Quantum Stochastic Evolution Research Articles

Existence and Uniqueness of Solutions of a Class of Quantum Stochastic Evolution Equations

We study the properties of the existence and uniqueness of solu-tions of a class of evolution quantum stochastic differential equations(QSDEs) dened on a locally convex space whose topology is gen-erated by a family of seminorms dened via the norm of the rangespace of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations denedon the space of operator processes where the topology is generated bythe family of seminorms dened via the inner product of the rangespace. The evolution operator generates a bounded semigroup. Weshow that under some more general conditions, the unique solutionis stable. These results extend some existing results in the literatureconcerning strong solutions of quantum stochastic differential equa-tions.

A Nonlocal Approach to the Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Experimental Quantum Stochastic Walks Simulating Associative Memory of Hopfield Neural Networks

With the increasing crossover between quantum information and machine learning, quantum simulation of neural networks has drawn unprecedentedly strong attention, especially for the simulation of associative memory in Hopfield neural networks due to their wide applications and relatively simple structures that allow for easier mapping to the quantum regime. Quantum stochastic walk, a strikingly powerful tool to analyze quantum dynamics, has been recently proposed to simulate the firing pattern and associative memory with a dependence on Hamming Distance. We successfully map the theoretical scheme into a three-dimensional photonic quantum chip and realize quantum stochastic walk evolution through well-controlled detunings of the propagation constant. We demonstrate a good match rate of the associative memory between the experimental quantum scheme and the expected result for Hopfield neural networks. The ability of quantum simulation for an important feature of a neural network, combined with the scalability of our approach through low-loss integrated chip and straightforward Hamiltonian engineering, provides a primary but steady step towards photonic artificial intelligence devices for optimization and computation tasks of greatly improved efficiencies.

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Quantum Filtering Equations for a System Driven by Nonclassical Fields

Using Gardiner and Collet’s input-output model and the concept of cascade system, we determine the filtering equation for a quantum system driven by light in some specific nonclassical states. The quantum system and electromagnetic field are described by making use of quantum stochastic unitary evolution. We consider two examples of the nonclassical states of field: a combination of vacuum and single photon states and a mixture of two coherent states. The stochastic evolution conditioned on the results of the photon counting and quadrature measurements is described.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

On Continuous Selection Sets of Non-Lipschitzian Quantum Stochastic Evolution Inclusions

We establish existence of a continuous selection of multifunctions associated with quantum stochastic evolution inclusions under a general Lipschitz condition. The coefficients here are multifunctions but not necessarily Lipschitz.

A Trotter-Kato theorem for quantum Markov limits

Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Itō evolution.

Canonical ensemble of an interacting Bose gas: Stochastic matter fields and their coherence

We present a novel quantum stochastic evolution equation for a matter field describing the canonical state of a weakly interacting ultracold Bose gas. In the ideal gas limit our approach is exact. This numerically very stable equation suppresses high-energy fluctuations exponentially, which enables us to describe condensed and thermal atoms within the same formalism. We present applications to ground-state occupation and fluctuations, density profile of ground-state and thermal cloud, and ground-state number statistics. Our main aim are spatial coherence properties which we investigate through the determination of interference contrast and spatial density correlations. Parameters are taken from actual experiments.

Non-abelian Weyl commutation relations and the series product of quantum stochastic evolutions

We show that the series product, which serves as an algebraic rule for connecting state-based input-output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born–Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

We consider the Hamiltonian operator associated to the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe a quantum mechanical evolution in the presence of a “quantum noise”. We characterize such a Hamiltonian in the case of arbitrary multiplicity and bounded coefficients: we find an essentially self-adjoint restriction of the operator and, in particular, we provide an explicit construction of a dense set of vectors belonging to its domain.

Control of Quantum Langevin Equations

The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.

Expressing stochastic unravellings using random evolution operators

We prove how the form of the most general invariant stochastic unravelling for Markovian (recently given in the literature by Wiseman and Diosi) and non-Markovian but Lindblad-type open quantum systems can be attained by imposing a single mathematical condition upon the random evolution operator of the system, namely a.s. trace preservation (a.s. stands for almost surely). The use of random operators ensures the complete positivity of the density operator evolution and characterizes the linear/non-linear character of the evolution in a straightforward way. It is also shown how three quantum stochastic evolution models-continuous spontaneous localization, quantum state diffusion and quantum mechanics with universal position localization-appear as concrete choices for the noise term of the evolution random operators are assumed. We finally conjecture how these operators may in the future be used in two different directions: both to connect quantum stochastic evolution models with random properties of space-time and to handle noisy quantum logical gates.

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

QSES's and the Quantum Jump

Abstract The stochastic methods in Hilbert space have been used both from a fundamental and a practical point of view. The result reported here concerns the application of these methods to model the evolution of quantum systems and does not enter into the question of their fundamental or practical status. It can easily be stated as follows: Once a quantum stochastic evolution scheme is assumed, the incompatibility of the Markov property with the notion of quantum jumps is easily established

On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

We prove that a single-jump unitary quantum stochastic evolution is unitarily equivalent to the Dirac boundary-value problem on the half-line in an extended space. It is shown that this solvable model can be derived from the Schrodinger boundary-value problem for a positive relativistic Hamiltonian on the half-line as the inductive ultrarelativistic limit corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation can be reduced to a quantum mechanical boundary-value problem in the extended space. The problem of microscopic time reversibility is also discussed in the paper.

Quantum stochastic Dirac boundary value problem, and the ultrarelativistic limit

We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half-line extra dimension. This amounts to the equivalence of the quantum measurement boundary value problem in infinite number of particles space to the stochastic calculus in Fock space. It is shown that this exactly solvable model can be obtained from a Schrödinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultra relativistic limit, corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the stochastic limit can be interpreted in terms of a quantum stochastic scheme for the time-continuous nondemolition observation. The question of microscopic time reversibility is also studied.

Quantum Stochastic Positive Evolutions:¶Characterization, Construction, Dilation

A characterization of the unbounded stochastic generators of quantum completely positive flows is given. This suggests the general form of quantum stochastic adapted evolutions with respect to the Wiener (diffusion), Poisson (jumps), or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is constructed. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is discovered, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators. A unitary quantum stochastic cocycle, dilating the subfiltering CP flows over L(H), is reconstructed.

On the generators of quantum stochastic evolution equations

A description is given for the stochastic bounded generators of quantum irreversible wave equations. The general form of quantum stochastic evolution with respect to Poisson (jumps), Wiener (diffusion), and general quantum noise is presented. The corresponding irreversible Heisenberg evolution is expressed in terms of completely positive stochastic maps. The general form of the stochastic equation for the completely dissipative equation is developed for the first time.