Mutually unbiased bases and related structures

A set of bases of a d dimensional complex vector space, each pair of which is unbiased, is a set of mutually unbiased bases (MUBs). MUBs have applications in quantum physics and quantum information theory. Although the motivation to study MUBs comes from physical properties, MUBs are a mathematical structure. This is a mathematical investigation. There are many open problems in the theory of MUBS, some with conjectured solutions. For example: What is the maximum number of MUBs in a d dimensional vector space? Do complete sets of MUBs exist in all dimensions? One such conjectured solution states that a complete set of MUBs exists in a d dimensional complex vector space if and only if a complete set of mutually orthogonal Latin squares (MOLS) of order d exists (Saniga et. al., Journal of Optics B, 6: L19-20, 2004). The aim of this research was to find evidence for or against this conjecture. Inspired by constructions of MUBs that use sets of MOLS, complete sets of MOLS were constructed from two complete sets of MUBs. It is interesting to note that the MOLS structure emerges not from the vectors, but from the inner products of the vectors. Analogous properties between Hjelmslev planes and MUBs, and gaps in this knowledge motivated investigation of Hjelmslev planes. The substructures of a Hjelmslev plane over a Galois ring, and a combinatorial algorithm for generating Hjelmslev planes were developed. It was shown that the analogous properties of Hjelmslev planes and MUBs occur only for odd prime powers, making a strong connection between MUBs and Hjelmslev planes unlikely. A construction of MUBs that uses planar functions was generalised by using an automorphism on the additive group of a Galois field. It is still unclear whether this generalisation is equivalent to the original construction. Relation algebras were constructed from the structure of MUBs which do not share any similarities with relation algebras constructed from MOLS. It is possible that further investigation may yield relation algebras that are similar. It was shown that a set of Wooters and Fields type MUBs, when represented as elements of a group ring, forms a commutative monoid, whereas a set of Alltop type MUBs when similarly represented does not form a closed algebraic structure. It is known that WF and Alltop MUBs are equivalent, thus the lack of a closed structure in the Alltop MUBs suggests that the monoid is not a property of MUBs in general. Complete sets of MOLS and complete sets of MUBs are `similar in spirit', but perhaps this is not an inherent feature of MUBs and MOLS. Since all the known constructions of MUBs rely on algebraic structures which exist only in prime power dimensions, the connection may not be with MOLS, but with algebraic structures which generate both MOLS and MUBs

Nice Complete Sets of Pairwise Quasi-Orthogonal Masas:From the Basics to a Unique Encoding

In der Quantenphysik hat der Begriff der "Mutually unbiased bases", im folgenden kurz als MUBs bezeichnet, in den letzten 25 Jahren zunehmende Bedeutung erlangt. Die vorliegende Arbeit behandelt paarweise quasi-orthogonale maximale *-Unteralgebren (Masas) von Matrixalgebren, die als algebraisches Gegenstück von MUBs verstanden werden können. Neben den Grundlagen dieses Themenkomplexes werden die bekanntesten Konstruktionsverfahren sog. vollständiger Familien von MUBs vorgestellt. Standardpaare von Masas werden besonders fokussiert, zudem zu sog. "normalen Paaren" verallgemeinert. Diese passen insoweit zum bekannten Konzept der "schönen Masa-Familien" von Aschbacher et al., als dass ein normales Paar stets auch eine schöne Familie ist. Das Hauptergebnis dieser Arbeit besagt, dass alle vollständigen schönen Masa-Familien auf eine einzige Weise codiert werden können. Ein äquivalentes Ergebnis findet sich schon bei Calderbank et al.; anders als dort wird es jedoch in der vorliegenden Arbeit mithilfe elementarer Matrixalgebra hergeleitet.Mutually unbiased bases (MUBs) have gained considerable importance in quantum physics over the past 25 years. The present thesis centres on pairwise quasi-orthogonal maximal abelian ∗-subalgebras (masas) of matrix algebras, which are an algebraic counterpart of MUBs. Starting from the basics, we first discuss the connections between equivalent pictures of MUBs, and then illustrate the most famous constructions of so-called complete sets of MUBs in prime power dimensions. We attach special importance on standard pairs of masas, and generalise this notion to pairs we call normal. Our concept of normal masa pairs is compatible with nice masa families defined by Aschbacher et al., in the sense that each normal masa pair is a nice family of length two. As the main result of this thesis, we prove that one unique method permits to encode all nice complete families. Calderbank et al. had established an equivalent result earlier; by contrast to their technique, the one presented here is based on elementary matrix algebra.<br

Mutually orthogonal Latin squares from the inner products of vectors in mutually unbiased bases

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in C-d are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Daki? and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350-354; Wootters and Fields 1989 Ann. Phys. 191 363-381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs

Mutually orthogonal Latin squares and mutually unbiased bases in dimensions of odd prime power: MOLS and MUBs in odd prime power dimensions

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19-L20, 2004) that the existence of a complete set of MUBs in ℂ is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case

From Incompatibility to Optimal Joint Measurability in Quantum Mechanics

This thesis is concerned with several topics related to concept of incompatibility of quan- tum observables. The operational description of quantum theory is given, in which incom- patibility is expressed in terms of joint measurability. A connection between symmetric informationally complete positive operator-valued measures and mutually unbiased bases is given, and examples of this connection holding based on investigations in Mathematica are presented. An extension of the Arthurs-Kelly measurement model is then given, where the measured observable is calculated, thereby generalising the results given previously in the literature. It is shown that in the case of prior correlations between measurement probes there exists the possibility that a measurement of both probes leads to marginal observables with smaller statistical spread than if measurements are performed on the individual probes. This concept is then highlighted by considering two probe states that allow for this reduction in spread, and the required conditions for success are given. Fi- nally, error-error relations for incompatible dichotomic qubit observables are considered in the case of state-dependent and independent error measures. Quantities that arise in the state-independent measures case, which were previously presented geometrically, have been given operational meaning, and optimal approximating schemes in both cases are compared. Limitations regarding the state-dependent optimal approximations, and experimental work built upon this construction are also discussed

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes