Editing Quantum decoherence (section)

===Semigroup approach===
A more general consideration for the existence of decoherence in a quantum system is given by the ''master equation'', which determines how the density matrix of the ''system alone'' evolves in time (see also the [[Belavkin equation]]<ref name=Belavkin89>{{cite journal
 | author = V. P. Belavkin
 | title = A new wave equation for a continuous non-demolition measurement
 | journal = Physics Letters A
 | volume = 140
 | number = 7–8
 | pages = 355–358
 | year = 1989
 | doi = 10.1016/0375-9601(89)90066-2
 | arxiv = quant-ph/0512136|bibcode = 1989PhLA..140..355B | s2cid = 6083856
 }}</ref><ref name=Carmichael93>{{cite book
 | author = Howard J. Carmichael
 | title = An Open Systems Approach to Quantum Optics
 | publisher = Springer-Verlag
 | year = 1993
 | location = Berlin Heidelberg New-York}}</ref><ref name=Bauer2012>{{cite tech report |author1=Michel Bauer |author2=Denis Bernard |author3=Tristan Benoist | title = Iterated Stochastic Measurements | arxiv = 1210.0425|bibcode=2012JPhA...45W4020B|doi=10.1088/1751-8113/45/49/494020}}</ref> for the evolution under continuous measurement). This uses the [[Quantum states#Schrödinger picture vs. Heisenberg picture|Schrödinger]] picture, where evolution of the ''state'' (represented by its density matrix) is considered. The master equation is

: <math>\rho'_S(t) = \frac{-i}{\hbar} \big[\tilde H_S, \rho_S(t)\big] + L_D \big[\rho_S(t)\big],</math>

where <math>\tilde H_S = H_S + \Delta</math> is the system Hamiltonian <math>H_S</math> along with a (possible) unitary contribution <math>\Delta</math> from the bath, and <math>L_D</math> is the ''Lindblad decohering term''.<ref name="Lidar and Whaley"/> The [[Lindblad equation|Lindblad decohering term]] is represented as
: <math>L_D\big[\rho_S(t)\big] = \frac{1}{2} \sum_{\alpha, \beta = 1}^M b_{\alpha\beta} \Big(\big[\mathbf F_\alpha, \rho_S(t)\mathbf F^\dagger_\beta\big] + \big[\mathbf F_\alpha \rho_S(t), \mathbf F^\dagger_\beta\big]\Big).</math>

The <math>\{\mathbf{F}_\alpha\}_{\alpha=1}^M</math> are basis operators for the ''M''-dimensional space of [[bounded operator]]s that act on the system Hilbert space <math>\mathcal H_S</math> and are the ''error generators''.<ref name="Lidar, Chuang, and Whaley">* {{cite journal |arxiv=quant-ph/9807004 |doi=10.1103/PhysRevLett.81.2594 |bibcode=1998PhRvL..81.2594L |title=Decoherence-Free Subspaces for Quantum Computation |year=1998 |last1=Lidar |first1=D. A. |last2=Chuang |first2=I. L. |last3=Whaley |first3=K. B. |journal=Physical Review Letters |volume=81 |issue=12 |pages=2594–2597 |s2cid=13979882}}</ref> The matrix elements <math>b_{\alpha\beta}</math> represent the elements of a [[Positive semidefinite matrix|positive semi-definite]] [[Hermitian matrix]]; they characterize the decohering processes and, as such, are called the ''noise parameters''.<ref name="Lidar, Chuang, and Whaley"/> The semigroup approach is particularly nice, because it distinguishes between the unitary and decohering (non-unitary) processes, which is not the case with the OSR. In particular, the non-unitary dynamics are represented by <math>L_D</math>, whereas the unitary dynamics of the state are represented by the usual [[Heisenberg commutator]]. Note that when <math>L_D\big[\rho_S(t)\big] = 0</math>, the dynamical evolution of the system is unitary. The conditions for the evolution of the system density matrix to be described by the master equation are:<ref name="Lidar and Whaley"/>
# the evolution of the system density matrix is determined by a one-parameter [[semigroup]]
# the evolution is "completely positive" (i.e. probabilities are preserved)
# the system and bath density matrices are ''initially'' decoupled