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arXiv:1605.08919 · Electronic Proceedings in Theoretical Computer Science

Constructing Mutually Unbiased Bases from Quantum Latin Squares

Benjamin Musto

Published Jan 1, 2017• 26 citations
arXivHTMLPDF

Abstract

We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.

DOI
10.4204/eptcs.236.8

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We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.
Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.
Quantum Latin Squares have emerged as mathematical structures of interest in quantum information theory. This paper provides a comprehensive overview of quantum Latin squares. We aim to synthesize the current knowledge for the benefit of researchers in quantum information theory. We begin by establishing the theoretical foundation linking classical Latin squares with their quantum counterparts. This provides insight into their potential applications, particularly in the realm of quantum error correction. Through a detailed examination, we outline known constructions and elucidate the properties of quantum Latin squares, highlighting their role within quantum computing frameworks. We then explore the relationships of quantum Latin squares with mutually unbiased bases and unitary error bases. This underscores their importance in the construction of quantum error correcting codes using mutually orthogonal quantum Latin squares. We give significant focus to a comparative analysis between classical and quantum Latin squares, particularly in their applications to error correcting codes. The applications are analogous to the application of classical Latin squares to classical error correcting codes. This overview aims to not only compile existing research but also to inspire further exploration into Quantum Latin squares and their applications in quantum information theory.
ABSTRACT Quantum Sudoku is the generalized version of the popular game Sudoku. A quantum Sudoku square is a special kind of quantum Latin square. A genuinely quantum solution of quantum Sudoku squares has been solved completely. In this paper, we introduce the notion of orthogonality of two quantum Sudoku squares. We establish three novel constructions of mutually orthogonal quantum Sudoku squares including Lorch construction, Pedersen‐Vis construction and direct product construction. As a result, we determine the genuinely quantum solution of a pair of orthogonal Sudoku squares of size for any positive integer . In general, we show that the resulting squares naturally define an absolutely maximally entangled state . Furthermore, we also show that the mutually orthogonal quantum Sudoku squares give rise to mutually unbiased bases based on maximally entangled states in for any prime power .
Mutually unbiased bases encapsulate the concept of complementarity---the impossibility of simultaneous knowledge of certain observables---in the formalism of quantum theory. Although this concept is at the heart of quantum mechanics, the number of these bases is unknown except for systems of dimension being a power of a prime. We develop the relation between this physical problem and the mathematical problem of finding the number of mutually orthogonal Latin squares. We derive in a simple way all known results about the unbiased bases, find their lower number, and disprove the existence of certain forms of the bases in dimensions different than power of a prime. Using the Latin squares, we construct hidden-variable models which efficiently simulate results of complementary quantum measurements.