A classical groupoid model for quantum networks

We give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution

Uniqueness of unitary structure for unitarizable fusion categories

We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.Comment: 14 pages; v2: generalized main theorem to semisimple C*-tensor categories with possibly infinitely many simple object

Higher linear algebra in topology and quantum information theory

We investigate categorifications of linear algebra, and their applications to the construction of 4-manifold invariants, to the construction of a variety of linear algebraic structures in quantum information theory, and to the classification of certain instances of `quantum pseudo-telepathy', a phenomenon in quantum physics where two non-communicating parties can use pre-shared entanglement to perform tasks classically impossible without communicating. This thesis is divided into four chapters, closely following arXiv:1812.11933, arXiv:1609.07775, and arXiv:1801.09705. In the first chapter, we introduce semisimple 2-categories, fusion 2-categories and spherical fusion 2-categories. We prove that every finite semisimple 2-category is the 2-category of finite semisimple module categories of a multifusion category, and give examples of fusion 2-categories. In the second chapter, we construct, for each spherical fusion 2-category, a state-sum invariant of oriented singular piecewise-linear 4-manifolds, and show that these invariants generalize various previous 4-manifold invariants, including the Crane-Yetter invariant and a recent invariant of Cui. In the third chapter, we use biunitary connections in the 2-category of 2-Hilbert spaces to generate many new construction schemes for linear algebraic quantities of relevance to quantum information, including complex Hadamard matrices and unitary error bases, and we use these techniques to construct a unitary error basis which cannot be built using any previously known method. In the fourth chapter, we classify quantum isomorphic graphs in terms of Morita equivalence classes of algebras in certain monoidal categories, give examples of such algebras arising from groups of central type, and discuss various applications to the study of quantum pseudo-telepathy in the graph isomorphism game

Biunitary constructions in quantum information

We present an infinite number of constructions involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary connections, algebraic objects which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method

Shaded tangles for the design and verification of quantum circuits

We give a scheme for interpreting shaded tangles as quantum circuits, with the property that if two shaded tangles are ambient isotopic, their corresponding computational effects are identical. We analyze 11 known quantum procedures in this way -- including entanglement manipulation, error correction and teleportation -- and in each case present a fully-topological formal verification, yielding generalized procedures in some cases. We also use our methods to identify 2 new procedures, for topological state transfer and quantum error correction. Our formalism yields in some cases significant new insight into how the procedures work, including a description of quantum entanglement arising from topological entanglement of strands, and a description of quantum error correction where errors are `trapped by bubbles' and removed from the shaded tangle.Comment: 35 pages. A short version of this paper can be found at arXiv:1701.03309. Final versio

Structural Features Underlying Raloxifene’s Biophysical Interaction with Bone Matrix

Raloxifene, a selective estrogen receptor modulator (SERM), reduces fracture risk at least in part by improving the mechanical properties of bone in a cell- and estrogen receptor-independent manner. In this study, we determined that raloxifene directly interacts with the bone tissue. Through the use of multiple and complementary biophysical techniques including nuclear magnetic resonance (NMR) and Fourier transform infrared spectroscopy (FTIR), we show that raloxifene interacts specifically with the organic component or the organic/mineral composite, and not with hydroxyapatite. Structure–activity studies reveal that the basic side chain of raloxifene is an instrumental determinant in the interaction with bone. Thus, truncation of portions of the side chain reduces bone binding and also diminishes the increase in mechanical properties. Our results support a model wherein the piperidine interacts with bone matrix through electrostatic interactions with the piperidine nitrogen and through hydrophobic interactions (van der Waals) with the aliphatic groups in the side chain and the benzothiophene core. Furthermore, in silico prediction of the potential binding sites on the surface of collagen revealed the presence of a groove with sufficient space to accommodate raloxifene analogs. The hydroxyl groups on the benzothiophene nucleus, which are necessary for binding of SERMs to the estrogen receptor, are not required for binding to the bone surface, but mediate a more robust binding of the compound to the bone powder. In conclusion, we report herein a novel property of raloxifene analogs that allows them to interact with the bone tissue through potential contacts with the organic matrix and in particular collagen