Relevant and irrelevant nonlinear Schrodinger equations

First, we summarize the argument against deterministic nonlinear Schrodinger equations. We recall that any such equation activates quantum non-locality in the sense that that information could be signalled in a finite time over arbitrarily large distances. Next we introduce a deterministic nonlinear Schrodinger equation. We justify it by showing that it is closest, in a precise sense, to the master equations for mixed states used to describe the evolution of open quantum systems. We also illustrate some interesting properties of this equation. Finally, we show that this equation can avoid the signalling problem if one adds noise to it in a precise way. Cases of both discrete and continuous noise are introduced explicitly and related to the density operator evolution. The relevance for the classical limit of the obtained stochastic equations is illustrated on a classically chaotic kicked anharmonic oscillator.