Abstract
Transformations of classical and quantum states and discrete dynamics in the set of quantum measurements
Karol Zyczkowski
Jagiellonian University
Discrete dynamics in the simplex of normalized probability vectors is given by classical stochastic matrices, while dynamics in the set of density operators is described by stochastic maps: completely positive, trace-preserving transformations.
Consider the set of generalized measurements (POVM), which consist of identity resolutions,
\( \sum_{i=1}^N E_i = \mathbb{I}_N \),
where non-negative effects \( E_i^{\dagger} = E_i \ge 0 \) play the role of non-negative probabilities. Transformations in this set are described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. 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, sometimes called quantum magic squares. Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices.
*Joint work with Albert Rico (Barcelona), Moisés Morán (Ankara), and Shmuel Fridland (Chicago).*
Consider the set of generalized measurements (POVM), which consist of identity resolutions,
\( \sum_{i=1}^N E_i = \mathbb{I}_N \),
where non-negative effects \( E_i^{\dagger} = E_i \ge 0 \) play the role of non-negative probabilities. Transformations in this set are described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. 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, sometimes called quantum magic squares. Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices.
*Joint work with Albert Rico (Barcelona), Moisés Morán (Ankara), and Shmuel Fridland (Chicago).*