A stochastic calculus of quantum input-output processes and quantum nondemolition filtering

A description is presented of the indefinite structure of quantum stochastic (QS) calculus in Fock space, as developed by Hudson and Parthasarathy, with the quantum stochastic integral defined as a continuous operator on the projective limit of Fock spaces. Differential conditions are found for QS calculus of input-output QS processes and nondemolition measurements, and it is proved that the nondemolition condition is necessary and sufficient for the existence of conditional expectations relative to the subalgebra of observables and any state vector. A stochastic calculus of posterior (conditional) expectations of quantum nondemolition processes is developed, and a general stochastic equation is derived for quantum nonlinear filtering, both in the Heisenberg picture (for posterior operators) and in the Schrödinger picture (for the posterior density matrix and wavefunction). It is shown that posterior dynamics, unlike prior dynamics, does not mix states if the nondemolition measurement is complete.