An optimal expression of a Kraus operator as a linear combination of unitary matrices

Abstract

Kraus operators are widely used in describing the evolution of an open quantum system. In this paper, we study the properties of a Kraus operator as a linear combination of unitary matrices and demonstrate that every single Kraus operator can be realized in an interference quantum circuit. We determine the minima of both l1 and l0 norm of the combination coefficients, where l1 norm means the sum of the absolute values of the coefficients and l0 norm means the number of non-zero coefficients. We find that both of them have clear physical meanings. The l1 minimum signifies the most constructive interference, and the l0 minimum provides the simplest way to construct a Kraus operator in a quantum circuit. These results may be useful in understanding interference and Kraus operators as well as in designing a quantum algorithm with a quantum computer in an open environment.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.