On a quantum martingale convergence theorem

TL;DR

This paper extends classical probability limit theorems to the quantum domain by establishing a quantum martingale convergence theorem, highlighting unique non-classical behaviors in quantum measurement and expectation limits.

Contribution

It introduces a quantum analogue of the Lebesgue dominated convergence theorem and proves a quantum martingale convergence theorem with partial classification of limits.

Findings

Quantum martingale limits are unique but not explicitly identifiable.

Established a quantum version of the Lebesgue dominated convergence theorem.

Provided a partial classification of quantum random variables with zero expectation.

Abstract

It is well-known in quantum information theory that a positive operator valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalised POVM, namely a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator valued function which is measurable with respect to a quantum probability. In the present work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.