On the simultaneous transformation of density operators by means of a completely positive, unity preserving linear map (Q1074072)

For a given pair of n-triples \(\omega_ 1...\omega_ n\) and \(\sigma_ 1...\sigma_ n\) of normal states over the algebra \(B({\mathcal H})\) of all bounded linar operators on a separable Hilbert space \({\mathcal H}\), the equivalence of the following conditions is proved: (1) There exists a unital, completely positive linear map T of B(\({\mathcal H})\) into itself such that \(\omega_ k=\sigma_ k\circ T\), \(k=1,...,n.\) (2) There is an irreducible UHF-algebra \({\mathcal A}\subset B({\mathcal H})\) such that \(\omega_ k| {\mathcal A}=\phi (\sigma_ k| {\mathcal A})\), \(k=1,...,n\) for \(\phi \in C_ u({\mathcal A})\) where \(C_ u({\mathcal A})\) is a set of bounded linear maps of \({\mathcal A}^*\) obtained as the weak closure of the convex hull of the set of all unitary transforms \(\phi\in {\mathcal A}^*\to \phi \circ AdU\) with a unitary \(u\in {\mathcal A}\) \(((AdU)x=UxU^*).\) (3) The preceding property holds for all irreducible UHF-algebras \({\mathcal A}B({\mathcal H}).\) (4) \(f(\omega_ 1,...,\omega_ n)\geq f(\sigma_ 1,...,\sigma_ n)\) for all quasi-concave, \(w^*\)-upper semicontinuous, unitarily invariant, real-valued functions f over n-tuples of normal states on \(B({\mathcal H}).\) For \(n=2\), the class of functions f in the condition (4) can be restricted by the following conditions: (a) \(f(\omega,\sigma)\in [0,1]\), (b) \(f(\omega,\sigma)=0\) if supports of \(\omega\) and \(\sigma\) are orthogonal, (c) \(f(\omega,\sigma)=1\) if and only if \(\omega =\sigma\) and (d) \(f(\omega_ x,\omega_ y)=| (x,y)|^ 2\) where \(\omega_ x(A)=(x,Ax)\). Such an f is called a generalized transition probability.