Online learning of a panoply of quantum objects

TL;DR

This paper develops an online learning framework with sublinear regret bounds for a wide range of quantum objects, advancing adaptive quantum state and process estimation techniques.

Contribution

It introduces a general regret bound for learning quantum objects using a regularized follow-the-leader algorithm, applicable to many quantum representations and including new matrix analysis results.

Findings

Sublinear regret bounds for learning quantum states, effects, channels, and more.

Generalization of Pinsker's inequality for positive semidefinite operators.

Applicable to diverse quantum objects with convex, compact representations.

Abstract

In many quantum tasks, there is an unknown quantum object that one wishes to learn. An online strategy for this task involves adaptively refining a hypothesis to reproduce such an object or its measurement statistics. A common evaluation metric for such a strategy is its regret, or roughly the accumulated errors in hypothesis statistics. We prove a sublinear regret bound for learning over general subsets of positive semidefinite matrices via the regularized-follow-the-leader algorithm and apply it to various settings where one wishes to learn quantum objects. For concrete applications, we present a sublinear regret bound for learning quantum states, effects, channels, interactive measurements, strategies, co-strategies, and the collection of inner products of pure states. Our bound applies to many other quantum objects with compact, convex representations. In proving our regret bound, we establish various matrix analysis results useful in quantum information theory. This includes a generalization of Pinsker's inequality for arbitrary positive semidefinite operators with possibly different traces, which may be of independent interest and applicable to more general classes of divergences.