Conditional Distributions for Quantum Systems

TL;DR

This paper explores the concept of conditional distributions within quantum systems using a categorical framework, introducing new methods for their construction and analyzing their properties through examples like the EPR scenario.

Contribution

It develops a categorical approach to quantum conditional distributions, providing criteria for positivity and comparing various quantum Bayesian inversion methods.

Findings

Conditional distributions as linear unital maps are constructed via Bayesian inversion.

Criteria for positivity of these maps are established.

EPR correlations are reproduced categorically in the quantum setting.

Abstract

Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, illustrating some similarities and key differences.