Weak mutually unbiased bases

Abstract

Quantum systems with variables in Z(d) are considered. The properties of lines in the Z(d) × Z(d) phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to d −1/2 or alternatively to one of the d −1/2 i , 0 (where di is a divisor of d apart from d, 1). They are designed for the geometry of the Z(d) × Z(d) phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime d, there are no divisors of d apart from 1, d and the weak mutually unbiased bases are mutually unbiased bases.form the group Sp(2, Z(d)). The cardinality of this group is dJ 2 (d).Acting with the matrix g(κ, λ|µ, ν) on a point (ρ, σ) ∈ Z(d) × Z(d) we get the point (κρ + λσ, µρ + νσ). Acting with the symplectic matrix g(κ, λ|µ, ν) on all the points of a line L(ρ, σ) we get all the points of the line L(κρ + λσ, µρ + νσ) for which we also use the notation g(κ, λ|µ, ν)L(ρ, σ).In [28] we have proved various properties for lines in Z(d) × Z(d). For completeness below we give these properties together with several new properties, but we prove only the new ones.Definition II.1. Lines L(ν, µ) through the origin, with exactly d points are called maximal lines.Proposition II.2.

Equiangular Vectors Approach to Mutually Unbiased Bases

“…To close, let us mention that it should be interesting to apply the developments above (especially Proposition 1) to the concept of weakly MUBs recently introduced for dealing in the Z/dZ × Z/dZ phase space [46]. In addition, the results presented in this paper might be of interest in studies involving the concept of constellations of MUBs introduced in [18].…”

Partial ordering of weak mutually unbiased bases

“…Non-near-linear geometries have non-trivial subgeometries (in analogous way to non-prime numbers which have non-trivial factors). For m|n, Z(m) is a subgroup of Z(n), and G(m) is a subgeometry of G(n) (we denote this as G(m) ≺ G(n)), in the sense of the following results [23,24] which we summarize in the following proposition without proof.…”

“…We next introduce the concept of weak mutually unbiased bases (WMUB) studied in [23,24]. They are products of mutually unbiased bases in each of the Hilbert spaces H(p i ):…”

“…In view of this, refs. [23,24] have introduced the concept of weak mutually unbiased bases (WMUB), where | X θ ; α|X φ ; β | 2 = 1 m ; m|n; α, β ∈ Z(n).…”

An analytic function approach to weak mutually unbiased bases

“…For simplicity, the case where d = p1 × p2,where p1, p2 are odd prime numbers different from each other, is considered. PACS numbers: 03.65.Aa, 02.10.De squares[21].Recent work [22,23] introduced a weaker concept called weak mutually unbiased bases (WMUB). It is a set of bases, for which the absolute value of the overlap of any two vectors in two different bases is 1/ √ k, where k|d (k is a divisor of d), or zero.…”