Asymptotic entanglement in quantum walks from delocalized initial states

Abstract

We study the entanglement between the internal (spin) and external (position) degrees of freedom of the one-dimensional discrete time quantum walk starting from local and delocalized initial states whose time evolution is driven by Hadamard and Fourier coins. We obtain the dependence of the asymptotic entanglement with the initial dispersion of the state and establish a way to connect the asymptotic entanglement between local and delocalized states. We find out that the delocalization of the state increases the number of initial spin states which achieves maximal entanglement from two states (local) to a continuous set of spin states (delocalized) given by a simple relation between the angles of the initial spin state. We also carry out numerical simulations of the average entanglement along the time to confront with our analytical results.Comment: One column, 15 pages, 6 figure

Trojan Quantum Walks

“…The initial qubits which lead to the maximal entanglement necessarily also lead to a symmetrical probability distribution regardless of the initial position state [18]. At the same time, while there are only two particular initial qubits over one position (local state) which lead the quantum walk to asymptotically time-evolve to the maximal entanglement condition, when a quantum walk starts from a delocalized state, there are infinite available initial qubits to reach this condition [19]. Figure 1 shows the probability distribution for Hadamard quantum walks starting from local and two Gaussian states after 3000 time steps.…”

Asymptotic reduced density matrix of discrete-time quantum walks

“…Obviously C(k) is symmetric under the interchanging part 1 and 2, so T r 1 (P 0 (k) ⊗ I C(k)) can replaced by T r 2 (I ⊗ P 0 (k) C(k)) in (20).…”

“…when we use projector of initial state as P 0 ⊗ I and it equals to summation of all diagonal blocks of C, similarly T r 2 (.) will be applied when we use I ⊗ P 0 in (20) in this case each element of result matrix is the trace of corresponding block. Therefore the action of T r 1 (.)…”

“…The study of asymptotic behavior of QW is another field that researchers are interested in. Asymptotic coinposition entanglement (CPE) [19,20], global chirality distribution (GCD) [21] or Thermodynamic behavior of QW [22] are a few examples in which asymptotic behavior of QW have been studied. Clearly to study asymptotic behavior, the numerical methods are not suitable or at least are not accurate, so we need exact analytical method.…”

“…Clearly to study asymptotic behavior, the numerical methods are not suitable or at least are not accurate, so we need exact analytical method. Although many analytical papers have been published, they are usually restricted to specific situations (specific initial state or coin operator) [20][21][22][23][24]), because analytical calculations in general form are very te- * Annabestani@shahroodut.ac.ir dious and difficult.…”