Mutually unbiased bases: polynomial optimization and symmetry

• URL: http://arxiv.org/abs/2111.05698v5
• Date: Mon, 15 Apr 2024 14:01:09 GMT
• Title: Mutually unbiased bases: polynomial optimization and symmetry
• Authors: Sander Gribling, Sven Polak,
• Abstract summary: A set of $k$ orthonormal bases of $mathbb Cd$ is called mutually unbiased $|langle e,frangle |2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them tractable.
• Score: 1.024113475677323
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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. Navascu\'es, Pironio, and Ac\'in 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~$12$.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Some new infinite families of non-$p$-rational real quadratic fields [0.0]
  We give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real fields, unramified above any of the $p_j$. One feature of these techniques is that they may be used to yield fields $K=mathbbQ(sqrtD)$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $pa
  arXiv  Detail & Related papers  (2024-06-20T18:00:51Z)
• Dimension-free Remez Inequalities and norm designs [48.5897526636987]
  A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• Mutually unbiased maximally entangled bases from difference matrices [0.0]
  Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We establish $q$ mutually unbiased bases with $q-1$ maximally entangled bases and one product basis in $mathbbCqotimes mathbbCq$ for arbitrary prime power $q$.
  arXiv  Detail & Related papers  (2022-10-04T10:45:22Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Dual bounds for the positive definite functions approach to mutually unbiased bases [6.6673883720496425]
  A long-standing open problem asks if there can exist 7 mutually unbiased bases (MUBs) in $mathbbC6$. We prove that such a method of a degree at most 6 cannot exist.
  arXiv  Detail & Related papers  (2022-02-27T01:06:56Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.