A Classical Measure for Quantum Entanglement. It is a challenging problem of theoretical quantum physics to find good measures for entanglement [26]. Corollary 3 above suggests the following measure, which is based on classical information theory. Definition 1 Let for a quantum state ρAB µ(ρAB):= min ( max (I(X; Y↓Z))) , {|z⟩} {|x⟩},{|y⟩} where the minimum is taken over all Ψ = Σ √p ψ ⊗ |z⟩ such that ρ = z z AB TrHE (PΨ ) holds and over all generating sets {|z⟩} of HE, the maximum is over all bases {|x⟩} of HA and {|y⟩} of HB, and where PXY Z(x, y, z):= |⟨x, y, z|Ψ⟩|2. ❢ The function µ has all the properties required from such a measure. If ρAB is pure, i.e., ρAB = |ψΣAB ⟩⟨ψAB|, then we have in the Xxxxxxx basis (see for example [24]) ψAB = j cj|xj, yj⟩, and µ(ρAB) = −Tr(ρA log ρA) (where ρA = TrB(ρAB)) as it should [26]. It is obvious that µ is convex, i.e., µ(λρ1 + (1 − λ)ρ2) ≤ λµ(ρ1)+ (1 − λ)µ(ρ2). (1 (−) √λ)/4 001 + 012 + 103 + 114 , where ψ = 10 01 / 2, and ρ = E√xample 5. This example is based on Xxxxxx’x states. Let Ψ = √λψ(−) ⊗ |0⟩ + − | ⟩ | − ⟩ AB λPψ(−) + ((1 − λ)/4)11. It is well-known that ρAB is separable if and only if λ ≤ 1/3. Then the classical distribution is P (010)= P (100)= λ/2 and P (001)= P (012)= P (103)= P (114)= (1 − λ)/4. If λ ≤ 1/3, then consider the channel PZ|Z (0, 0)= PZ|Z (2, 2)= PZ|Z (3, 3)= 1 , PZ|Z (0, 1) = PZ|Z (0, 4) = ξ, PZ|Z (1, 1) = PZ|Z (4, 4) = 1 − ξ, where ξ = 2λ/(1 −λ) ≤ 1. Then µ(ρAB)= I(X; Y↓Z)= I(X; Y |Z)= 0 holds, as it should. If λ > 1/3, then consider the (obviously optimal) channel PZ|Z (0, 0) = PZ|Z (2, 2)= PZ|Z (3, 3)= PZ|Z (0, 1)= PZ|Z (0, 4)= 1. Then µ(ρAB)= I(X; Y↓Z)= I(X; Y |Z)= PZ (0) · I(X; Y |Z = 0) = 1+ λ · (1 − q log2 q − (1 − q)log2(1 − q)) , where q = 2λ/(1 + λ). ♦
A Classical Measure for Quantum Entanglement. It is a challenging problem of theoretical quantum physics to find good measures for entanglement [26]. Corollary 3 above suggests the following measure, which is based on classical information theory.
A Classical Measure for Quantum Entanglement. It is a challenging problem of theoretical quantum physics to find good measures for entanglement [26]. Corollary 3 above suggests the following measure, which is based on classical information theory. Definition 1 Let for a quantum state ρAB {|z)} {|x)},{|y)} µ(ρAB) := min ( max (I(X; Y ↓Z))) , z z Σ z where the minimum is taken over all Ψ = √p ψ ⊗ |z) such that ρ = AB TrHE (PΨ) holds and over all bases {|z)} of HE, the maximum is over all bases {|x)} of HA and {|y)} of HB, and where PXY Z(x, y, z) := |(x, y, z|Ψ)|2 .
