Quantum Decoherence - Mathematical Details

Mathematical Details

We assume for the moment the system in question consists of a subsystem being studied, A and the "environment", and the total Hilbert space is the tensor product of a Hilbert space describing A, H_A and a Hilbert space describing E, : that is,

.

This is a reasonably good approximation in the case where A and are relatively independent (e.g. there is nothing like parts of A mixing with parts of or vice versa). The point is, the interaction with the environment is for all practical purposes unavoidable (e.g. even a single excited atom in a vacuum would emit a photon which would then go off). Let's say this interaction is described by a unitary transformation U acting upon H. Assume the initial state of the environment is and the initial state of A is the superposition state

where and are orthogonal and there is no entanglement initially. Also, choose an orthonormal basis for H_A, . (This could be a "continuously indexed basis" or a mixture of continuous and discrete indexes, in which case we would have to use a rigged Hilbert space and be more careful about what we mean by orthonormal but that's an inessential detail for expository purposes.) Then, we can expand

and

uniquely as

and

respectively. One thing to realize is that the environment contains a huge number of degrees of freedom, a good number of them interacting with each other all the time. This makes the following assumption reasonable in a handwaving way, which can be shown to be true in some simple toy models. Assume that there exists a basis for such that and are all approximately orthogonal to a good degree if i is not j and the same thing for and and also and for any i and j (the decoherence property).

This often turns out to be true (as a reasonable conjecture) in the position basis because how A interacts with the environment would often depend critically upon the position of the objects in A. Then, if we take the partial trace over the environment, we'd find the density state is approximately described by

(i.e. we have a diagonal mixed state and there is no constructive or destructive interference and the "probabilities" add up classically). The time it takes for U(t) (the unitary operator as a function of time) to display the decoherence property is called the decoherence time.