arXiv subjects

V. P. Belavkin

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

At least 19 recordsLinked to original sources

arXiv2007

A Game Theoretic Approach to Quantum Information

This work is an application of game theory to quantum information. In a state estimate, we are given observations distributed according to an unknown distribution $P_θ$ (associated with award $Q$), which Nature chooses at random from the set $\{P_θ: θ\in Θ\}$ according to a known prior distribution $μ$ on $Θ$, we produce an estimate $M$ for the unknown distribution $P_θ$, and in the end, we will suffer a relative entropy cost $\mathcal{R}(P;M)$, measuring the quality of this estimate, therefore the whole utility is taken as $P \cdot Q -\mathcal{R}(P; M)$. In an introduction to strategic game, a sufficient condition for minimax theorem is obtained; An estimate is explored in the frame of game theory, and in the view of convex conjugate, we reach one new approach to quantum relative entropy, correspondingly quantum mutual entropy, and quantum channel capacity, which are more general, in the sense, without Radon-Nikodym (RN) derivatives. Also the monotonicity of quantum relative entropy and the additivity of quantum channel capacity are investigated.

quant-ph↗

arXiv2007

Eventum Mechanics of Quantum Trajectories: Continual Measurements, Quantum Predictions and Feedback Control

Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.

math-ph↗

arXiv2007

Additivity of Entangled Channel Capacity for Quantum Input States

An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki-Umegaki (a-type) and of Belavkin-Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of V. P. Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.

quant-ph↗

arXiv2006

Notes on the equality in SSA of entropy on CAR algebra

We prove a necessary an sufficient condition for the states which satisfy strong subadditivity of von Neumann entropy with equality on CAR algebra and we show an example when the equality holds but the state is not separable.

math-ph↗

arXiv2005

On Stochastic Schroedinger Equation as a Dirac Boundary-value Problem, and an Inductive Stochastic Limit

We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. It is shown that this exactly solvable model can be obtained from a Schroedinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultrarelativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation is reduced to the to the quantum-mechanical boundary value problem in the extra dimension. The question of microscopic time reversibility is also studied for this paper.

math-ph↗

arXiv2005

A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock scale

A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and basis form in terms of Malliavin derivative on a projective Fock scale, and their uniform continuity and QS differentiability with respect to the inductive limit convergence is proved. A new form of QS calculus based on an inductive *-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and the unitary and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the pseudounitary and *-homomorphism property of the QS integrable generators, is proved.

math-ph↗

arXiv2005

On the Dynamical Interpretation for the Quantum-Measurement Projection Postulate

An apparatus model with discrete momentum space suitable for the exact solution of the problem is considered. The special Hamiltonian of its interaction with the object system under consideration is chosen. In this simple case it is easy to illustrate how difficulties in constructing the dynamical interpretation of selective collapse could be overcomed without any limiting procedure. For this purpose one can apply either averaging with respect to a non-quantum parameter or reducing the algebra of the joint system operators (i. e. passing to the quantum system subalgebra). The latter procedure effectively implies averaging with respect to apparatus quantum variables not belonging to the system.

quant-ph↗

arXiv2005

The Nondemolition Measurement of Quantum Time

The problem of time operator in quantum mechanics is revisited. The unsharp measurement model for quantum time based on the dynamical system-clock interaction, is studied. Our analysis shows that the problem of the quantum time operator with continuous spectrum cannot be separated from the measurement problem for quantum time.

quant-ph↗

arXiv2005

Quantum Stochastics, Dirac Boundary Value Problem, and the Ultra Relativistic Limit

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

quant-ph↗

arXiv2005

Quantum Noise, Bits and Jumps: Uncertainties, Decoherence, Trajectories and Filtering

It is shown that many dissipative phenomena of "old" quantum mechanics which appeared 100 years ago in the form of the statistics of quantum thermal noise and quantum spontaneous jumps, have never been explained by the "new" conservative quantum mechanics discovered 75 years ago by Heisenberg and Schroedinger. This led to numerous quantum paradoxes which are reconsidered in this paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators from the algebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule, together with the self-compatibility of measurements insuring the consistency of histories, is called the nondemolition principle. The application of this causality condition in the form of the dynamical commutation relations leads to the derivation of the generalized von Neumann reductions, usharp, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic 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. The simplest Markovian quantum stochastic model for the continuous in time measurements involves a boundary-value problem in second quantization for input "offer" waves in one extra dimension, and a reduction of the algebra of "actual" observables to an Abelian subalgebra for the output waves.

quant-ph↗

arXiv2005

Quantum Stochastic Differential Equation

A semi-classical non-Hamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes non-Hamiltonian at random instants of transition of pure states to reduced ones, given by a contraction C. The counting trajectories are assumed to satisfy the Poisson law. A unitary dilation of the concractive stochastic dynamics is found. In particular, in the limit of frequent detection corresponding to the large number limit we obtain the Ito-Schroedinger stochastic unitary evolution for the pure state of unstable quantum system providing a new stochastic version of the quantum Zeno effect.

math-ph↗

arXiv2005

Measurements continuous in time and a posteriori states in quantum

Measurements continuous in time were consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after the measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.

quant-ph↗

arXiv2005

A Stochastic Hamiltonian Approach for Quantum Jumps, Spontaneous Localizations, and Continuous Trajectories

We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with "bubbles" which admit a continual counting observation. This model allows to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localisations of the momentums of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e., a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yelds Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.

quant-ph↗

arXiv2005

A Dynamical Theory of Quantum Measurement and Spontaneous Localization

We develop a rigorous treatment of discontinuous stochastic unitary evolution for a system of quantum particles that interacts singularly with quantum "bubbles" at random instants of time. This model of a "cloud chamber" allows to watch and follow with a quantum particle along the trajectory in the cloud chamber by sequential unsharp localization of spontaneous scatterings of the bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained as the result of quantum filtering theory, i.e., a theory describing the conditioning of the a priori quantum state by the measurement data. We show that in the case of indistinguishable particles the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yields Gaussian mixing fluctuations described by stochastic reduction equations of diffusive type.

math-ph↗

arXiv2005

On Quantum Ito Algebras and their decompositions

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom 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. Two basic examples of non-commutative Ito finite group algebras are considered.

math-ph↗

arXiv2005

Quantum Ito B*-algebras, their Classification and Decomposition

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.OA↗

arXiv2005

A nonadapted stochastic calculus and non stationary evolution in Fock scale

A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of Malliavin derivative on a projective Fock space, and their uniform continuity with respect to the inductive limite convergence is proved. A new form of QS calculus based on an inductive *-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and isometry and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the peseudounitary and *-homomorphism property of the QS integrable generators is proved.

math.PR↗

arXiv2005

On the General Form of Quantum Stochastic Evolution Equation

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

math.PR↗