Orientation Statistics and Quantum Information

TL;DR

This paper explores the connection between orientation statistics and quantum information, introducing probability distributions on Stiefel manifolds to analyze quantum states and operations, and examining their geometric properties.

Contribution

It applies exponential family theory on Stiefel manifolds to quantum information, enabling new inference and sampling methods, and investigates the convex geometry of quantum operations.

Findings

Identifies natural probability distributions for quantum systems

Links exponential families on Stiefel manifolds to quantum inference

Characterizes quantum operations as averages of random operations

Abstract

Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered in the field of orientation statistics. This approach identifies natural probability distributions that can be used in the analysis, simulation, and inference of quantum information systems. The theory of exponential families on Stiefel manifolds provides the appropriate generalization to the classical case, and fortunately there are many existing techniques for inference and sampling that exist for these distributions. Furthermore, this viewpoint motivates a number of additional questions into the convex geometry of quantum operations relative to both the differential geometry of Stiefel manifolds as well as the information geometry of exponential families defined upon them. In particular, we draw on results from convex geometry to characterize which quantum operations can be represented as the average of a random quantum operation.