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.
Highlights
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 into the one of finding d(d + 1) vectors in the d2 -dimensional space Cd
The concept of unbiased bases and, more generally, of a set of mutually unbiased bases (MUBs) takes its origin in the work by Schwinger [1] on unitary operator bases. It is of paramount importance in quantum mechanics
We have found a transformation that allows to replace the search of d + 1 MUBs in Cd by the determination of d(d + 1) vectors in Cd
Summary
Algebras, finite geometries, combinatorics and graph theory, frames and 2-designs, and Hopf fibrations This can be illustrated by some of the numerous ways of constructing sets of MUBs. Most of them are based on discrete Fourier transform over Galois fields and Galois rings [4,6,7,23], discrete Wigner distribution [3,7,8], generalized Pauli spin operators [24,25,26,27,28], generalized Hadamard matrices [15,17], mutually orthogonal Latin squares [15,29,30], finite geometry methods [30,31,32], projective. A preliminary note on some of the results of this paper was posted on arXiv [40]
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