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arXiv:1604.04797

An infinite family of strongly unextendible mutually unbiased bases in $\mathbb{C}^{2^{2h}}$

Jonathan Jedwab, Lily Yen

Published Apr 16, 2016• 1 citation
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Abstract

A set of $b$ mutually unbiased bases (MUBs) in $\mathbb{C}^d$ (for $d > 1$) comprises $bd$ vectors in $\mathbb{C}^d$, partitioned into $b$ orthogonal bases for $\mathbb{C}^d$ such that the pairwise angle between all vectors from distinct bases is $\arccos(1/\sqrt{d})$. The largest number $μ(d)$ of MUBs that can exist in $\mathbb{C}^d$ 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 $\mathbb{C}^d$ 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 $\mathbb{C}^d$ satisfying $b(d) < μ(d)$ was known, this family, due to Szántó, is asymptotically "large" in the sense that $b(d)/μ(d) \to 1$ as $d \to \infty$. However, the existence of $2^{m-1}+1$ strongly unextendible MUBs in $\mathbb{C}^{2^m}$ 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.

DOI
10.48550/arxiv.1604.04797

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A set of kkk orthonormal bases of Cd\mathbb C^dCd is called mutually unbiased if ∣⟨e,f⟩∣2=1/d|\langle e,f\rangle |^2 = 1/d∣⟨e,f⟩∣2=1/d whenever eee and fff are basis vectors in distinct bases. A natural question is for which pairs (d,k)(d,k)(d,k) there exist~kkk mutually unbiased bases in dimension ddd. The (well-known) upper bound k≤d+1k \leq d+1k≤d+1 is attained when~ddd is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗C^*C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of SdS_dSd​ and SkS_kSk​. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀SkS_d \wr S_kSd​≀Sk​-module C([d]×[k])t\mathbb C^{([d]\times [k])^t}C([d]×[k])t into irreducible modules. We present numerical results for small d,kd,kd,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2d+2d+2 mutually unbiased bases in dimensions~d=2,3,4,5,6,7,8d=2,3,4,5,6,7,8d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 333 MUBs in dimension 666 requires polynomials of total degree at least~121212.
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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.
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