arXiv:1210.8173

On mutually unbiased bases: Passing from d to d**2

M. Kibler

Published Oct 30, 2012• 1 citation

Abstract

We show how to transform the problem of finding d+1 mutually unbiased bases in the d-dimensional Hilbert space into the one of finding d(d+1) vectors in the N-dimensional Hilbert space with N=d**2. The transformation formulas admit a solution when d is a prime number.

DOI: 10.48550/arxiv.1210.8173

Related papers

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.

EntropyMay 8, 20135 citations

Mutually unbiased unitary bases of operators on d-dimensional hilbert space

Rinie N. M. Nasir, Jesni Shamsul Shaari, Stefano Mancini

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

arXiv:2003.12201International Journal of Quantum InformationFeb 1, 20204 citations

Mutually unbiased unitary bases of operators on d-dimensional hilbert space

Rinie N. M. Nasir, Shamsul Shaari, Jesni, Stefano Mancini

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, Mðd; CÞ, acting on such Hilbert spaces. The notion of MUUB re°ects the equiprobable guesses of unitary operators in one basis of Mðd; CÞ when estimating a unitary operator in another. Though, for prime dimension d, the maximal number of MUUBs is known to be d2 1, there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of d. MUUBs can also exist for some d-dimensional subspace of Mðd; CÞ with the maximal number being d.

Jan 1, 20201 citation

Mutually unbiased bases and finite projective planes

Метод Санига, Michel R. P. Planat, Haret Codratian Rosu

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.

arXiv:math-ph/0403057Journal of Optics B Quantum and Semiclassical OpticsJul 24, 200493 citations

Mutually unbiased bases

TR Thijs Timmerman

Two orthogonal bases A and B of an n-dimensional Hilbert space are called Mutually Unbiased if for ai ∈ A and bj ∈ B holds: |(ai, bj)| = 1 n for each i, j ∈ {1, . . . , n}. A set B of Mutually Unbiased Bases (MUBs) is a set of bases such that each two distinct bases Bi, Bj ∈ B are mutually unbiased. The bounds on the number of MUBs, dependent on the dimension n, will be derived. Also a construction for MUBs in Cp, with p prime, will be showed. Finally, some open problems will be discussed.

Jan 1, 20090 citations

Mutually Unbiased Bases and Their Symmetries

G. Alber, Christopher Charnes

We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces.

Quantum ReportsNov 8, 20192 citations