Generalized channels: channels for convex subsets of the state space

TL;DR

This paper introduces and characterizes generalized channels for convex subsets of state spaces in finite-dimensional $C^*$-algebras, extending the concept of quantum channels and defining a framework for generalized supermaps.

Contribution

It defines generalized channels as restrictions of completely positive maps on convex subsets, characterizes their structure, and introduces generalized supermaps with decomposition theorems.

Findings

Generalized channels are restrictions of completely positive maps.

The set of generalized channels forms a convex subset under certain conditions.

Decomposition theorems for generalized supermaps are established.

Abstract

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.