Discrete dynamics in the set of quantum measurements

• URL: http://arxiv.org/abs/2308.05835v2
• Date: Fri, 27 Dec 2024 17:57:40 GMT
• Title: Discrete dynamics in the set of quantum measurements
• Authors: Albert Rico, Karol Życzkowski,
• Abstract summary: A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_jdaggeq 0$ summing to identity. We describe discrete transformations in the set of quantum measurements by em blockwise matrices. We formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope.
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• 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.

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