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This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

The notion of quantoid is an analogy to the notion of matroid in the context of quantum realm. This thesis summarizes basic properties of quantoids and the correspondence between quantoids and selfdual matroids. A new set of axioms is derived as an alternative to the set which was used as the original definition of quantoid. A catalog enumerating all quantoids with the size of their ground set up to 5 elements is attached in the appendix and a larger database of quantoids (up to 7 elements in the ground set) is enclosed as an attachment of this thesis.

We explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings. Error correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts to the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity. Lastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result. / The Natural Sciences and Engineering Research Council of Canada (NSERC)

Recent results from cognitive psychology suggest that human behavior can be, in some situations, better described by laws known from physical quantum theory. This work responds to this development and seeks to answer the question of how quantum models of cognition and decision-making can contribute to our understanding of the social world. We introduce the approaches to the application of quantum theory in social sciences - quantum consciousness, quantum reasoning and quantum metaphor and further develop the second, in the context of strategic interaction in game theory. We show that the quantum model corresponds to both the psychological intuition and many empirically known anomalies such as the order effect, disjunction effect, or context effect. In this work, we derive a specific model of strategic interaction between two players, and we test it in the experimental Prisoner's Dilemma game. The model is based on the rational choice theory, which is nevertheless re-defined when it is shown how the strategic choice is changed if the quantum probability (C*-algebra) is used as the underlying theory instead of classical probability (σ-algebra). The results obtained support the quantum model and showed a different representation of the game among various groups of players. The quantum model of...