K. R. Parthasarathy

Quantum Ito's formula and stochastic evolutions

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

Unification of Fermion and Boson Stochastic Calculus

Fermion annihilation and creation processes are explicitly realised in Boson Fock space as functions of the corresponding Boson processes and second quantisations of reflections. Conversely, Boson annihilation and creation processes can be constructed from the Fermion processes. The existence of unitary stochastic evolutions driven by Fermion and gauge noise is thereby reduced to an equivalent Boson problem, which is then solved.

On the maximal dimension of a completely entangled subspace for finite level quantum systems

AbstractLetHibe a finite dimensional complex Hilbert space of dimensiondi associated with a finite level quantum system Ai for i = 1, 2, ...,k. A subspaceS ⊂

Cohomology of power sets with applications in quantum probability

{\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k

On Estimating the State of a Finite Level Quantum System

is said to becompletely entangled if it has no non-zero product vector of the formu1⊗u2 ⊗ ... ⊗uk with ui inHi for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

Coding theorems of classical and quantum information theory

\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1