Approximating quantum channels by completely positive maps with small Kraus rank

TL;DR

This paper demonstrates that any quantum channel can be approximated by a simpler one with significantly fewer Kraus operators, reducing complexity while maintaining output accuracy, especially for highly mixed outputs.

Contribution

It introduces a method to approximate quantum channels with a small number of Kraus operators, improving efficiency in quantum information processing.

Findings

Any quantum channel can be approximated with O(d log d) Kraus operators.

For channels with highly mixed outputs, approximation requires only O(d) Kraus operators.

The results are close to optimal, with discussions on potential improvements.

Abstract

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.