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ON TWO WAYS TO LOOK FOR MUTUALLY UNBIASED BASES

M. Kibler

Published Aug 16, 2016• 5 citations
OpenAlex

Abstract

Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.

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Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.

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Equiangular Vectors Approach to Mutually Unbiased Bases

M. Kibler

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.

Entropy•May 8, 2013•5 citations
Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
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.