On the von Neumann entropy of certain quantum walks subject to decoherence

TL;DR

This paper analyzes how decoherence affects the von Neumann entropy in quantum walks on cycles, showing that any decoherence causes the system to become fully disentangled over time.

Contribution

It proves that decoherence leads the von Neumann entropy to reach its maximum, causing the bipartite system to become completely disentangled.

Findings

Von Neumann entropy converges to maximum under decoherence.

Mutual information diminishes to zero with decoherence.

System becomes fully disentangled due to decoherence.

Abstract

Consider a discrete-time quantum walk on the $N$-cycle governed by the following condition: at every time step of the walk, the option persists, with probability $p$, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems, corresponding respectively to the space of the coin and the space of the walker, eventually must diminish to zero. To put it plainly, any level of decoherence greater than zero forces the system eventually to become completely "disentangled".}