UK Project Leader: Professor V. P. Belavkin

During the second year in the frame of the project we studied:

1. Stochastic analysis and representation theory of the exponentials of quantum white noise. In particular, Levy-Khinchin type decomposition theorem for infinite-dimensional quantum processes with independent increments.

2. The quasi-classical stochastic inductive limit of the self-adjoint boundary value problem in Hilbert space corresponding to classical and quantum stochastic differential equations (SDE).

3. The analysis of causality and decoherence for quantum measurement as the dynamical Dirac type boundary value problem with superselection rule for the observables.

4. Extension and deformation of the algebraic theory of quantum stochastic product integrals to double and higher order multiple product integrals and universal solution of the quantum Yang-Baxter equation constructed using these integrals.

5. Mathematical description and classification of entanglement and mutual information in the quantum compound states on semifinite and general von Neumann algebras.

Carrying out this program we achieved the following results:

1. During the visit to Japan in Spring 2000 supported by this grant, Belavkin was invited to a participate in the German-Japan conference on Harmonic Analysis and related topics. While preparing his talk he discovered a characterization of the quantum stochastic white noise exponents which leads to the deep results on the Levy-Khinchin type decompositions for the quantum white noise. The preliminary publication on these important results has already appeared in the proceedings of this conference [1], and it will be followed up by other publications.

2. The discussions with Professor Obata of quantum white noise analysis during his visit to Nottingham in May 2000 heavily stimulated Belavkin's work on the equivalence of white noise quantum Schroedinger dynamics and the secondary quantized Dirac boundary value problems in one extra-dimension [2,3]. Thus the classical continuous stochastics corresponding to quantum state diffusions has been also derived from a quantum continuous deterministic unitary evolution in Fock Hilbert modules by the second quantization of the Dirac boundary value problem.

3. The stochastic limit reconsidered as an ultra relativistic approximation of positive free relativistic evolution Hamiltonians for the Schroedinger equations has been extended to the arbitrary free evolutions by a WKB methods. It has been proved in collaboration with Kolokol'tsov [4] that the rapidly oscillating solutions to the Schroedinger input and output equations, which are connected by the boundary condition in the half-space, strongly converge to the corresponding solutions of the Dirac one-dimensional boundary value problem in the inductive Hardy class pre-Hilbert module in second quantization.

4. A theory is developed of product integrals over disjoint finite subintervals of real line for formal power series whose coefficients are elements of tensor powers of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series whose coefficients are finite sums of iterated stochastic integrals. It is conjectured that for suitable choice of the formal power series the product integral is a universal solution of the quantum Yang-Baxter equation whose linear term satisfies the classical Yang-Baxter equation. These results were finalised by Hudson during his visit to Japan and will be published in [5].

5. Belavkin in collaboration with Professor Ohya extended the operational description of quantum entanglements in terms of completely positive maps to semifinite and general von Neumann algebras {6-10]. As in the case of the simple algebras the mutual information for the c-compound (classical) and for q-compound (entangled) states leads to two different types of entropies for a given quantum state. The first one is the von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the second one is the dimensional entropy. It is achieved at the standard entanglement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-capacity of a quantum noiseless channel, defined as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semi-quantum couplings (d-entanglements, or encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra.

The results obtained and methods developed during this discussions and collaborations opened a new interesting area in the intersection of stochastic analysis, mathematical physics and quantum information theory. This area has various applications: in classical and quantum probability, in the theory of open systems, in quantum measurement and filtering theory, in detection of gravitation waves and communication. The results of this programme need to be considered as a first step in the realization of a long term project related to the these fields.

The scientific results of the project are reported in six conferences held in Japan and elsewhere and have been published as:

1. V P Belavkin: On Stochastic Generators of Positive Definite Exponents. In: Infinite Dimensional Harmonic Analysis, Transactions of a Japan-German Symposium, Kyoto, September 1999.
2. V P Belavkin: On the Equivalence of Quantum Stochastics and a Dirac Boundary Value Problem, and Inductive Stochastic Limit. September 1999, RIMS conference on New Development of Infinite-Dimensional Analysis and Quantum Probability. Kyoto University, Kyoto, 2000, 54-73.
3. V P Belavkin: On Stochastic Schroedinger Equation as Dirac Boundary-Value Problem. In: evolution equations and their applications in physical and life sciences, Ed G Lumer and L. Weis. Lecture Notes in Pure and Applied Mathematics (2000) 311-327.
4. V P Belavkin and V N Kolokol'tsov: Stochastic Evolutions as Boundary Value Problems. November 2000, RIMS conference on New Development of Infinite-Dimensional Analysis and Quantum Probability. Kyoto University, Kyoto, 2001, 83-95.
5. R L Hudson: Multiple Quantum Stochastic Product Integrals. February 2000, to be published elsewhere.
6. V P Belavkin: Quantum Entanglements, epsilon-Entropy, and Value of Quantum Information. Program of the 2000 IEICE General Conference, March 2000, Hiroshima University, Higashi-Hiroshima, p. 38.
7. V P Belavkin and M Ohya: Quantum Entropy and Information in Discrete Entangled States. Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 4 No. 2 (2001) 137-160.
8. V P Belavkin: On Entangled Information and Quantum Capacity. Open Sys & Information Dyn. Vol. 8 (2001) 1-18.
9. V P Belavkin: Quantum Sex and Mutual Information. November 2000, RIMS conference on New Development of Infinite-Dimensional Analysis and Quantum Probability. Kyoto University, Kyoto, 2001, 61-82.
10. V P Belavkin and M Ohya: Entanglement, Quantum Entropy and Mutual Information. Proc. R. Soc. Lond. A, Vol. 458 (2002) 209-231.

Acknowledgement: The UK project leader, other members of the team and Japanese vitors acknowledge the Royal Society funding of the UK-Japan joint research project.