Physics – Quantum Physics
Scientific paper
2011-05-10
Journal of Mathematical Physics 53, 012201 (2012)
Physics
Quantum Physics
37 pages, published version. Theorems 3 and 4 were replaced by Theorem 3, which was proved by using Arveson's extension theore
Scientific paper
Let $K$ be a convex subset of the state space of a finite dimensional $C^*$-algebra. We study the properties of channels on $K$, which are defined as affine maps from $K$ into the state space of another algebra, extending to completely positive maps on the subspace generated by $K$. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on $K$. Moreover, if $K$ contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process POVMs.
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