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Quantum Information and Computation

New construction of mutually unbiased bases in square dimensions

WocjanPaweł, BethThomas

Published Mar 1, 2005• 1 citation
Publisher

Abstract

We show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines th...
DOI
10.5555/2011626.2011627

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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.

arXiv:1605.08919•Electronic Proceedings in Theoretical Computer Science•Jan 1, 2017•26 citations
We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (s,k)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^{1/14.8} for all s but finitely many exceptions. Furthermore, our construction gives more mutually unbiased bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.
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.