Editing Quantum decoherence (section)

==Mathematical details==
Assume for the moment that the system in question consists of a subsystem ''A'' being studied and the "environment" <math>\epsilon</math>, and the total [[Hilbert space]] is the [[tensor product]] of a Hilbert space <math>\mathcal H_A</math> describing ''A'' and a Hilbert space <math>\mathcal H_\epsilon</math> describing <math>\epsilon</math>, that is,

:<math>\mathcal H = \mathcal H_A \otimes \mathcal H_\epsilon.</math>

This is a reasonably good approximation in the case where ''A'' and <math>\epsilon</math> are relatively independent (e.g. there is nothing like parts of ''A'' mixing with parts of <math>\epsilon</math> or conversely). 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 <math>\mathcal H</math>. Assume that the initial state of the environment is <math>|\text{in}\rangle</math>, and the initial state of ''A'' is the superposition state

: <math>c_1 |\psi_1\rangle + c_2 |\psi_2\rangle,</math>

where <math>|\psi_1\rangle</math> and <math>|\psi_2\rangle</math> are orthogonal, and there is no entanglement initially. Also, choose an orthonormal basis <math>\{ |e_i\rangle \}_i</math> for <math>\mathcal H_A</math>. (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

: <math>U\big(|\psi_1\rangle \otimes |\text{in}\rangle\big)</math>

and

: <math>U\big(|\psi_2\rangle \otimes |\text{in}\rangle\big)</math>

uniquely as

: <math>\sum_i |e_i\rangle \otimes |f_{1i}\rangle</math>

and

: <math>\sum_i |e_i\rangle \otimes |f_{2i}\rangle</math>

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 <math>\mathcal H_\epsilon</math> such that <math>|f_{1i}\rangle</math> and <math>|f_{1j}\rangle</math> are all approximately orthogonal to a good degree if ''i'' ≠ ''j'' and the same thing for <math>|f_{2i}\rangle</math> and <math>|f_{2j}\rangle</math> and also for <math>|f_{1i}\rangle</math> and <math>|f_{2j}\rangle</math> 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 would find the density state{{clarify|date=October 2018}} is approximately described by

:<math>\sum_i \big(\langle f_{1i}|f_{1i}\rangle + \langle f_{2i}|f_{2i}\rangle\big) |e_i\rangle \langle e_i|,</math>

that is, we have a diagonal [[Mixed state (physics)|mixed state]], 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'''.