Evidence for and against Zauner's MUB Conjecture in $\mathbb{C}^6$
- URL: http://arxiv.org/abs/2103.08703v1
- Date: Mon, 15 Mar 2021 20:29:36 GMT
- Title: Evidence for and against Zauner's MUB Conjecture in $\mathbb{C}^6$
- Authors: Gary McConnell, Harry Spencer and Afaq Tahir
- Abstract summary: Zauner predicted that there can exist no more than three MUBs.
In $mathbbC6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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) - Quantum charges of harmonic oscillators [55.2480439325792]
We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$.
We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
arXiv Detail & Related papers (2024-04-02T09:16:18Z) - Provably learning a multi-head attention layer [55.2904547651831]
Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models.
In this work, we initiate the study of provably learning a multi-head attention layer from random examples.
We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
arXiv Detail & Related papers (2024-02-06T15:39:09Z) - 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) - 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) - 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) - Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$.
It is assumed that a density matrix $rho$ can be determined from the expectation values of observables.
Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
arXiv Detail & Related papers (2021-12-14T16:29:53Z) - Mutually unbiased bases: polynomial optimization and symmetry [1.024113475677323]
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.
arXiv Detail & Related papers (2021-11-10T14:14:53Z) - Mutually Unbiased Unitary Bases of Operators on $d$-dimensional Hilbert
Space [0.0]
We consider mutually unbiased unitary bases (MUUB) for the space of operators, $M(d, mathbbC)$, acting on such Hilbert spaces.
The notion of MUUB reflects the equiprobable guesses of unitary in one bases of $M(d, mathbbC)$ when estimating a unitary operator in another.
arXiv Detail & Related papers (2020-03-27T01:41:03Z)
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.