arXiv subjects

V. P. Belavkin

Publications and source records attributed to V. P. Belavkin.

At least 37 records · Page 2Linked to original sources

arXiv2005

On the Dynamical Solution of Quantum Measurement Problem

The development of quantum measurement theory, initiated by von Neumann, only indicated a possibility for resolution of the interpretational crisis of quantum mechanics. We do this by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or beables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the past beables with the potential future. This rule, together with the self-compatibility of the measurements insuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of the dynamical commutation relations leads in particular to the derivation of the von Neumann projection postulate. This gives a quantum stochastic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schroedinger and Bohr.

quant-ph↗

arXiv2005

Nondemolition Principle of Quantum Measurement Theory

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The "collapsed state-vector" after the "objectification" is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of "continuous spontaneous localization" has been derived from the Schroedinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments.

quant-ph↗

arXiv2005

Quantum Continual Measurements and a Posteriori Collapse on CCR

A quantum theory for the Markovian dynamics of an open system under the unsharp observation which is continuous in time, is developed within the CCR stochastic approach. A stochastic classical equation for the posterior evolution of quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the general linear case of a quasi--free Hamiltonian in the initial CCR algebra with a fixed output observable field, and the posterior Kalman dynamics coresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of quantum unstable open system that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one.

math-ph↗

arXiv2005

Reconstruction theorem for quantum stochastic processes

Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered operator field in an arbitrary space-time region T of an open quantum system under a sequential observation at a discrete space-time localization. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered discrete T=Z to the construction of the inductive limit of Lindblad's canonical representations, corresponds to Kolmogorov's classical reconstruction if the order on T is ignored and leads to Lewis construction if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically preordered on T. The approach presented encompasses both nonrelativistic and relativistic irreversible dynamics of open quantum systems and fields satisfying the conditions of local commutativity and semigroup covariance. Also given are necessary and sufficient conditions of dynamicity (or conditional Markovianity) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP.

math.PR↗

arXiv2005

Semilogics, Quasilogics and Other Quantum Structures

We give an axiomatic formulation of quantum structures like semilogics and quasilogics which generalize the boolean semirings of events and fuzzy logics. The notions of distributions, states, representations observables and semiobservables are introduced and their Hilbert space realizations are found. The closed and open structures in semilogics are introduced and the regular distributions on the semilogics are studied.

math.LO↗

arXiv2005

Quantum Probabilities and Paradoxes of the Quantum Century

A history and drama of the development of quantum probability theory is outlined starting from the discovery of the Plank's constant exactly a 100 years ago. It is shown that before the rise of quantum mechanics 75 years ago, the quantum theory had appeared first in the form of the statistics of quantum thermal noise and quantum spontaneous jumps which have never been explained by quantum mechanics. Moreover, the only reasonable probabilistic interpretation of quantum theory put forward by Max Born was in fact in irreconcilable contradiction with traditional mechanical reality and classical probabilistic causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schroedinger. They are reconsidered in this paper from the modern quantum probabilistic point of view.

math.PR↗

arXiv2005

A New Wave Equation For a Continuous Nondemolition Measurement

A stochastic model for nondemolition continuous measurement in a quantum system is given. It is shown that the posterior dynamics, including a continuous collapse of the wave function, is described by a nonlinear stochastic wave equation. For a particle in an electromagnetic field it reduces the Schroedinger equation with extra imaginary stochastic potentials.

quant-ph↗

arXiv2005

A Quantum Particle Undergoing Continuous Observation

A stochastic model for the continuous nondemolition ohservation of the position of a quantum particle in a potential field and a boson reservoir is given. lt is shown that any Gaussian wave function evolving according to the posterior wave equation with a quadratic potential collapses to a Gaussian wave packet given by the stationary solution of this equation.

quant-ph↗

arXiv2005

Nondemolition observation of a free quantum particle

A stochastic model of a continuous nondemolition observation of a free quantum Brownian motion is presented. The nonlinear stochastic wave equation describing the posterior dynamics of the observed quantum system is solved in a Gaussian case for a free particle of mass m. It is shown that the dispersion of the wave packet does not increase to infinity like for the free unobserved particle but tends to the finite limit.

quant-ph↗

arXiv2005

Uniformization of Nonlinear Hamiltonian Systems of Vlasov and Hartree Type

Nonlinear Hamiltonian systems describing the abstract Vlasov and Hartree equations are considered in the framework of algebraic Poissonian theory. The concept of uniformization is introduced; it generalizes the method of second quantization of classical systems to arbitrary Hamiltonian (Lie-Jordan, or Poissonian) algebras, in particular to the algebras of operators in indefinite spaces. A functional calculus is developed for the uniformized observables, which unifies the calculi of generating functionals for classical, Bosonian and Fermionian multiparticle variables. The nonlinear kinetic equation of Vlasov and Hartree type are derived for both classical and quantum multiparticle Hamiltonian systems in the mean field approximation. The possibilities for finding approximate solutions of these equations from the solutions of uniformized linear equations are investigated.

math-ph↗

arXiv2005

Quantum Stochastic Semigroups and Their Generators

A rigged Hilbert space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is described, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators is constructed.

math.PR↗

arXiv2005

Quantum Stochastic Calculus and Quantum Nonlinear Filtering

A *-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given.

math.PR↗

arXiv2005

On the Generators of Quantum Stochastic Master Equation

A characterisation of the stochastic bounded generators of quantum irreversible Master equations is given. This suggests the general form of quantum stochastic evolution with respect to the Poisson (jumps), Wiener (diffusion) or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is found and the general form of the stochastic completely dissipative (CD) operator equation is discovered.

quant-ph↗

arXiv2005

Chaotic States and Stochastic Integration in Quantum Systems

Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel-operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space.

math.PR↗

arXiv2005

Infinite Dimensional Ito Algebras of Quantum White Noise

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.

math.PR↗

arXiv2005

Positive Definite Germs of Quantum Stochastic Processes

A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Ito algebra. A representation theorem, giving the pseudo-Hilbert dilation for the germ-matrix of the differential, is proved. This suggests the general form of quantum stochastic evolution equations with respect to the Poisson (jumps), Wiener (diffusion) or general quantum noise.

math.PR↗

arXiv2005

On Stochastic Generators of Positive Definite Exponents

A characterisation of quantum stochastic positive definite (PD) exponent is given in terms of the conditional positive definiteness (CPD) of their form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The structure of quasi-Poisson stochastic generators giving rise to a quantum stochastic birth processes is studied.

math.PR↗

arXiv2005

Quantum Filtering of Markov Signals with White Quantum Noise

Time-continuous non-anticipating quantum processes of nondemolition measurements are introduced as the dynamical realizations of the causal quasi-measurements, which are described in this paper by the adapted operator-valued probability measures on the trajectory spaces of the generalized temporal observations in quantum open systems. In particular, the notion of physically realizable quantum filter is defined and the problem of its optimization to obtain the best a posteriori quantum state is considered. It is proved that the optimal filtering of a quantum Markovian Gaussian signal with the Gaussian white quantum noise is described as a coherent Markovian linear filter generalizing the classical Kalman filter. As an example, the problem of optimal measurement of complex amplitude for a quantum Markovian open oscillator, loaded to a quantum wave communication line, is considered and solved.

quant-ph↗