Editing Quantum decoherence (section)

=== Density-matrix approach ===
The effect of decoherence on [[density matrix|density matrices]] is essentially the decay or rapid vanishing of the [[off-diagonal element]]s of the [[partial trace]] of the joint system's density matrix, i.e. the [[trace (linear algebra)|trace]], with respect to ''any'' environmental basis, of the density matrix of the combined system ''and'' its environment. The decoherence irreversibly converts the "averaged" or "environmentally traced-over"<ref name="zurek03"/> density matrix from a pure state to a reduced mixture; it is this that gives the ''appearance'' of [[wave-function collapse]]. Again, this is called "environmentally induced superselection", or einselection.<ref name="zurek03"/> The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

Initially, the density matrix of the combined system can be denoted as

: <math>\rho = |\text{before}\rang \lang\text{before}| = |\psi\rang \lang\psi| \otimes |\epsilon\rang \lang\epsilon|,</math>

where <math>|\epsilon\rang</math> is the state of the environment.
Then if the transition happens before any interaction takes place between the system and the environment, the environment subsystem has no part and can be [[Quantum entanglement#Reduced density matrices|traced out]], leaving the reduced density matrix for the system:

: <math>\rho_\text{sys} = \operatorname{Tr}_\textrm{env}(\rho) = |\psi\rang \lang\psi| \lang\epsilon|\epsilon\rang = |\psi\rang \lang\psi|.</math>

Now the transition probability will be given as

: <math>\operatorname{prob}_\text{before}(\psi \to \phi) = \lang\phi| \rho_\text{sys} |\phi\rang = \lang\phi|\psi\rang \lang\psi|\phi\rang = \big|\lang\psi|\phi\rang\big|^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.

Now the case when transition takes place after the interaction of the system with the environment. The combined density matrix will be

: <math> \rho = |\text{after}\rang \lang\text{after}| = \sum_{i,j} \psi_i \psi_j^* |i, \epsilon_i\rang \lang j, \epsilon_j| = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \otimes |\epsilon_i\rang \lang\epsilon_j|.</math>

To get the reduced density matrix of the system, we trace out the environment and employ the decoherence/einselection condition and see that the off-diagonal terms vanish (a result obtained by Erich Joos and H. D. Zeh in 1985):<ref name="JZ">E. Joos and H. D. Zeh, "The emergence of classical properties through interaction with the environment", ''Zeitschrift für Physik B'', '''59'''(2), pp. 223–243 (June 1985): eq. 1.2.</ref>

: <math>\rho_\text{sys} = \operatorname{Tr}_\text{env}\Big(\sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \otimes |\epsilon_i\rang \lang\epsilon_j|\Big) = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \lang\epsilon_j|\epsilon_i\rang  = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \delta_{ij} = \sum_i |\psi_i|^2 |i\rang \lang i|.</math>

Similarly, the final reduced density matrix after the transition will be

: <math>\sum_j |\phi_j|^2 |j\rang \lang j|.</math>

The transition probability will then be given as

: <math>\operatorname{prob}_\text{after}(\psi \to \phi) = \sum_{i,j} |\psi_i|^2 |\phi_j|^2 \lang j|i\rang \lang i|j\rang = \sum_i |\psi_i^* \phi_i|^2,</math>

which has no contribution from the interference terms

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

The density-matrix approach has been combined with the [[De Broglie–Bohm theory|Bohmian approach]] to yield a ''reduced-trajectory approach'', taking into account the system [[reduced density matrix]] and the influence of the environment.<ref>{{Cite journal |last1=Sanz |first1=A. S. |last2=Borondo |first2=F. |date=2007 |title=A quantum trajectory description of decoherence |journal=The European Physical Journal D |volume=44 |issue=2 |pages=319–326 |doi=10.1140/epjd/e2007-00191-8 |arxiv=quant-ph/0310096 |bibcode=2007EPJD...44..319S |s2cid=18449109 |issn=1434-6060}}</ref>