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Entropy 2013, 15(5), 1726-1737; doi:10.3390/e15051726
Article

Equiangular Vectors Approach to Mutually Unbiased Bases

1,2,3
Received: 26 April 2013; Accepted: 6 May 2013 / Published: 8 May 2013
(This article belongs to the Special Issue Quantum Information 2012)
Download PDF [239 KB, uploaded 8 May 2013]
Abstract: Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.
Keywords: finite-dimensional quantum mechanics; mutually unbiased bases; projection operators; positive-semidefinite Hermitian operators; equiangular lines; Gauss sums finite-dimensional quantum mechanics; mutually unbiased bases; projection operators; positive-semidefinite Hermitian operators; equiangular lines; Gauss sums
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Kibler, M.R. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy 2013, 15, 1726-1737.

AMA Style

Kibler MR. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy. 2013; 15(5):1726-1737.

Chicago/Turabian Style

Kibler, Maurice R. 2013. "Equiangular Vectors Approach to Mutually Unbiased Bases." Entropy 15, no. 5: 1726-1737.


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