Designing Stochastic Channels

TL;DR

This paper explores the properties of general stochastic quantum channels, establishing their relationship with process infidelity, unitality, and the effects of twirling with unitary 1-designs, revealing new insights into their structure.

Contribution

It proves the equivalence of diamond distance and process infidelity for stochastic channels, and analyzes the limitations of twirling with unitary 1-designs in generating stochastic channels.

Findings

Diamond distance equals process infidelity for stochastic channels.

Existence of non-unital multi-qubit stochastic channels.

Twirling by a unitary 1-design yields stochastic channels, but not all stochastic channels are obtainable this way.

Abstract

Stochastic channels are ubiquitous in the field of quantum information because they are simple and easy to analyze. In particular, Pauli channels and depolarizing channels are widely studied because they can be efficiently simulated in many relevant quantum circuits. Despite their wide use, the properties of general stochastic channels have received little attention. In this paper, we prove that the diamond distance of a general stochastic channel from the identity coincides with its process infidelity to the identity. We demonstrate with an explicit example that there exist multi-qubit stochastic channels that are not unital. We then discuss the relationship between unitary 1-designs and stochastic channels. We prove that the twirl of an arbitrary quantum channel by a unitary 1-design is always a stochastic channel. However, unlike with unitary 2-designs, the twirled channel depends upon the choice of unitary 1-design. Moreover, we prove by example that there exist stochastic channels that cannot be obtained by twirling a quantum channel by a unitary 1-design.