arXiv subjects

V P Belavkin

Publications and source records attributed to V P Belavkin.

4 recordsLinked to original sources

arXiv2007

Representations of homogeneous quantum Lévy fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.

math.PR↗

arXiv2005

Contravariant Densities, Complete Distances and Relative Fidelities for Quantum Channels

Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace-densities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for the generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states.

math-ph↗

arXiv2005

Bellman equations for optimal feedback control of qubit states

Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.

quant-ph↗

arXiv2004

Towards the theory of control in observable quantum systems

An operational description of the controlled Markov dynamics of quantum-mechanical system is introduced. The feedback control strategies with regard to the dynamical reduction of quantum states in the course of quantum real-time measurements are discribed in terms of quantum filtering of these states. The concept of sufficient coordinates for the description of the a posteriori quantum states from a given class is introduced, and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schroedinger equation.

quant-ph↗