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arXiv:2301.04501 · Physical Review A

Recurrent generation of maximally entangled single-particle states via quantum walks on cyclic graphs

Dinesh Kumar Panda, Colin Benjamin

Published Aug 25, 2023• 7 citations
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Abstract

While earlier research has predicted that maximally entangled single-particle states (MESPS), i.e., entanglement between different degrees of freedom of the same particle, can occur using quantum walk schemes with two or more coins or with a coin and identity operation, the authors analytically predict that MESPS can be generated using only a single coin on cyclic graphs. The generated MESPS occurs at recurring time steps, an advantage the authors exploit in a quantum cryptographic protocol.

DOI
10.1103/physreva.108.l020401

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While earlier research has predicted that maximally entangled single-particle states (MESPS), i.e., entanglement between different degrees of freedom of the same particle, can occur using quantum walk schemes with two or more coins or with a coin and identity operation, the authors analytically predict that MESPS can be generated using only a single coin on cyclic graphs. The generated MESPS occurs at recurring time steps, an advantage the authors exploit in a quantum cryptographic protocol.
We present a mathematical formalism for the description of un- restricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, with two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on two and three qubit coins leads to different position probability distributions in one- and two-dimensional graphs.
Quantum states exhibiting single-particle entanglement (SPE) can encode and process quantum information more robustly than their multiparticle analogs. Understanding the vulnerability and resilience of SPE to disorder is therefore crucial. This letter investigates phase, coin, and position disorder via discrete-time quantum walks on odd and even cyclic graphs to study their effect on SPE. The reduction in SPE is insignificant for low levels of phase or coin disorder, showing the resilience of SPE to minor perturbations. However, SPE is seen to be more vulnerable to position disorder. We analytically prove that maximally entangled single-particle states (MESPS) at time step t=1 are impervious to phase disorder regardless of the choice of the initial state. Further, MESPS at timestep t=1 is also wholly immune to coin disorder for phase-symmetric initial states. Position disorder breaks odd-even parity and distorts the physical time cone of the quantum walker, unlike phase or coin disorder. SPE saturates toward a fixed value for position disorder, irrespective of the disorder strength at large timestep t. Furthermore, SPE can be enhanced with moderate to significant phase or coin disorder strengths at specific time steps. Interestingly, disorder can revive single-particle entanglement from absolute zero in some instances, too. These results are crucial in understanding single-particle entanglement evolution and dynamics in a laboratory setting.
We present a mathematical formalism for the description of unrestricted quantum walks with entangled coins and one walker. Furthermore, we examine the numerical behaviour of such walks using Bell states as initial coin states, two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins. We find the behavior of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also find that simply by changing the shift operator, we can generate widely different distributions.
We study maximal coin-position entanglement generation via a discrete-time quantum walk, in which the coin operation is randomly selected from one of two coin operators set at each step. We solve maximal entanglement generation as an optimization problem with quantum process fidelity as the cost function. Then we determine the maximal entanglement that can be rigorously generated for any step beyond the second regardless of initial conditions with appropriate coin sequences. The simplest coin sequence comprising Hadamard and identity operations is equivalent to the generalized elephant quantum walk, which exhibits an increasingly faster spreading in terms of probability distribution. Experimentally, we demonstrate a ten-step quantum walk driven by such coin sequences with linear optics and thereby show the desired high-dimensional bipartite entanglement as well as the transport behavior of faster spreading.
Entanglement with single-particle states is advantageous in quantum technology because of the states' ability to encode and process information more securely than their multiparticle analogs. Three-way- and nonlocal two-way-entangled single-particle states are desirable in this context. Herein we generate genuine three-way entanglement from an initially separable state involving three degrees of freedom of a quantum particle, which evolves via a two-dimensional alternate quantum walk employing a resource-saving single-qubit coin. We achieve maximum possible values for the three-way entanglement quantified by the $\ensuremath{\pi}$-tangle between the three degrees of freedom. We also generate optimal nonlocal two-way entanglement, quantified by the negativity between the nonlocal position degrees of freedom of the particle. This prepared architecture using quantum walks can be experimentally realized with a photon.
We introduce the concept of a quantum walk with two particles and study it for the case of a discrete time walk on a line. A quantum walk with more than one particle may contain entanglement, thus offering a resource unavailable in the classical scenario and which can present interesting modifications on quantum walks with single particles. In this work, we show both numerically and analytically how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, the probability to find at least one particle in a certain position after NNN steps of the walk, as well as the average distance and the squared distance between the two particles, can be larger or smaller than the case of two unentangled particles, depending on the initial conditions we choose. This resource can then be tuned according to our needs to modify the features of a quantum walk. Experimental implementations are briefly discussed.