Editing Quantum decoherence (section)

===Loss of interference and the transition from quantum to classical probabilities===
The utility of decoherence lies in its application to the analysis of probabilities, before and after environmental interaction, and in particular to the vanishing of [[quantum interference]] terms after decoherence has occurred. If we ask what is the probability of observing the system making a [[Quantum state|transition]] from <math>\psi</math> to <math>\phi</math> ''before'' <math>\psi</math> has interacted with its environment, then application of the [[Born probability]] rule states that the transition probability is the [[squared modulus]] of the scalar product of the two states:

:<math>\operatorname{prob}_\text{before}(\psi \to \phi) = \left|\lang\psi|\phi\rang\right|^2 = \left|\sum_i \psi^*_i \phi_i\right|^2 = \sum_i |\psi_i^* \phi_i|^2 + \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i ,</math>

where <math>\psi_i = \lang i|\psi\rang</math>, <math>\psi_i^* = \lang\psi|i\rang</math>, and <math>\phi_i = \lang i|\phi\rang</math> etc.

The above expansion of the transition probability has terms that involve <math>i \ne j</math>; these can be thought of as representing ''interference'' between the different basis elements or quantum alternatives. This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives.

To calculate the probability of observing the system making a quantum leap from <math>\psi</math> to <math>\phi</math> ''after'' <math>\psi</math> has interacted with its environment, then application of the [[Born probability]] rule states that we must sum over all the relevant possible states <math>|\epsilon_i\rang</math> of the environment ''before'' squaring the modulus:

: <math>\operatorname{prob}_\text{after}(\psi \to \phi) = \sum_j \,\left|\lang\text{after}\right| \phi, \epsilon_j \rang|^2 = \sum_j \,\left|\sum_i \psi_i^* \lang i, \epsilon_i|\phi, \epsilon_j\rang\right|^2 = \sum_j\left|\sum_i \psi_i^* \phi_i \lang\epsilon_i|\epsilon_j\rang \right|^2.</math>

The internal summation vanishes when we apply the decoherence/einselection condition <math>\lang\epsilon_i|\epsilon_j\rang \approx \delta_{ij}</math>, and the formula simplifies to

: <math>\operatorname{prob}_\text{after}(\psi \to \phi) \approx \sum_j |\psi_j^* \phi_j|^2 =
\sum_i |\psi^*_i \phi_i|^2.</math>

If we compare this with the formula we derived before the environment introduced decoherence, we can see that the effect of decoherence has been to move the summation sign <math>\textstyle\sum_i</math> from inside of the modulus sign to outside. As a result, all the cross- or [[quantum interference]]-terms

: <math>\sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i</math>

have vanished from the transition-probability calculation. The decoherence has [[Reversible process (thermodynamics)|irreversibly]] converted quantum behaviour (additive [[probability amplitude]]s) to classical behaviour (additive probabilities).<ref name="zurek03"/><ref name="zurek91">[[Wojciech H. Zurek]], "Decoherence and the transition from quantum to classical", ''Physics Today'', 44, pp. 36–44 (1991).</ref><ref name=Zurek02>{{cite journal
 | last = Zurek
 | first = Wojciech
 | title = Decoherence and the Transition from Quantum to Classical—Revisited
 | journal = Los Alamos Science
 | volume = 27
 | year = 2002
 | url = https://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf
| bibcode = 2003quant.ph..6072Z
 | arxiv = quant-ph/0306072
 }}</ref> 
However, Ballentine<ref>{{Cite journal |last=Ballentine |first=Leslie |date=October 2008 |title=Classicality without Decoherence: A Reply to Schlosshauer |journal=Foundations of Physics |language=en |volume=38 |issue=10 |pages=916–922 |doi=10.1007/s10701-008-9242-0 |issn=0015-9018|doi-access=free |bibcode=2008FoPh...38..916B }}</ref> shows that the significant impact of decoherence to reduce interference need not have significance for the transition of quantum systems to classical limits.

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.<ref name="zurek03"/>