2 April 8, / 13

9 Review: A Recipe for Entanglement Monotones The Convex Roof Extension to Mixed States 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 a convex function. Theorem: µ is an entanglement monotone. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

10 Entanglement Entropy The first and most important entanglement measure we will consider is called the entanglement entropy (or entropy of entanglement). Definition The Shannon entropy of a probability distributions {p 1, p 2,, p d } is the number given by H({p i }) = d p i log 2 p i. i=1 The Shannon entropy is one of the most fundamental quantities in the subject of classical information theory. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

11 Properties of Shannon Entropy Fact: The entropy H({p i }) is a concave function. Proof: Follows from convexity of the function f (x) = x log x. Fact: H({p i }) log d, with equality iff p i = 1 d for i = 1,, d. Proof: Use Lagrange Multipliers. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

12 Entanglement Entropy The von Neumann entropy of a density matrix ρ is defined by S(ρ) = tr[ρ log ρ] = H({λ i }), where the λ i are the eigenvalues of ρ. Let Ψ = d i=1 λi α i β i be a Schmidt decomposition of Ψ. The Entanglement Entropy of Ψ is defined as E( Ψ ) = S(tr A ( Ψ Ψ )) = H({λ i }). The Entanglement of Formation of a mixed state ρ is the convex roof extension of the Entanglement Entropy: E F (ρ) = min p i E( ϕ i ). {p i, ϕ i } i PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13

13 Entanglement Entropy Example Compute the Entanglement Entropy of the two-qubit state Ψ = cos θ 00 + sin θ 11. Example Compute the Entanglement Entropy of the two-qubit state Ψ = a 00 + b 01 + c 10 + d 11. PHYS Southern Illinois University Entanglement Measures and Monotones Pt. 2 April 8, / 13