Stochastic Schrödinger equations

Abstract

A derivation of stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence.

Optimality of Feedback Control Strategies for Qubit Purification

“…Here Ω t is the control input and W t is the innovations Wiener process [6,7,8]. It is related to the measurement record Y t on which ρ t is conditioned by…”

Quantum Risk-Sensitive Estimation and Robustness

“…1.11): For any observable and density matrices and , the following relation holds: (8) Proof: The proof is straightforward. Defining a density matrix , we obtain Then, as the quantum relative entropy is always nonnegative and takes zero only when , we observe that (8) holds and the maximum is attained only when .…”

“…Fortunately, there exists a quantum filtering theory as a beautiful parallel to the classical one. The theory was pioneered by Belavkin in the remarkable papers [4], [5], [6], and the quantum filtering equation or stochastic master equation is now widely used in the physics community [1], [8], [16], [21], [31], [38], [40], [47]. Moreover, as in the classical theory, it is possible to show that a separation principle holds in the quantum case [10].…”

Unifying decoherence and the Heisenberg Principle

“…If we use λ −2 as a unit of time, then the restricted evolution R t * (r ) := tr F ⊗FTt * (r ) of the two-level system is known (see [3]) to satisfy the Master equation…”