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

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 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 have 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.

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

“…Furthermore, the entanglement of the DTQW has been analyzed in Refs. [1,9,49,54] for the coin and space state of a single particle and in Refs. [62,72] for multi particles.…”

Extraordinary behaviors in a two-dimensional decoherent alternative quantum walk

“…When considering the real experimental implementation, the physical system will have an inevitable interaction with the surrounding environment. Many studies of DTQW report that due to the decoherence induced by the environment, the position distribution pattern of QW change to a binomial distribution that is similar to the distribution of the classical walk [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. For the coherent QW, the variance of position distribution in the QW grows quadratically with time.…”

A Perron–Frobenius Type of Theorem for Quantum Operations

“…The next lemma follows a similar argument about the convergence of convex combinations of operators in [LP11b] and [LP11a], as well as in [XY13]: Proof. If x ∈ V can be written with generalized eigenvectors of A as…”