UK Project Leader: Professor V. P. Belavkin

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

1. Mathematical description and classification of entanglement in quantum theory and quantum compound states from the point of view of information theory.

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

3. The derivation and analysis of the dynamical boundary value models in this framework for quantum measurement, and the decoherence problem.

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

Carrying out this program we achieved the following results:

1. Belavkin in collaboration with Professor Ohya [1] has found an operational description of quantum entanglements in terms of completely positive maps. The mutual information for the c-compound (classical) and for q-compound (entangled) states lead 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.

2. As a result of consultations with Professors Arimitsu and Obata, Belavkin proved that the simplest discontinuous single-jump classical SDE satisfying the unitarity conditions in a Hilbert space is equivalent to a Dirac boundary value problem on the half-line in one extra-dimension [2]. Thus the classical discontinuous stochastics generated by such jumps can be 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 is reviewed in an inductive convergence as the ultra relativistic approximation of positive free evolution Hamiltonians for the relativistic Schroedinger equations. It has been proved in [2] 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.

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 [3].

5. The representation theory for quantum Ito algebras developed by Belavkin gives a possibility to obtain a Levy-Khinchin type decomposition, for quantum infinitely-divisible states, of processes with independent increments. This result was presented during the workshop in Kyoto and will be published elsewhere. It has received an interesting development into the field of Markov chains during the recent visit of Professor Matsui from Kyushu University to Nottingham.

The results obtained and methods developed during the first year open a new interesting area belonging to mathematical physics, stochastic analysis and quantum information theory. In the next year one can expect further results to be established by the collaboration. It is worth stressing that 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 realisation of a long term project related to the these fields.

The results of the project are included in three new papers submitted for publication:

1. V P Belavkin and M Ohya: Entanglements and Compound States in Quantum Information Theory. July 1999, Submitted to Reviews in Mathematical Physics.
2. V P Belavkin: On the Equivalence of Quantum Stochastics and a Dirac Boundary Value Problem, and an Inductive Stochastic Limit. September 1999, to be published in the proceedings of RIMS conference.
3. R L Hudson: Multiple Quantum Stochastic Product Integrals. February 2000, to be published elsewhere.

Acknowledgement: The UK project leader and other members of the team acknowledge the Royal Society funding for the first year of the UK-Japan joint research project.