Dirac brackets and the Lindblad equation: A correspondence

The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindblad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the correlations between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, among other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints that arise either because they are introduced by hand or because of singular Lagrangians. We derive an intriguing but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts such as quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.