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Quantum Information and Computation

Evidence for and against Zauner's MUB conjecture in C^6

Gary McConnell, Harry Spencer, Afaq Tahir

Published Jul 14, 2021• 1 citation
Publisher

Abstract

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, 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 $\CC^6$, 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.

DOI
10.26421/qic21.9-10-1

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The problem of finding provably maximal sets of mutually unbiased bases in \CCd\CC^d\CCd, 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 ind=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 $\CC^6$, 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.
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≡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 C6\mathbb{C}^6C6, 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.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′\mathbb C^d, B {and} B'Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′\forall b\in B, b'\in B'∀b∈B,b′∈B′ the scalar product b⋅b′b\cdot b'b⋅b′ has modulus d−1/2d^{-1/2}d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cdψ\in \mathbb C^dψ∈Cd when measured in the mutually unbiased bases B_jofCdB\_{j} {of} \mathbb C^dB_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when ddd is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension ddd. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the ddd-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for ddd even or odd number, and to recover the known result of existence of d+1d+1d+1 MUB's for ddd a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
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.