Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation

Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements

The Wigner phase operator (WPO) is identified as an operator valued measure (OVM) and its eigenstates are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner phase distribution. It is then argued that the WPO cannot represent a projective measurement of the phase; but is in fact to be interpreted as an operator valued measure for the phase. The non-positivity of the latter can be overcome by defining a positive operator valued measure (POVM) via a proper filter function, based on the view that phase measurements are coarse-grained in phase space, leading to the well known Q-distribution. The identification of the Q phase operator as a POVM is in good agreement with the earlier observation regarding the relation between operational phase measurement schemes and the Q-distribution. The Q phase POVM can be dilated in the sense of Gelfand–Naimark, to an operational setting of interference at a beam-splitter with another coherent state – this results in a von Neumann projector with well-defined phase in the expanded Hilbert space of the two modes.

This chapter, and the three succeeding it, constitute a mathematical interlude, preparing the ground for the formal definition of a coherent state in Chapter 7 and the subsequent development of the general theory. As should be clear already, from a look at the last chapter, in order to define CS mathematically and obtain a synthetic overview of the different contexts in which they appear, it is necessary to understand a bit about positive operator-valued (POV) measures on Hilbert spaces and their close connection with certain types of group representations. In Chapter 2, we have also encountered examples of reproducing kernels and reproducing kernel Hilbert spaces, which in turn are intimately connected with the notion of POV measures and, hence, coherent states. In this chapter, we gather together the relevant mathematical concepts and results about POV measures. In the next chapter, we will do the same for the theory of groups and group representations. Chapters 5 and 6 will then be devoted to a study of reproducing kernel Hilbert spaces. The treatment is necessarily condensed, but we give ample reference to more exhaustive literature. Although the mathematically initiated reader may wish to skip these four mathematical chapters, the discussion of many of the topics here is sufficiently different from their treatment in standard texts to warrant at least a cursory glance at it.KeywordsHilbert SpaceCoherent StateBorel MeasureReproduce Kernel Hilbert SpaceTight FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

Non-orthogonal positive operator valued measure (POVM) phase distributions are analysed for one-mode and two-mode electromagnetic (em) waves. For a one-mode em field the non-orthogonal POVM phase distribution is given indirectly from the one-mode density matrix which is obtained by measurements of the quadrature probability distribution and a certain integration based on sampling pattern functions. The compensation for non-unitary detection is demonstrated by making explicit calculations for coherent states. For a two-mode em field the non-orthogonal POVM relative phase distribution is related to an angular momentum basis and two-mode angular momentum phase states. We compare our analysis with two-mode phase measurements made by other authors. We demonstrate the use of the present methods by analysing the phase distribution for the em field output from a Mach?Zehnder interferometer. For the case of coherent state input the phase distribution is obtained using the one-mode non-orthogonal POVM method, since in this case we get two coherent state outputs which are not correlated. For the case of a number state input the phase distribution is analysed by the two-mode non-orthogonal POVM method as there are strong correlations between the two em wave outputs.

The space of positive operator-valued measures on the Borel sets of a compact\n(or even locally compact) Hausdorff space with values in the algebra of linear\noperators acting on a d-dimensional Hilbert space is studied from the\nperspectives of classical and non-classical convexity through a transform\n$\\Gamma$ that associates any positive operator-valued measure with a certain\ncompletely positive linear map of the homogeneous C*-algebra $C(X)\\otimes B(H)$\ninto $B(H)$. This association is achieved by using an operator-valued integral\nin which non-classical random variables (that is, operator-valued functions)\nare integrated with respect to positive operator-valued measures and which has\nthe feature that the integral of a random quantum effect is itself a quantum\neffect. A left inverse $\\Omega$ for $\\Gamma$ yields an integral representation,\nalong the lines of the classical Riesz Representation Theorem for certain\nlinear functionals on $C(X)$, of certain (but not all) unital completely\npositive linear maps $\\phi:C(X)\\otimes B(H) \\rightarrow B(H)$. The extremal and\nC*-extremal points of the space of POVMS are determined.\n

Chapter 3 is a mathematical analysis of two related notions, namely, positive operator-valued (POV) measures and frames, and some generalizations. We start with continuous frames and introduce two generalizations, called upper, resp. lower, semi-frames, for which only the upper, resp. lower, frame bound is satisfied. Then we turn to the more familiar discrete frames and their generalizations.KeywordsHilbert SpaceCoherent StateBorel MeasureReproduce Kernel Hilbert SpaceFrame OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Quantum correlations in composite quantum systems are at the origin of the most peculiar features of quantum mechanics such as the violation of Bells inequalities and non-locality. In quantum information theory, they are viewed as quantum resources used by quantum algorithms and communication protocols to outperform their classical analogs. In this paper, we define a new measure of quantum correlation based on von Neumann entropy and positive operator-valued measurement,which has clear physical meaning and we can prove that it satisfying many good property for a measure of quantumness.

This thesis examines random walks on finite abelian groups.In particular, we ask the number of steps for convergence to uniformity for a random walk with step probability supported on a fixed number of points (with some necessary restriction on the support).Some notation, concepts, and results commonly used in such a study are discussed.A few previously known results are presented showing that the number of steps is about n 2/(k-1) for certain groups of order n where k is the size of the support.A result given by Greenhalgh shows that a constant times n 2/(k-1) steps are necessary for a random walk to approach uniform.A result from Hildebrand shows that, for a typical choice of support on Z/nZ where n is prime, a constant times n 2/(k-1) steps are sufficient.The other results are for a typical choice of support on a finite abelian group of order n = n 1 n 2 . . .n t where n 1 n 2 n t and n 1 , n 2 , . . ., n t are prime numbers.A result from Dou proves that with t L, if n 1 < An t for some constants A and L, then for some function f (n) as n , f (n)n 2/(k-1) steps suffice.McCollum extended this result to finite abelian groups such that n t > n 1/(k-1)+ for some < 1/(t 2 + t).The main result of this thesis shows that a constant times n 2/(k-1) steps are sufficient if n t as n .There are a wealth of people who have made this work possible.It gives me pleasure to be able to take this opportunity to show my appreciation for a few of them.I would first like to acknowledge the mathematicians on whose shoulders I stand.I am grateful to all the men and women whom I have never met, but whose books, papers, and results stand as a foundation for this dissertation.There have been many professors who have guided me into and through my mathematical career.

We propose a scheme to implement the quantum walk for SU(1,1) in the phase space, which generalizes those associated with the Heisenberg-Weyl group. The movement of the walker described by the SU(1,1) coherent states can be visualized on the hyperboloid or the Poincar\'{e} disk. In both one-mode and two-mode realizations, we introduce the corresponding coin-flip and conditional-shift operators for the SU(1,1) group, whose relations with those for Heisenberg-Weyl group are analyzed. The probability distribution, standard deviation and the von Neumann entropy are employed to describe the walking process. The nonorthogonality of the SU(1,1) coherent states precludes the quantum walk for SU(1,1) from the idealized one. However, the overlap between different SU(1,1) coherent states can be reduced by increasing the Bargmann index $k$, which indicates that the two-mode realization provides more possibilities to simulate the idealized quantum walk.

The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.

An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff–von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to define locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented.

Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators, eigenvalues 1.4 Four questions about Cayley graphs 1.5 Finite Euclidean graphs and three questions about their spectra 1.6 Random walks on Cayley graphs 1.7 Applications in geometry and analysis 1.8 The quadratic reciprocity law 1.9 The fast Fourier transform 1.10 The DFT on finite Abelian groups - finite tori 1.11 Error-correcting codes 1.12 The Poisson sum formula on a finite Abelian group 1.13 Some applications in chemistry and physics 1.14 The uncertainty principle Part II. Introduction 2.1 Fourier transform and representations of finite groups 2.2 Induced representations 2.3 The finite ax + b group 2.4 Heisenberg group 2.5 Finite symmetric spaces - finite upper half planes Hq 2.6 Special functions on Hq - K-Bessel and spherical 2.7 The general linear group GL(2, Fq) 2.8. Selberg's trace formula and isospectral non-isomorphic graphs 2.9 The trace formula on finite upper half planes 2.10 The trace formula for a tree and Ihara's zeta function.

Positron position operators. I. A natural option

The evolution of a discrete-time Markov Chain (MC) is determined by the evolution equation p T (t) = p T (t − 1) · P, where p(t) stands for the stochastic state vector at time t, t∈ ℕ, P interprets the stochastic transition matrix of the MC, and the superscript Tdenotes transposition of the respective column vector (or matrix). The present chapter examines under which conditions concerning the stochastic matrix P, a set of stochastic vectors, { p(t − 1)}, representing a hypersphere on the set of the attainable structures of the MC, is transformed into a stochastic set { p(t) } also representing a hypersphere of the MC. The results concerning the form of the transition matrix Pare derived by means of the product PP T . The set of the matrices P turns out to be a subset of the set of the doubly stochastic matrices.