Multipartite entanglement measures

the structure of entanglement (interesting for more than two subsystems) the qualification of entanglement (separability criteria) the quantification of entanglement (entanglement measures) These are difficult for mixed states of more than two subsystems. Here we present well motivated answers for these questions. We recall the bipartite case, we show the tripartite case in details, and we refer mainly to [1] for the n-partite case. 2. Setting the stage state vector: |ψ 〉 ∈ H (normalized) pure state: pi = |ψ〉〈ψ | ∈ P mixed state (of an ensemble): % = j pjpij ∈ D = ConvP mixedness: e.g. von Neumann entropy S(%) = − tr % ln % distinguishability: e.g. relative entropy D(%‖ω) = tr %(ln % − lnω) 3. Bipartite systems and entanglement Pure states |ψ 〉 ∈ H = H1 ⊗H2 |ψ〉〈ψ | = pi ∈ P separable: |ψ 〉 = |ψ1 〉 ⊗ |ψ2 〉 pi = pi1 ⊗ pi2 ∈ Psep ⊂ P else it is entangled (P \ Psep). Then measurement on subsystem 1 “causes” the collapse of the state of subsystem 2. (worry of EPR) state of subsystems (e.g. pi1 = tr2 pi ∈ D1) are not necessary pure (|ψ〉) pi is entangled if (and only if) pi1 and pi2 are mixed. In this case, “the best possible knowledge of the whole does not involve the best possible knowledge of its parts. ” (Schrödinger) the mixedness of the subsystems E (pi) = S(pi1) “Entanglement Entropy” is a measure of entanglement (entanglement monotone) a different motivation can also be given, from statistical distinguishability, see later vanishes exactly for separable states (indicator function) Mixed states a mixed state is separable if there exists separable pure decomposition: i pipii,1 ⊗ pii,2 ∈ Dsep ⊂ D motivation: classically correlated sources produce states of this kind (Werner) else it is entangled (D \ Dsep) the decomposition is not unique, decision of its existence is difficult the average entanglement of the optimal decomposition (convex roof extension) E (pi) = S(pi1) E: P → [0, Smax] E∪(%) = mi