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On the Maximal Number of Real Mutually Unbiased Bases

Paweł Wocjan

Published Feb 3, 2005• 1 citation
OpenAlex

Abstract

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.

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arXiv:quant-ph/0502024•Feb 3, 2005•29 citations

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In this paper, we prove that the existence of a complete set of mutually unbiased bases (MUBs) in N-dimensional Hilbert space implies the existence of a complete set of mutually orthogonal Latin squares (MOLSs) of order N. In particular, we prove that a complete set of MUBs does not exist in dimension six (the first dimension which is not a power of prime).

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Mean king’s problem with mutually unbiased bases and orthogonal Latin squares

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Constructions of Mutually Unbiased Bases

Andreas Klappenecker, Martin Roetteler

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.

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On the connection between mutually unbiased bases and orthogonal Latin squares

Tomasz Paterek, M Paw Lowski, Markus Grassl +1 more

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure that has been shown to relate the two problems and generates complete sets of MUB in power-of-prime dimensions and three MUB in dimension six. For these cases, every square from an augmented set of MOLS has a corresponding MUB. We show that this no longer holds for certain composite dimensions.

•Aug 18, 2016•11 citations
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.
We tabulate bounds on the optimal number of mutually unbiased bases in R^d. For most dimensions d, it can be shown with relatively simple methods that either there are no real orthonormal bases that are mutually unbiased or the optimal number is at most either 2 or 3. We discuss the limitations of these methods when applied to all dimensions, shedding some light on the difficulty of obtaining tight bounds for the remaining dimensions that have the form d=16n^2, where n can be any number. We additionally give a simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets by Cameron and Seidel, Delsarte, and Calderbank et al.
On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions d≢0 mod 4d \not\equiv 0 \bmod 4d≡0mod4
It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension d>2d > 2d>2 which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct >⌈d⌉> \lceil \sqrt{d} \rceil>⌈d​⌉ many ARMUBs for certain odd dimensions ddd of the form d=(4n−t)sd = (4n-t)sd=(4n−t)s, t=1,2,3t = 1, 2, 3t=1,2,3, where nnn is a natural number and sss is an odd prime power. Our method exploits any available 4n×4n4n \times 4n4n×4n real Hadamard matrix H4nH_{4n}H4n​ (conjectured to be true) and uses this to construct an orthogonal matrix Y4n−t{Y}_{4n-t}Y4n−t​ of size (4n−t)×(4n−t)(4n - t) \times (4n - t)(4n−t)×(4n−t), such that the absolute value of each entry varies a little from 14n−t\frac{1}{\sqrt{4n-t}}4n−t​1​. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be ≤1d(1+O(d−14))<2\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2≤d​1​(1+O(d−41​))<2, for proper choices of parameters, the class of dimensions ddd being infinitely large.
The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension ddd is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1)(d+1)(d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6d=6d=6 or d=10d=10d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
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.