The structure of state space with respect to imbedding (Q1410194)

The paper is devoted to study of entanglement of formation and the corresponding structure of leaves in the state space. Given a subalgebra \(A\) of a finite-dimensional von Neumann algebra \(M\), the entanglement of the state \(\omega\) on \(M\) with respect to \(A\) is defined by \[ E(\omega, M,A)=\inf\sum_ i \lambda_ i\, S(\omega_ i)| A\,, \] where the infimum is taken over all possible decompositions \(\omega= \sum_ i\,\lambda_ i\, \omega_ i\) of the state \(\omega\) into states on \(M\). It is shown that the state space of \(M\) decomposes into leaves given by entanglement. The result due to Benatti and Nardhofer (2001) describing when extremal points belong to the same leaf is recalled. In the subsequent part the case of infinite algebras is considered. Let \(A\) be a Type \(II_ 1\) factor and \(\alpha\) a free automorphism of \(A\) with \(\alpha^2=1\). Let \(M\) be the cross product of \(A\) with \(Z_ 2\) according to \(\alpha\). The entanglement of the embedding of \(A\) into \(M\) is defined and its universal upper bound is found. The final part deals with the conditional entropy and gives a counterexample to the additivity of the this quantity.