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

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Nice Complete Sets of Pairwise Quasi-Orthogonal Masas —From the Basics to a Unique Encoding— Mathematik Dissertationsthema Nice Complete Sets of Pairwise Quasi-Orthogonal Masas —From the Basics to a Unique Encoding— Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakultät der Westfälischen Wilhelms-Universität Münster vorgelegt von Sebastian Krusekamp aus Paderborn – 2014 – Dekan : Prof. Dr. Martin Stein Erster Gutachter : Prof. Dr. Dr. h.c. Joachim Cuntz Zweiter Gutachter : Prof. Dr. Andreas Winter Tag der mündlichen Prüfung : 14.8.2014 Tag der Promotion : 14.8.2014 PhD Thesis Nice Complete Sets of Pairwise Quasi-Orthogonal Masas —From the Basics to a Unique Encoding— Sebastian Krusekamp Münster 2014 Supervised by Prof. Dr. Dr. h.c. Joachim Cuntz Abstract Mutually unbiased bases (MUBs) have gained considerable importance in the field of quantum physics over the past twenty-five years. Various applications have been found (see e.g. the survey article [35]), and connections to different mathematical domains of interest, such as unitary Hadamard matrices ([13]), Ga- lois fields ([114]), Latin squares and design theory ([79, 112]), and finite geometry ([89]), are numerous. The counterpart of mutually unbiased bases on the level of matrix algebras are pairwise quasi-orthogonal maximal abelian ∗-subalgebras, briefly called masas. Whereas physicists’ accounts on the subject mainly focus on the picture of MUBs (e.g. [27, 35]), the present mathematical thesis centres on the algebraic framework of quasi-orthogonal masas. Starting from the very basics, we first thoroughly discuss the connections be- tween the equivalent pictures of (mutually) unbiased bases, unbiased unitary Ha- damard matrices, and quasi-orthogonal masas, and illustrate two of the most fa- mous constructions of so-called complete sets of pairwise quasi-orthogonal masas in prime power dimensions. Though this is a mathematical thesis, we also turn our attention to basic physical aspects of MUBs. We attach special importance on standard pairs of masas ([44]), and generalise this notion to pairs which we call normal. While standard pairs in dimension d model actions of the cyclic group of order d on the complex functions on d different points, normal masa pairs model actions of direct sums Z/d0 × · · · × Z/dm on the same function space, where d = d0 ··· dm is any factorisation of the dimension. We shortly address a further generalisation of normal masa pairs, and demonstrate that this generalised class comprises all quasi-orthogonal pairs in dimension four. Our concept of normal masa pairs is compatible with the existing definition of nice masa families by Aschbacher, Childs, and Wocjan ([3]), in the sense that each normal masa pair is a nice family of length two. Moreover, each masa pair in a complete nice family is automatically normal. According to Godsil and Roy ([40]), all known constructions of complete quasi-orthogonal masa sets produce nice families. We show that the generalised Clifford algebra, as defined by Yamazaki and Morris in the sixties ([74,75,115]), is an appropriate framework for the construction of nice complete masa families. As the main result of this thesis, we prove that, although various constructions of nice complete families have been proposed so far (e.g. [1, 9, 49, 60, 114]), one unique method permits to encode all nice complete families. Calderbank, Cameron, Kantor, and Seidel ([23]) had established an equivalent result earlier, in terms of so-called symplectic spreads, as has later been observed by Godsil and Roy ([40]). By contrast to this approach, the one presented here is based on more or less elementary means of matrix algebra. The unique encoding of nice complete masa families opens the theoretical door to classify the latter. While the complete nice masa families in the smallest dimen- sions turn out to be essentially unique, it is known that there are many prime power dimensions admitting inequivalent complete families. However, the classi- fication of these families appears to be an extraordinarily complicated matter ofits own, and is not dealt with in the present work. Danksagung (Acknowledgements) Herzlich möchte ich mich an erster Stelle bei meinem Betreuer Prof. Dr. Dr. h.c. Joachim Cuntz bedanken. Er erlaubte mir frei und eigenständig zu arbeiten, nahm sich zugleich aber stets Zeit für ausführliche Besprechungen, die für mich nicht nur mathematisch sehr anregend und lehrreich, sondern oft genug auch persönlich ermutigend waren, gerade wenn sich Erfolge mitunter nur schleppend einstellten. Meinen Freunden und Kollegen am Fachbereich Mathematik der Wilhelms- Universität Münster, vor allem in der Arbeitsgruppe Funktionalanalysis, Operatoral- gebren und nichtkommutative Geometrie, danke ich für das menschlich wie mathema- tisch hervorragende Arbeitsumfeld. Zum Glück gehören Elke und Gabi dazu, die immer einen klaren Kopf und ein gutes Wort haben, vom richtigen Formular ganz zu schweigen. Meine Freunde Klaus und Walther waren mir zugleich wunderbare Büro-Kol- legen. Beiden bin ich darüber hinaus, wie auch Thomas und Moritz, zu Dank verpflichtet für konkrete fachliche Hilfe und fruchtbare Diskussionen. Prof. Dr. Andreas Winter gab mir wertvolle Impulse bei meiner Suche nach einem interesssanten Thema. Zudem erklärte er sich bereit, Zweitgutachter dieser Arbeit zu sein. Für beides danke ich ihm herzlich. Prof. Dr. Urs Hartl nahm sich die Zeit, Fragestellungen meiner Arbeit mit mir zu diskutieren. Ihm bin ich ebenso dankbar wie Prof. Dr. William M. Kantor, der mir mit großer Klarheit und Freund- lichkeit eine Forschungsfrage beantwortete, die mich so lange verfolgte. Großen Dank schulde ich Franziska für die Korrektur meiner sprachlichen Mängel in wichtigen Textpassagen. Meiner Familie danke ich; meinen Freundinnen und Freunden, neben den be- reits genannten vor allem Thomas, Doris und Stefan mit Alida, Marika und Frank, Barbara und Thomas, Jule und Didi, Robert, Carl und schließlich Bruno, für ihre vielfältige Unterstützung in den vergangenen Jahren. Zu guter Letzt danke ich meiner Freundin Johanna, für alles. Contents Introduction 1 Motivation and subject classification . 1 Content and organisation of the present thesis . 2 Notations and conventions . 5 1 Maximal abelian *-subalgebras (masas) in the complex d×d-matrices 9 1.1 Motivation: Finite dimensional quantum systems and maximal abelian *-subalgebras . 9 1.2 Masas in the complex d×d-matrices, and associated orthonormal bases and unitaries . 12 1.3 Complex Hadamard matrices . 17 1.4 Unitary Hilbert-Schmidt orthonormal bases of masas . 21 2 Quasi-orthogonality of masas 27 2.1 Motivation: Pairs of mutually unbiased bases in quantum physics . 27 2.2 Quasi-orthogonal masas, unbiased bases and unbiased unitaries . 29 2.3 Measures of quasi-orthogonality . 41 2.4 Equivalent, standard, and non-standard pairs of quasi-orthogonal masas 53 2.5 Excursion: Standard pairs of masas and crossed products . 63 3 Families of pairwise quasi-orthogonal masas 71 3.1 Motivation: Optimal state-determination of quantum systems . 71 3.2 Maximal and complete families of pairwise quasi-orthogonal masas and the MUB-Problem . 78 3.3 Constructions of complete quasi-orthogonal families of masas in prime power dimensions . 88 3.4 A construction of quasi-orthogonal masa families in square dimensions 100 3.5 Excursion: Mutually unbiased bases and design theory . 104 4 Nice masa families and the generalised Clifford algebra 115 4.1 Nice unitary error bases . 115 4.2 Normal pairs of quasi-orthogonal masas . 124 4.3 Nice families of pairwise quasi-orthogonal masas . 134 4.4 Nice complete masa families in the generalised Clifford algebra . 142 4.5 Excursion: Concatenated normal pairs of masas . 154 5 Smid families 161 5.1 Equivalence classes of smid families and nice sets of masas . 162 5.2 Complete linear smid families in the matrix algebras M2(Fp) . 182 5.3 Smid families in M2(Fp), permutation polynomials, and Latin squares . 190 5.4 Smid families in small dimensions—some computer algebraic results . 204 Conclusion 209 Appendix 211 Bibliography 235 List of figures 245 List of symbols 247 Index 251 Introduction In the present thesis, we study families of pairwise quasi-orthogonal maximal abelian ∗-subalgebras—called masas in short—of the complex d×d-matrices, with a special emphasis on so-called nice masa families. Though doubtlessly of an intrinsic, purely mathematical interest, a very strong motivation for this topic stems from the field of quantum physics, since pairwise quasi-orthogonal masas in Md(C) can be considered as the algebraic counterpart of mutually unbiased bases (MUBs) in the Hilbert space Cd. Motivation and subject classification d A pair of orthonormal bases (x0,..., xd−1), (y0,..., yd−1) of C is said to be unbiased 2 if the identity j(xijyj)j = 1/d holds for all 0 ≤ i, j < d. Pairs of unbiased bases cor- respond to so-called complementary pairs of quantum physical measurements (observ- ables). Performing such measurements one after the other in any order, the second outcome is perfectly independent of the first. As one of the basic principles of quan- tum mechanics, this is not true in general. A set of MUBs for the Hilbert space Cd containing d + 1 members is called complete, for it is known that no larger sets can exist. Families of MUBs, complete families in par- ticular, have met a number of important applications in quantum information theory in the recent decades, such as optimal state determination, quantum teleportation, or quantum cryptography (see [35] for a good
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