# Exponential Families for Bayesian Quantum Process Tomography

## TL;DR

This paper develops new Bayesian probability distributions on quantum channels, enabling comprehensive Bayesian quantum process tomography with prior information and credible interval estimation.

## Contribution

It introduces the first continuous, non-unitary prior distributions on CPTP maps using the Choi matrix and Stiefel manifold, facilitating Bayesian analysis of quantum processes.

## Key findings

- First continuous non-unitary CPTP priors introduced
- Enables Bayesian point and full tomography with credible intervals
- Connects prior distributions to experimental process data

## Abstract

A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is a distribution resulting from binomial likelihood data that enables a simple connection to data gathered through process tomography experiments. To our knowledge these are the first examples of continuous, non-unitary random CPTP maps, that capture meaningful prior information for use in Bayesian estimation. We show how these distributions can be used for point estimation using either maximum a posteriori estimates or expected a posteriori estimates, as well as full Bayesian tomography resulting in posterior credibility intervals. This approach will enable the full power of Bayesian analysis in all forms of quantum characterization, verification, and validation.