Schrödinger’s Stochastic Variational Dynamics

Jean-Claude Zambrini

J.C. Zambrini

Tetsuya Misawa

A quantal-dissipative system described by the Schr\"odinger equation with an additional nonlinear term is formulated in the Lagrangian stochastic mechanics on the basis of the symmetric stochastic calculus of variations (SSCV). A certain class of stochastic Lagrangian dynamical systems defined through the least-action principle in SSCV is associated with solutions of the nonlinear Schr\"odinger equations, and thereby the quantal-dissipative system described by the equation is connected with a stochastic Lagrangian dynamical system. The long-time behavior of momentum, angular momentum, and energy, and the virial theorem for the stochastic Lagrangian system corresponding to the quantal-dissipative system, are investigated through the conserved quantities and the first variations of the functional induced by one-parameter transformations. Examples are given for a free particle and for the harmonic oscillator.

Fabrizio Illuminati, Lorenza Viola

We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schrodinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials.

Hans-Jürgen Mann

Einar Gabbassov

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.