Robin L. Hudson

Bio: Robin L. Hudson is an academic researcher from Loughborough University. The author has contributed to research in topics: Quantum stochastic calculus & Quantum probability. The author has an hindex of 22, co-authored 78 publications receiving 3194 citations. Previous affiliations of Robin L. Hudson include Nottingham Trent University & University of Nottingham.

Quantum Ito's formula and stochastic evolutions

Abstract: Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

Locally normal symmetric states and an analogue of de Finetti's theorem

Abstract: Let 9 be a separable Hilbert space and let 9 , be the Hilbert space tensor product ofn copies ofg. Let ~3, ~3, be respectively the C*-algebras of all bounded operators on 9, 9 , . The infinite C*-tensor product of countably many copies of ~3, denoted . by @ ~ , is defined as the C*-inductive limit of the sequence {~3,}, equipped with the injections (P,m: A,~--~ A,| from ~3, into ~3,,, where for m > n, llm_ . is the identity operator on the Hilbert space tensor product of m n copies of 9. A state co of@~3 determines a hierarchy {co,} of states of the C*-algebras ~ , by

A quantum-mechanical central limit theorem

Abstract: A quantum-mechanical central limit theorem for sums of pairwise anti- commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved. p(j) p(k) p(k)p(j) = qj)q(k) q(k)(j) = 0, pjq(k) _ (k)p(j) = ijk for finitely many degrees of freedom. The theorem is 'non-commutative' in so far as canonically conjugate observables are incompatible and therefore do not possess a joint probability distribution in the usual sense, but is 'commutative' in so far as observables drawn from two different summands are required to be compatible. In the present work a quantum-mechanical central limit theorem is constructed which is non-commutative in both these senses, in which the sum- mands are representations of the canonical anti-commutation relations over a finite-dimensional space such that elements drawn from two different summands anti-commute and are thus incompatible. The possibility of formulating the theorem of (3) depended on two properties

Fermion Ito's Formula and Stochastic Evolutions*

Abstract: An Ito product formula is proved for stochastic integrals against Fermion Brownian motion, and used to construct unitary processes satisfying stochastic differential equations. As in the corresponding Boson theory [10, 11] these give rise to stochastic dilations of completely positive semigroups.

Time-frequency distributions-a review

Abstract: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

Gaussian quantum information

Abstract: The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

Linear and quadratic time-frequency signal representations

Abstract: A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

Gaussian quantum information

Abstract: The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

Finitely correlated states on quantum spin chains

Abstract: We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.