# Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups

## TL;DR

This paper introduces transferred channels derived from classical Markov kernels to estimate quantum decoherence times and capacities, providing new bounds and extending analysis techniques to non-ergodic quantum channels.

## Contribution

It develops a transference method linking classical Markov processes to quantum channels, enabling capacity and decoherence time estimates for complex, non-ergodic channels.

## Key findings

- New estimates for channels that randomly swap subsystems.
- Extension of transference techniques to non-ergodic channels.
- Application of bounds to entanglement breaking times.

## Abstract

Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels, which are built through representation theory from a classical Markov kernel defined on a compact group. We study two subclasses of such kernels: H\"ormander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on $d$ qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of various capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.