Doubly stochastic matrices in quantum mechanics

Jerzy Stryla

I propose to treat quantum evolution as a stochastic process consisting from a sequence of doubly stochastic matrices, which naturally arise in a generalized quantum evolution. Then it is proved that the law of non-decreasing entropy is fulfilled and that the law characterizes doubly stochastic matrices. An application of the model to support the generalized second law of black hole thermodynamics appears.

Victor Yu. Podlipchuk

Ammar Daskin

A matrix can be converted into a doubly stochastic matrix by using two diagonal matrices. And a doubly stochastic matrix can be written as a sum of permutation matrices. In this paper, we describe a method to write a given generic matrix in terms of quantum gates based on the block encoding. In particular, we first show how to convert a matrix into doubly stochastic matrices and by using Birkhoff's algorithm, we express that matrix in terms of a linear combination of permutations which can be mapped to quantum circuits. We then discuss a few optimization techniques that can be applied in a possibly future quantum compiler software based on the method described here.

Alladi Ramakrishnan, Nagarajan Ranganathan

Alexander Karabegov, Piotr Kielanowski, Anatol Odzijewicz +2 more

We prove that the mapping from the unitary to doubly stochastic matrices that maps a unitary matrix (ukl) to the doubly stochastic matrix (|ukl|2) is a submersion at a generic unitary matrix. The proof uses the framework of operator symbols on a finite set.