Discrete dynamics in the set of quantum measurements

TL;DR

This paper introduces a framework for understanding discrete transformations of quantum measurements using blockwise stochastic matrices, leading to a quantum analog of classical majorization and a resource theory for quantum measurement dynamics.

Contribution

It develops a novel mathematical framework for quantum measurement transformations using blockwise stochastic matrices and establishes a quantum majorization theory.

Findings

Defined blockwise stochastic and bistochastic matrices for quantum measurements.

Established a quantum analog of the Birkhoff polytope and majorization.

Formulated a resource theory for quantum measurement transformations.

Abstract

A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_j^\dag\geq 0$ summing to identity, $\sum_jP_j=\mathbb{1}$. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe discrete transformations in the set of quantum measurements by {\em blockwise stochastic matrices}, composed of positive blocks that sum columnwise to identity, using the notion of {\em sequential product} of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity, we arrive at blockwise bistochastic matrices (also called {\em quantum magic squares}). Analyzing their dynamical properties, we formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope, which introduces the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.