# How do you properly analyze decoherence when the system-environment interaction is strong?

The standard decoherence analyses work something like this. Split the universe into a system and its environment, and split the Hamiltonian as the sum of the system Hamiltonian, environment Hamiltonian and the system-environment interaction Hamiltonian. Assume an initial stosszahlansatz factorization into the product state of a pure environment state and a pure system state. Evolve over time, then take the partial trace over the environment to get the reduced density matrix for the system.

This works fairly well if the magnitude of the interaction Hamiltonian is small in comparison with the system Hamiltonian. Under decoherence, in the pointer state basis, the off-diagonal entries get suppressed, and the pointer states are obtained from maximizing predictability via minimizing the von Neumann entropy over time. But what if the interaction isn't weak? Then, assuming we are dealing with an example with decoherence and predictability, stable pointer states do exist but are not product states, but rather entangled pure states between the system and the environment.

An example might help to clarify matters here. With a spatially sharp boundary between the system and the environment, quantum fluctuations across the boundary over length scales smaller than the correlation length become significant. There are still pure stably predictable pointer states. It's just that they are entangled up to distances of the order of the correlation length to both sides of the boundary. A conventional decoherence analysis will mangle this up, though. The partial trace will give a mixed state with a von Neumann entropy proportional to the area of the boundary in Planck units, assuming a Planck scale cutoff. This is a fake entropy, though, and falsely suggests the actual predictability is much less than it actually is. Contrast this with a different scenario where the system actually thermalizes and the entangled correlations are dumped into multiple fragments scattered all over the environment spreading ever farther and farther away. No one will consider this highly predictable. The difference is in the former, the entanglement is confined to a layer the thickness of the correlation length from the boundary, while in the latter, the entanglement spreads farther and farther outward. How should we improve decoherence measures to adapt to these sorts of scenarios?

I suppose you could say all we have to do is choose a better system-environment cut, but is there a general prescription for this? Can this always be done? What about the example of a strongly interacting conformal field theory with infinite correlation length? Do we invoke S-duality or the AdS/CFT correspondence to come up with an alternative weakly coupled description? What if this isn't possible? Such an example need not be irreducibly holistic, and might still be partially analyzed by breaking down into subsystems, but how do we do that?