MUBs, Polytopes, and Finite Geometries

2013

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p 2 and rank r. The q vectors of a basis of Hq correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

International Journal of Quantum Information, 2010

matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime.

2005

The Regular Polytope has been shown to be a promising candidate for the rigorous representation of geometric objects, in a form that is computable using the finite arithmetic available on digital computers. It is also apparent that the approach can be used in the formulation of repeatable and verifiable definitions of geometric objects with the aim of defining robust information interchange protocols. In the definition and investigation of the properties of the Regular Polytope representation, several propositions have been made, and the proofs developed. These proofs can be quite long and complex, and not suitable for presentation at conferences or publication in journals. They are gathered here to make them available for scrutiny in conjunction with the various papers and publications that refer to them. The major assertions that this document addresses are: The set of Regular Polytopes forms and spans a Topological Space. There is a useful correspondence between the definition of equality of Regular Polytopes and the natural "point set" definition of equality.. A simplified "programming shortcut" can be used to reduce significantly the algorithmic complexity of implementation.

Journal of Combinatorial Designs, 2020

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called k-nets (and in particular, between complete collections of MUBs and finite affine-or equivalently: finite projective-planes). Here we introduce the notion of a k-net over an algebra A and thus provide a common framework for both objects. In the commutative case, we recover (classical) k-nets, while choosing A := M d (C) leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for k-nets that can be completed to affine planes using a completely different, combinatorial argument. For k-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but k ≤ √ d bases of a complete collection of MUBs in C d are the elements of the remaining k bases (up to phase factors). In general, this is false when k is just the next integer after √ d; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

Journal of Physics A: Mathematical and General, 2006

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H q , q = p r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p 2 and rank r. The q vectors of a basis of H q correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

2003

2003

Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if |<b|b'>|^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

AIP Conference Proceedings, 2007

This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of "mubness" is introduced, and applied to some explicit calculations in six dimensions (partly done by Björck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.

Algorithmica, 2002

We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions that are powers of primes is presented. It is also proved that in any dimension d the number of mutually unbiased bases is at most d + 1. An explicit representation of mutually unbiased observables in terms of Pauli matrices are provided for d = 2 m .

Discrete &amp; Computational Geometry, 2017

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter diam(K) is given in Euclidean d-dimensional space, where d is a constant. Given an error parameter ε > 0, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most ε • diam(K). By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that O(1/ε (d-1)/2 ) facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is O(1/ε (d-1)/2 ), where O conceals a polylogarithmic factor in 1/ε. This is a significant improvement upon the best known bound, which is roughly O(1/ε d-2 ). Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Bárány and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

Israel Journal of Mathematics, 1984

Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to a d-polytope with n vertices, given combinatorially, one gets many combinatorial d-polytopes-and polytopes only-with n + I vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.

Journal of Mathematical Psychology, 1997

Reviewed by Reinhard Suck Gu nter Ziegler is Professor of Mathematics at the Technische Universita t Berlin. Previously he held positions at Augsburg University and at ZIB Berlin. He received his Ph.D. in Mathematics from MIT in 1987. He was awarded the Gerhard Hess-Prize of the German Science Foundation (DFG) in 1994. His primary research interests are topological combinatorics, discrete geometry, and linear and combinatorial optimization. He is a coauthor (with A. Bjo rner, M. Las Vergnas, B. Sturmfels, and N. White) of the book Oriented Matroids (Cambridge University Press, 1993). Reinhard Suck of Akademischer Rat at the University of Osnabru ck. His primary research interests are measurement and probabilistic scaling with a bias towards discrete mathematics. He is coeditor with Eddy Roskam of Progress in Mathematical Psychology (Amsterdam: North Holland, 1987).

Journal of Combinatorial Theory, Series B, 1971

Computing Research Repository, 2009

We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.

2017

I introduce a new notion, that extends the mutually unbiased bases (MUB) conditons to more than two bases. These, I call the nUB conditions, and the corresponding bases n-fold unbiased. They naturally appear while optimizing generic n-to-one quantum random access code (QRAC) strategies. While their existence in general dimensions is an open question, they nevertheless give close-to-tight upper bounds on QRAC success probabilities, and raise fundamental questions about the geometry of quantum states.