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

Abstract

In this paper, we consider a discrete-time quantum walk on the N-cycle governed by the condition that 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 to the space of the coin and the space of the walker must eventually diminish to zero. Put plainly, any level of decoherence greater than zero forces the system to become completely 'disentangled' eventually. † The authors acknowledge, with due gratitude, the support provided to us by the NSF-funded HBCU-UP/BETTER Project at Bowie State University.

Discrete-time quantum walks in random artificial gauge fields

“…The entanglement of the averaged dynamics can also be quantified by the Shannon entropy S r of the reduced density operator ρ r in spin space. To be precise [29][30][31], ρ r = m ρ m,m ′ =m and the Shannon entropy S r = −tr(ρ r log(ρ r )). The time-evolution of S r is prensented is Figure 3, together with the entanglement entropy of the pure Hadamard walk with the same initial condition, which admits 0.872 as asymptotic value [32].…”

On Limiting Distributions of Quantum Markov Chains

“…It can be verified that the bit-phase flip channel (p377 in [19]) also belongs to category (1). A less trivial example of this sort can be found in [32].…”

On Some Questions of C. Ampadu Associated with the Quantum Random Walk

“…If the system is pure, the von Neumann entropy is used as a measure to quantify the entanglement. The measure of entanglement in this case is usually given by ρ has dimension two, see [161]- [163] for examples, the case where C ρ has dimension higher than two has been given little attention in the literature, the known studies in this case include [164] [165].…”