paper

Smooth manifold structure for extreme channels

arXiv:1610.02513 · doi:10.1063/1.5019837

Abstract

A quantum channel from a system $A$ of dimension $d_{A}$ to a system $B$ of dimension $d_{B}$ is a completely positive trace-preserving map from complex $d_{A} \times d_{A}$ to $d_{B} \times d_{B}$ matrices, and the set of all such maps with Kraus rank $r$ has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the $r d_{B} \times d_{A}$ dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank $r$ . These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system $A$ to system $B$ of a fixed Kraus rank $r$ is a smooth submanifold of dimension $2 r d_{A} d_{B} - d_{A}^{2} - r^{2}$ of the set of all Choi-states of rank $r$ . As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from $A$ to $B$ arbitrarily well.

9 pages, v2: a few minor corrections to match journal version