Robustness of Fixed Points of Quantum Channels and Application to Approximate Quantum Markov Chains

TL;DR

This paper investigates the stability of fixed points in quantum channels and states, providing conditions under which approximate fixed points can be closely approximated by exact ones, with applications to quantum Markov chains.

Contribution

It establishes general conditions and explicit bounds for fixing approximate fixed points of quantum channels, including special cases like unitary and unital channels, and applies these results to quantum Markov chains.

Findings

Approximate fixed points can be closely approximated by exact fixed points under general conditions.

Explicit bounds are derived for specific structures like unitary and unital channels.

Approximate fixed point equations are not rapidly fixable for bipartite channels acting trivially on one subsystem.

Abstract

Given a quantum channel and a state which satisfy a fixed point equation approximately (say, up to an error $\varepsilon$), can one find a new channel and a state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? It is interesting to ask this question for different choices of constraints on the structures of the original channel and state, and requiring that these are also satisfied by the new channel and state. We affirmatively answer the above question, under fairly general assumptions on these structures, through a compactness argument. Additionally, for channels and states satisfying certain specific structures, we find explicit upper bounds on the distances between the pairs of channels (and states) in question. When these distances decay quickly (in a particular, desirable manner) as $\varepsilon\to 0$, we say that the original approximate fixed point equation is rapidly fixable. We establish rapid fixability, not only for general quantum channels, but also when the original and new channels are both required to be unitary, mixed unitary or unital. In contrast, for the case of bipartite quantum systems with channels acting trivially on one subsystem, we prove that approximate fixed point equations are not rapidly fixable. In this case, the distance to the closest channel (and state) which satisfy an exact fixed point equation can depend on the dimension of the quantum system in an undesirable way. We apply our results on approximate fixed point equations to the question of robustness of quantum Markov chains (QMC) and establish the following: For any tripartite quantum state, there exists a dimension-dependent upper bound on its distance to the set of QMCs, which decays to zero as the conditional mutual information of the state vanishes.