The Quantum Stochastic Differential Equation Is Unitarily Equivalent to a Symmetric Boundary Value Problem for the Schr\"odinger Equation

TL;DR

This paper demonstrates that solutions to quantum stochastic differential equations can be equivalently represented as boundary value problems for the Schrödinger equation, linking stochastic quantum dynamics with deterministic boundary conditions.

Contribution

It establishes a unitary equivalence between quantum stochastic differential equations and symmetric boundary value problems for the Schrödinger equation, providing a new analytical perspective.

Findings

Equivalence between quantum stochastic differential equations and Schrödinger boundary value problems.

Derivation of the Lindblad equation from the boundary value problem.

Description of phase and amplitude jumps in Fock space components.

Abstract

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schr\"odinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schr\"odinger equation.