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arXiv:2103.08703

Evidence for and against Zauner's MUB Conjecture in $\mathbb{C}^6$

Gary McConnell, Spencer, Harry, Afaq Tahir

Published Mar 15, 2021• 2 citations
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Abstract

The problem of finding provably maximal sets of mutually unbiased bases in $\mathbb{C}^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case, $d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in $d=6$ by looking at their `shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers $q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which -- at least formally -- points in the opposite direction. In $\mathbb{C}^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

DOI
10.48550/arxiv.2103.08703

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Evidence for and against Zauner's MUB Conjecture in C6\mathbb{C}^6C6
The problem of finding provably maximal sets of mutually unbiased bases in Cd\mathbb{C}^dCd, for composite dimensions ddd which are not prime powers, remains completely open. In the first interesting case, d=6d=6d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in d=6d=6d=6 by looking at their shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers $q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which -- at least formally -- points in the opposite direction. In $\mathbb{C}^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions ddd which are not prime powers, remains completely open. In the first interesting case,~d=6d=6d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~d=6d=6d=6 by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~q≡5 mod 12q \equiv 5 \bmod 12q≡5mod12, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in Cd\mathbb{C}^dCd, a set of MUBs of the theoretical maximal size d+1d + 1d+1 exists only if ddd is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers n,d≥2 that can be maximally violated in dimension d if and only if n MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the ``four most distant bases'' in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. Lü, B.-G. Englert, {Phys. Rev. A}, { 83} 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions ddd that are powers of prime numbers, there exists a set of up to d+1d+1d+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 kkk MUBs in a given dimension ddd. 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.
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.