A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

Abstract

The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs.This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion.Some directions for proving non-existence results are sketched as well.

Constructions on Real Approximate Mutually Unbiased Bases

“…and R(v i ), I(v i ) denote the real and imaginary parts of v i ∈ C respectively. The mapping from v to v (1) and v (2) has the following properties. For odd integer d, to construct real AMUB in R d with larger size seems to be…”

“…For v, u ∈ C (d) and 1 ≤ i, j ≤ 2, |(v (i) , u ( j) )| ≤ |(v, u)|. Particularly, if (v, u) = 0, then v (1) , v (2) , u (1) and u (2) are orthogonal to each others.…”

“…MUB is one of the basic notions widely used in quantum information processing and quantum cryptography [1,8,9,14]. One of the basic problems is to construct MUB in F d with larger size n. More precisely, for d ≥ 2, let N F (d) = max{n : there exists MUB B = {B 1 , .…”

Morphophoric POVMs, generalised qplexes, and 2-designs

“…For t = 0 we get the Hesse configuration and the above equation takes a particularly nice form [77]: The fact that a complete set of MUBs constitutes a 2-design POVM was first shown explicitly in [50], however, this also follows easily [5] from the fact that in this case Π fulfils both the Welch bound [82] and the Levenshtein bound [54] for the angles between 2-design lines. Now, the system of linear equations defining the primal affine space A takes a particularly nice form since, instead of the initial n = d(d + 1) equations, it suffices to consider the following d + 1 equations expressing the simple fact that the probabilities over any basis (k = 0, .…”

The algebraic structure of Mutually Unbiased Bases

“…Complete sets of MUBs are shown to be equivalent to Complex Projective t−designs with angle set {0, 1 d } [11,16], which has lead to some work using Fourier matrices [4]. A construction using generalized Hadamard matrices and mutually orthogonal Latin squares yields larger (though incomplete) sets of MUBs in some nonprime powered dimensions [19].…”