Asymptotic evolution of quantum walks on the $N$-cycle subject to decoherence on both the coin and position degrees of freedom

TL;DR

This paper analyzes the long-term behavior of quantum walks on an N-cycle with decoherence, revealing how the system's density matrix evolves towards a maximally mixed state or a well-defined asymptotic form, depending on N.

Contribution

It provides explicit asymptotic descriptions of quantum walks with decoherence on both coin and position, including the structure of eigenspaces for generalized random unitary operations.

Findings

For odd N, the density matrix tends to the maximally mixed state.

For even N, the asymptotic dynamics are explicitly characterized.

The mutual information between coin and walker converges to a limit.

Abstract

Consider a discrete-time quantum walk on the $N$-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some new conclusions about the asymptotic behavior of the system. When $N$ is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When $N$ is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.