Over at PhysicsOverflow, Daniel Ranard asked a question that’s near and dear to my heart:

How deterministic are large open quantum systems (e.g. with humans)?

Consider some large system modeled as an open quantum system — say, a person in a room, where the walls of the room interact in a boring way with some environment. Begin with a pure initial state describing some comprehensible configuration. (Maybe the person is sitting down.) Generically, the system will be in a highly mixed state after some time. Both normal human experience and the study of decoherence suggest that this state will be a mixture of orthogonal pure states that describe classical-like configurations. Call these configurations branches.

How much does a pure state of the system branch over human time scales? There will soon be many (many) orthogonal branches with distinct microscopic details. But to what extent will probabilities be spread over macroscopically (and noticeably) different branches?

I answered the question over there as best I could. Below, I’ll reproduce my answer and indulge in slightly more detail and speculation.

This question is central to my research interests, in the sense that completing that research would necessarily let me give a precise, unambiguous answer. So I can only give an imprecise, hand-wavy one. I’ll write down the punchline, then work backwards.

##### Punchline

**The instantaneous rate of branching, as measured in entropy/time (e.g., bits/s), is given by the sum of all positive Lyapunov exponents for all non-thermalized degrees of freedom.**

Most of the vagueness in this claim comes from defining/identifying degree of freedom that have thermalized, and dealing with cases of partial/incomplete thermalization; these problems exists classically.

##### Elaboration

The original question postulates that the macroscopic system starts in a quantum state corresponding to some comprehensible classical configuration, i.e., the system is initially in a quantum state whose Wigner function is localized around some classical point in phase space.… [continue reading]