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arXiv:2402.05080 · Physical Review A

Designing three-way-entangled and nonlocal two-way-entangled single-particle states via alternate quantum walks

Dinesh Kumar Panda, Colin Benjamin

Published Jan 13, 2025• 6 citations
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Abstract

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.

DOI
10.1103/physreva.111.012420

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ-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.
Abstract A single-photon entangled state (or single-particle entangled state (SPES) in general) can offer a more secure way of encoding and processing quantum information than their multi-photon (or multi-particle) counterparts. The SPES generated via a 2D alternate quantum-walk setup from initially separable states can be either 3-way or 2-way entangled. This letter shows that the generated genuine three-way and nonlocal two-way SPES can be used as cryptographic keys to securely encode two distinct messages simultaneously. We detail the message encryption–decryption steps and show the resilience of the 3-way and 2-way SPES-based cryptographic protocols against eavesdropper attacks like intercept-and-resend and man-in-the-middle. We also detail the experimental realization of these protocols using a single photon, with the three degrees of freedom being orbital angular momentum, path, and polarization. We have proved that the protocols have unconditional security for quantum communication tasks. The ability to simultaneously encode two distinct messages using the generated SPES showcases the versatility and efficiency of the proposed cryptographic protocol. This capability could significantly improve the throughput of quantum communication systems.
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of xxx-yyy spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.
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.
The nonlocalized case of the spatial density probability of the two-dimensional Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions. This significantly reduces the resources necessary for its feasible experimental realization. We present a formal proof of this correspondence and analyze the behavior of the coin-position entanglement as well as the x-y spatial entanglement in our scheme with respect to the Grover one. Our scheme allows us to entangle the two orthogonal directions of the walk more efficiently.
The generation and control of quantum correlations in high-dimensional systems is a major challenge in the present landscape of quantum technologies. Achieving such non-classical high-dimensional resources will potentially unlock enhanced capabilities for quantum cryptography, communication and computation. We propose a protocol that is able to attain entangled states of d-dimensional systems through a quantum-walk (QW)-based transfer & accumulate mechanism involving coin and walker degrees of freedom. The choice of investigating QW is motivated by their generality and versatility, complemented by their successful implementation in several physical systems. Hence, given the cross-cutting role of QW across quantum information, our protocol potentially represents a versatile general tool to control high-dimensional entanglement generation in various experimental platforms. In particular, we illustrate a possible photonic implementation where the information is encoded in the orbital angular momentum and polarization degrees of freedom of single photons.