A collection of pairwise mutually unbiased bases (in short: MUB) in d>1 dimensions may consist of at most d+1 bases. Such "complete" collections are known to exists in C^d when d is a power of a prime. However, in general little is known about the maximal number N(d) of bases that a collection of MUBs in C^d can have. In this work it is proved that a collection of d MUBs in C^d can be always completed. Hence N(d) cannot be d and when d>1 we have a dichotomy: either N(d)=d+1 (so that there exists a complete collection of MUBs), or N(d)\leq d-1. In the course of the proof an interesting new characterization is given for a linear subspace of M_d(C) to be a subalgebra.
A collection of pairwise mutually unbiased bases (in short: MUB) in d>1 dimensions may consist of at most d+1 bases. Such "complete" collections are known to exists in C^d when d is a power of a prime. However, in general little is known about the maximal number N(d) of bases that a collection of MUBs in C^d can have. In this work it is proved that a collection of d MUBs in C^d can be always completed. Hence N(d) cannot be d and when d>1 we have a dichotomy: either N(d)=d+1 (so that there exists a complete collection of MUBs), or N(d)\leq d-1. In the course of the proof an interesting new characterization is given for a linear subspace of M_d(C) to be a subalgebra.
A collection of pairwise mutually unbiased bases (in short: MUB) in d > 1 d>1 dimensions may consist of at most d + 1 d+1 bases. Such “complete” collections are known to exist in C d \mathbb {C}^d when d d is a power of a prime. However, in general, little is known about the maximum number N ( d ) N(d) of bases that a collection of MUB in C d \mathbb {C}^d can have. In this work it is proved that a collection of d d MUB in C d \mathbb {C}^d can always be completed. Hence N ( d ) ≠ d N(d)\neq d , and when d > 1 d>1 we have a dichotomy: either N ( d ) = d + 1 N(d)=d+1 (so that there exists a complete collection of MUB) or N ( d ) ≤ d − 1 N(d)\leq d-1 . In the course of the proof an interesting new characterization is given for a linear subspace of
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions d that are powers of prime numbers, there exists a set of up to d+1 bases that form an MUB set. However, the maximum number of MUBs in dimensions that are not powers of prime numbers is not known. To address this issue, we introduce three algorithms based on First-Order Logic that can determine the maximum number of bases in an MUB set without numerical approximation. Our algorithms can prove this result in finite time, although the required time is impractical. Additionally, we present a heuristic approach to solve the semi-decision problem of determining if there are k MUBs in a given dimension d. As a byproduct of our research, we demonstrate that the maximum number of MUBs in any dimension can be achieved with definable complex parameters, computable complex parameters, and other similar fields.
Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if ||^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.
An infinite family of strongly unextendible mutually unbiased bases in C22h
A set of b mutually unbiased bases (MUBs) in Cd (for d>1) comprises bd vectors in Cd, partitioned into b orthogonal bases for Cd such that the pairwise angle between all vectors from distinct bases is arccos(1/d). The largest number μ(d) of MUBs that can exist in Cd is at most d+1, but constructions attaining this bound are known only when d is a prime power. A set of b MUBs in Cd that cannot be enlarged, even by the first vector of a potential (b+1)-th MUB, is called strongly unextendible. Until now, only one infinite family of dimensions d containing b(d) strongly unextendible MUBs in Cd satisfying b(d)<μ(d) was known, this family, due to Szántó, is asymptotically "large" in the sense that b(d)/μ(d)→1 as d→∞. However, the existence of 2m−1+1 strongly unextendible MUBs in C2m for each integer m>1 has been conjectured by Mandayam et al. We prove their conjecture for all even values of m, using only elementary linear algebra. The existence of this "small" new infinite family suggests, contrary to widespread belief, that μ(d) for non-prime-powers d might be significantly larger than the size of particular unextendible sets.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.