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arXiv:1909.09214 · Journal of Physics A Mathematical and Theoretical

Asymptotic reduced density matrix of discrete-time quantum walks

Mostafa Annabestani

Published Feb 28, 2020• 0 citations
arXivHTMLPDF

Abstract

Abstract In this article we show that for any quantum walker with m -dimensional coin subspace, we have m 2 × m 2 specific constant matrix C where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P 0 projector, the reduced density matrix can be obtained by T r 1 P 0 ⊗ I C or equivalently T r 2 I ⊗ P 0 C . It is worth mentioning that the characteristic matrix C is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix C for the general coin operator, U 2 , as well as the exact form of this matrix for the local initial state.

DOI
10.1088/1751-8121/ab7b20

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Abstract In this article we show that for any quantum walker with m -dimensional coin subspace, we have m 2 × m 2 specific constant matrix C where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P 0 projector, the reduced density matrix can be obtained by T r 1 P 0 ⊗ I C or equivalently T r 2 I ⊗ P 0 C . It is worth mentioning that the characteristic matrix C is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix C for the general coin operator, U 2 , as well as the exact form of this matrix for the local initial state.
The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the 'coin' and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well-defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to the maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.
Abstract. The coin-position entanglement generated by the evolution operator of a discrete–time quantum walk is quantified, using the von Neumann entropy of the reduced density operator (entropy of entanglement). In the case of a single walker, the entropy of entanglement converges, in the long time limit, to a well defined value which depends on the initial state. Exact expressions are obtained for local and non-local initial conditions. We also discuss the asymptotic bi-partite entanglement generated by non-separable coin operations for two coherent quantum walkers. In this case, the entropy of entanglement is observed to increase logarithmically with time. 1.
We calculated reduced density characteristic matrix (RDCM) for quantum walk on cycles (QWC) to study asymptotic properties of the most general form of quantum walk on cycles with general U(2)U(2)U(2) coin operator. As an example, entanglement temperature for general initial state has been calculated and compared to previous results. Also, we have modified RDCM to derive analytical expression for general form of limiting distribution (LD).
Consider a discrete-time quantum walk on the NNN-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 NNN is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When NNN 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.
The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumman entropy of the reduced density operator (entropy of entanglement). We show analytically that for a Hadamard walk with local initial conditions the asymptotic entanglement is 0.872 for all initial coin states. When nonlocal initial conditions are considered, the asymptotic entanglement varies smoothly between almost complete entanglement and no entanglement (product state). An exact expression for the asymptotic (long-time) entanglement is obtained for initial conditions in the position subspace spanned by $\ensuremath{\mid}\ifmmode\pm\else\textpm\fi{}1⟩$.
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.