Multi-Photon Multi-Channel Interferometry for Quantum Information Processing

This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on the state of light in $n_{s}$ spatial and $n_{p}$ internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU$(n)$ basis states, which reduce the canonical subgroup chain, for arbitrary $n$. The boson realizations are employed to construct $\mathcal{D}$-functions, which are the matrix elements of arbitrary irreducible representations, of SU$(n)$ in the canonical basis. I show that immanants of principal submatrices of a unitary matrix $T$ are a sum of the diagonal $\mathcal{D}(\Omega)$-functions of group element $\Omega$ over $t$ determined by the choice of submatrix and over the irrep $(\lambda)$ determined by the immanant under consideration. The algorithm for $\mathrm{SU}(n)$ $\mathcal{D}$-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.Comment: PhD thesis submitted and defended successfully at the University of Calgary. This thesis is based on articles arXiv:1403.3469, arXiv:1507.06274, arXiv:1508.00283, arXiv:1508.06259 and arXiv:1511.01851 with co-authors. 145 pages, 31 figures, 11 algorithms and 4 tables. Comments are welcom

Combinatorial aspects of Hecke algebra characters

Iwahori-Hecke algebras are deformations of Coxeter group algebras. Their origins lie in the theory of automorphic forms but they arise in the representation theory of Coxeter groups and Lie algebras and in quantum group theory. The Kazhdan-Lusztig bases of these algebras, originally introduced in the late 1970s in connection with representation-theoretic concerns, has turned out to have deep connections to Schubert varieties, intersection cohomology, and related topics.Matrix immanants were originally introduced by Littlewood as a generalization of determinants and permanants. They remained obscure until the 1980s when their connections to symmetric function and representation theory as well as their surprising algebraic and combinatorial properties came to light. In particular, it was discovered that they have a fruitful connection to the theory of total positivity. More recently, a theory of quantum immamants was developed, providing a bridge to the quantum group theory.In this paper we develop the theory of certain planar networks, which provide a unified combinatorial setting for these fields of study. In particular, we use these networks to evaluate certain characters of the symmetric group algebra. We give new combinatorial interpretations of the quantum induced sign and trivial characters of the type A Iwahori-Hecke algebras

Path Tableaux and the Combinatorics of the Immanant Function

Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant. The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant. We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the $R$-weighted planar network construction, where $R$ is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices

Gaussian limit for determinantal random point fields

We prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized Gaussian random process.Comment: This is the revised version accepted for publication in the Annals of Probability. The results of Theorems 1 and 2 are extended, minor misprints are correcte

Statistics of Extreme Spacings in Determinantal Random Point Processes

We study translation-invariant determinantal random point fields on the real line. We prove, under quite general conditions, that the smallest nearest spacings between the particles in a large interval have Poisson statistics as the length of the interval goes to infinity.Comment: 16 page