Asymptotic Evolution of Random Unitary Operations

TL;DR

This paper investigates the long-term behavior of quantum systems under repeated random unitary operations, revealing that their asymptotic dynamics are governed by a low-dimensional, diagonalizable superoperator independent of the applied probability distribution.

Contribution

It demonstrates that the asymptotic dynamics of iterated random unitary operations are confined to a low-dimensional attractor space spanned by eigenvectors with eigenvalues of modulus one, regardless of the probability distribution.

Findings

Asymptotic dynamics are described by a diagonalizable superoperator.

The attractor space is low-dimensional and independent of probability distribution.

Eigenvectors with eigenvalues of modulus one determine long-term behavior.

Abstract

We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.