Smooth manifold structure for extreme channels

TL;DR

This paper characterizes the smooth manifold structure of extreme quantum channels with fixed Kraus rank, providing insights into their geometric properties and implications for quantum circuit approximation.

Contribution

It introduces two equivalent descriptions of the set of quantum channels as smooth manifolds and derives a lower bound on parameters for circuit approximation of extreme channels.

Findings

The set of extreme channels forms a smooth submanifold of known dimension.

Two topologically equivalent descriptions of the channel set are provided.

A lower bound on the number of parameters for quantum circuit approximation is established.

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 $rd_B\times d_A$ 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 $2rd_Ad_B-d_A^2-r^2$ 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.