Mutually unbiased bases: polynomial optimization and symmetry

Sander Gribling and Sven Polak

Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands

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A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. A natural question is for which pairs $(d,k)$ there exist $k$ mutually unbiased bases in dimension $d$. The (well-known) upper bound $k \leq d+1$ is attained when $d$ 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 $C^*$-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 $S_d$ and $S_k$.
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 $S_d \wr S_k$-module $\mathbb C^{([d]\times [k])^t}$ into irreducible modules. We present numerical results for small $d,k$ and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist $d+2$ mutually unbiased bases in dimensions $d=2,3,4,5,6,7,8$. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than $3$ MUBs in dimension $6$ requires polynomials of total degree at least $12$.

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[7] Maria Prat Colomer, Luke Mortimer, Irénée Frérot, Máté Farkas, and Antonio Acín, "Three numerical approaches to find mutually unbiased bases using Bell inequalities", Quantum 6, 778 (2022).

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