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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

Quantum probabilities in open dissipative systems: a path integral derivation

T. Sami, J. Richert

Published May 1, 1986• 3 citations
Publisher
DOI
10.1007/bf02819988

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Abstract In Bohmian mechanics it is assumed that quantum objects are particles that follow real physical paths. These Bohmian trajectories are obtained by integrating the velocity field that occurs in the continuity equation. However, in the course of deriving these trajectories, the independent position variable x, appearing in the velocity field, is arbtirarily replaced by the time-dependent trajectory q(t). We show under which restrictions such a replacement can be justified, leading to a totally different interpretation of the Bohmian trajectories. They are not real paths of physical particles but borders of regions of constant probability, so-called quantiles, and therefore must be seen in the context of descriptive statistics. Thus they provide a tool complementary to the conventional statistical formulation of quantum mechanics. A possible extension to include open dissipative systems is also discussed, showing that the inclusion of an environment does not change our interpretation of the Bohmian trajectories.
We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
We investigate open-loop quantum state preparation for a class of open quantum systems whose dynamics follow a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation that admits a trajectory-based stochastic representation. The deterministic control objective is reformulated as a stochastic optimal control problem -- interpreting stochasticity as a methodological tool akin to stochastic Schrödinger equation unravelings -- which situates the problem within the path integral control framework. For the class of GKLS generators under consideration, this reformulation leads to an explicit expression for the optimal control as a weighted average over stochastic quantum trajectories, thereby eliminating the need for gradient evaluations. Building on this theoretical result, we derive a control update rule for piecewise-constant control pulses and demonstrate that adaptive importance sampling progressively enhances the control estimator during optimization, culminating in the algorithm we term Path integral Quantum Control (PiQC). We further introduce an annealed variant of PiQC, wherein a synthetic noise schedule gradually steers open-system trajectories toward closed-system dynamics, enabling high-fidelity unitary state preparation. Numerical studies on a dissipative single-qubit system and a multi-qubit Nuclear Magnetic Resonance model verify that PiQC yields precise open-loop controls and displays robustness to Hamiltonian perturbations. We propose PiQC as a trajectory-based alternative to gradient-based approaches, which might offer a viable solution in quantum control problems where gradient computation is infeasible or computationally demanding.