Is entropy production of a decohering chaotic system set by Lyapunov exponents?
In plain words
When a quantum system whose classical version is chaotic (extremely sensitive to initial conditions) is weakly disturbed by its surroundings, Zurek and Paz predicted that it loses information at a rate fixed by its chaos, independent of how strong the disturbance is. Whether and when this holds is not settled.
Precise statement
For a quantum system with a classically chaotic limit, positive Lyapunov exponents $\lambda_i$, and weak Markovian coupling to an environment with momentum-diffusion constant $D$, Zurek and Paz (PRL 1994) conjectured that after a transient the von Neumann entropy $S$ of the reduced state grows as $dS/dt = \text{sum of positive } \lambda_i$, independent of $D$ over a wide range. Determine the range of $D$, effective Planck constant $\hbar_{\mathrm{eff}}$ and time for which this holds, and its corrections, in standard models (kicked rotor, kicked top); a related coupling-independent Lyapunov decay of the Loschmidt echo was found by Jalabert and Pastawski (PRL 2001).
What would settle it
An analytic derivation or controlled numerics at decreasing $\hbar_{\mathrm{eff}}$ showing a $D$-independent plateau of $\mathrm{d}S/\mathrm{d}t$ equal to the Lyapunov sum, or its absence.