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arXiv:1810.05161 · Journal of Physics A Mathematical and Theoretical

Equiangular quantum key distribution in more than two dimensions

Radhakrishnan Balu, Paul J. Koprowski, Kasso A. Okoudjou, Justin Park, George Siopsis

Published Dec 11, 2018• 1 citation

Abstract

Abstract We extend the spherical code-based key distribution protocols to qudits with dimensions 4 and 16 by constructing equiangular frames and their companions. We provide methods for equiangular frames in arbitrary dimensions for Alice to use and the companion frames, that have one antipode to eliminate one of the possibilities, made up of qudits with as part of Bob’s code. Non-orthogonal bases that form positive operator-valued measures can be constructed using the tools of frames (overcomplete bases of a Hilbert space), and here we apply them to key distribution that is robust due to the large size of the bases, making it hard to eavesdrop. We demonstrate a method to construct a companion frame for an equiangular tight frame for generated from the discrete Fourier transform, where p is any odd prime. The security analysis is based on the assumption of restricting possible attacks to an intercept/resend scenario, highlighting the advantages of qudit-based over qubit-based protocols.

DOI: 10.1088/1751-8121/aaf7bc

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Mutually unbiased bases have been extensively studied in the literature and are simple and effective in quantum key distribution protocols, but they are not optimal. Here equiangular spherical codes are introduced as a more efficient and robust resource for key distribution. Such codes are sets of states that are as evenly spaced throughout the vector space as possible. In the case of qubits, the two parties can make use of three equally-spaced states, called a trine, to outperform the original four-state BB84 protocol in both speed and reliability. This points toward the optimality of spherical codes in arbitrary dimensions.

Quantum Key Distribution (QKD) provides a secure way of generating shared cryptographic keys between a sender (Alice) and a receiver (Bob). The original BB84 protocol uses a two-dimensional polarization basis, limiting the information content of a single photon to 1 bit. Using the transverse position of single photons as one basis and the Fourier space as a second basis, one can construct a pair of mutually unbiased higher-dimensional bases. This improves not only the security of the protocol, but also the key generation rate. We present experimental results with a Spatial-Light-Modulator-based encoding scheme employing two nearly orthogonal alphabets with on the order of 103symbols each and an information content of about 10 bit.

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