Quantum Latin squares and unitary error bases

TL;DR

This paper introduces quantum Latin squares and demonstrates their utility in constructing unitary error bases, providing a new method that generalizes existing techniques and yields novel UEBs.

Contribution

The paper presents quantum Latin squares and a new construction method for UEBs that extends and unifies previous approaches, producing previously unattainable bases.

Findings

New quantum Latin squares concept introduced

A novel UEB construction method developed

Explicit example of a UEB not obtainable by existing methods

Abstract

In this paper we introduce quantum Latin squares, combinatorial quantum objects which generalize classical Latin squares, and investigate their applications in quantum computer science. Our main results are on applications to unitary error bases (UEBs), basic structures in quantum information which lie at the heart of procedures such as teleportation, dense coding and error correction. We present a new method for constructing a UEB from a quantum Latin square equipped with extra data. Developing construction techniques for UEBs has been a major activity in quantum computation, with three primary methods proposed: shift-and-multiply, Hadamard, and algebraic. We show that our new approach simultaneously generalizes the shift-and-multiply and Hadamard methods. Furthermore, we explicitly construct a UEB using our technique which we prove cannot be obtained from any of these existing methods.