Stochastic optimal control theory for quantum systems

In 1966, E. Nelson established a new interpretation of quantum mechanics, whereby the particles follow some conservative diffusion process, i.e. forward-backward stochastic differential equations (FBSDEs), which are equivalent to the Schrödinger equation (1). Until now, this equivalence has been applied in such a way that a known solution to the Schrödinger equation is used to integrate the stochastic differential equations numerically and analyze the statistical properties of the sample paths. Compared to the options available to treat classical systems this is limited, both in methods and in scope.

However, in analogy to classical mechanics, we show that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two aspects (2): i) the Schrödinger equation as the Hamilton-Jacobi-Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the $n$-dimensional, non stationary and the stationary case.

The resulting forward-backward stochastic differential equations can be solved numerically without using the solution of the Schrödinger equation, which is done for many different systems, e.g. one- and two-dimensional harmonic oscillator, one-dimensional double-well potential or hydrogen atom.

1. E. Nelson (1966). Derivation to the Schrödinger Equation from Newtonian Mechanics. Phys. Rev. 150(4), 1079--1085
2. J. Köppe, W. Grecksch, W. Paul (2017). Derivation and application of quantum Hamilton equations of motion. Ann. Phys. 529(3), 1600251