Quantum Stochastic Flows Research Articles

Feynman–Kac perturbation of $$C^*$$ quantum stochastic flows

Feynman–Kac perturbation of $$C^*$$ quantum stochastic flows

A quantum algorithm for uniform sampling of models of propositional logic based on quantum probability

We describe a class of quantum algorithms to generate models of propositional logic with equal probability. We consider quantum stochastic flows that are the quantum analogues of classical Markov chains and establish a relation between fixed points on the two flows. We construct chains inspired by von Neumann algorithms using uniform measures as fixed points to construct the corresponding irreversible quantum stochastic flows. We formulate sampling models of propositions in the framework of adiabatic quantum computing and solve the underlying satisfiability instances. Satisfiability formulation is an important and successful technique in modeling the decision theoretic problems in a classical context. We discuss some features of the proposed algorithms tested on an existing quantum annealer D-Wave II extending the simulation of decision theoretic problems to a quantum context.

A Homomorphism Theorem and a Trotter Product Formula for Quantum Stochastic Flows with Unbounded Coefficients

We give a new method for proving the homomorphic property of a quantum stochastic ow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic ows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of [5]

Quantum Feynman-Kac perturbations

We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.

Itô calculus and quantisation of Lie bialgebras

Motivated by classical and quantum stochastic flows, particularly the distinction between inner and outer flows in the quantum case, we develop analogous notions which live in the Itô Hopf algebra, which is got by replacing the shuffle product in a shuffle Hopf algebra by a noncommutative multiplication which abstracts the product rule for iterated quantum stochastic integrals. Inner flows got by conjugation by ordered double product integrals ∏ → ← ( 1 + d r [ h ] ) are used to quantise Lie bialgebras.

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

We consider a Markovian approximation, of weak coupling type, to an open system perturbation involving emission, absorption and scattering by reservoir quanta. The result is the general form for a quantum stochastic flow driven by creation, annihilation and gauge processes. A weak matrix limit is established for the convergence of the interaction-picture unitary to a unitary, adapted quantum stochastic process and of the Heisenberg dynamics to the corresponding quantum stochastic flow: the convergence strategy is similar to the quantum functional central limits introduced by Accardi, Frigerio and Lu$^{[ 1]}$. The principal terms in the Dyson series expansions are identified and re-summed after the limit to obtain explicit quantum stochastic differential equations with renormalized coefficients. An extension of the Pul\UNICODE{0xe9} inequalities$^{[ 2]}$ allows uniform estimates for the Dyson series expansion for both the unitary operator and the Heisenberg evolution to be obtained.

A stochastic Stinespring Theorem

Completely positive Markovian cocycles on a von Neumann algebra, adapted to a Fock filtration, are realised as conjugations of \(^*\)-homomorphic Markovian cocycles. The conjugating processes are affiliated to the algebra, and are governed by quantum stochastic differential equations whose coefficients evolve according to the \(^*\)-homomorphic process. Some perturbation theory for quantum stochastic flows is developed in order to achieve the above Stinespring decomposition.

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form govern stochastic flows on a C *-algebra ?. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on ?. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when ? is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and *-homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations: in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions.

Deformations of Algebras Constructed using Quantum Stochastic Calculus

Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.

Generators of quantum stochastic flows and cyclic cohomology

This paper relates cyclic cohomology, as obtained from traces with suitable domains on free product algebras, to the construction, by using the quantum stochastic calculus of Hudson and Parthasarathy, of quantum (stochastic) flows. As an application of the methods of the present article, we show how the characters of finitely summable Fredholm modules, as constructed by Connes, can be recovered from quantum flows in both the specialisation to the pure gauge and to the Brownian case. We also relate the regularized trace of Connes, used in the construction of these characters, to Lindblad generators.

Non-Abelian cohomology and Fermion flows over ?2-Graded *-algebras

We extend to the case of Fermion flows the description in terms of Lue's non-Abelian cohomology of the generators of quantum stochastic flows.

Fermionic quantum stochastic flows

Existence, uniqueness and, using a higher-order Fermionic Ito formula, multiplicativity are proved for even quantum stochastic flows over ℤ2graded C*-algebras driven by Fermionic noise with bounded structure maps.

Fermion stochastic calculus in Dirac-Fock space

A quantum stochastic calculus for fermions is developed where the basic integrators are based on Dirac fields and the charge operator. The associated Ito formula has seven non-trivial correction terms. Conditions are found for the solutions of stochastic differential equations to be unitary and it is shown that the corresponding quantum stochastic flow manifests a broken symmetry whereby the particle and antiparticle noises no longer balance each other. An abstract theory of such flows is then developed. By employing the unification between boson and fermion stochastic calculi, we are able to develop the entire theory using boson Fock spaces.

Deformations of cocycles, quantum Lévy processes and quantum stochastic flows

We obtain a quantum analogue of a classical stochastic process with stationary and independent increments via the Weyl quantisation of certain deformed cocycles. A class of quantum stochastic flows driven by such processes is then constructed.

On a class of stochastic flows driven by quantum Brownian motion

A class of quantum stochastic flows on*-algebras is introduced which includes both classical flows on Riemannian manifolds and flows induced by Lie group actions onC*-algebras. Criteria are established to determine those flows which are unitarily equivalent to ones driven by classical Brownian motion. It is shown that taking complex combinations of the driving coefficients of such flows gives rise to flows which are not of Evans-Hudson type (i.e., all driving coefficients do not preserve the relevant algebra).