Generalized model-independent approach to intrinsic decoherence
A formalism is presented to express decoherence both in the markovian and nonmarkovian regimes and both dissipative and nondissipative in isolated systems. The main physical hypothesis, already contained in the literature \cite{Mil91a,BonOliTomVit00a}, amounts to allowing some internal parameters of the system to evolve in a random fashion. This formalism may also be applicable to open quantum systems.
quant-ph0204141.tex
%% ****** Start of file template.aps ****** %
%%
%%
%%   This file is part of the APS files in the REVTeX 4 distribution.
%%   Version 4.0 of REVTeX, August 2001
%%
%%
%%   Copyright (c) 2001 The American Physical Society.
%%
%%   See the REVTeX 4 README file for restrictions and more information.
%%
%
% This is a template for producing manuscripts for use with REVTEX 4.0
% Copy this file to another name and then work on that file.
% That way, you always have this original template file to use.
%
% Group addresses by affiliation; use superscriptaddress for long
% author lists, or if there are many overlapping affiliations.
% For Phys. Rev. appearance, change preprint to twocolumn.
% Choose pra, prb, prc, prd, pre, prl, prstab, or rmp for journal
%  Add 'draft' option to mark overfull boxes with black boxes
%  Add 'showpacs' option to make PACS codes appear
%  Add 'showkeys' option to make keywords appear
%\documentclass[aps,prl,preprint,groupedaddress]{revtex4}
%\documentclass[aps,prl,preprint,superscriptaddress]{revtex4}
\documentclass[aps,pra,twocolumn,groupedaddress,showpacs,showkeys]{revtex4}

% You should use BibTeX and apsrev.bst for references
% Choosing a journal automatically selects the correct APS
% BibTeX style file (bst file), so only uncomment the line
% below if necessary.
\bibliographystyle{/usr/share/texmf.local/bibtex/bst/aps/apsrev}

\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}

\allowdisplaybreaks

\begin{document}

% Use the \preprint command to place your local institutional report
% number in the upper righthand corner of the title page in preprint mode.
% Multiple \preprint commands are allowed.
% Use the 'preprintnumbers' class option to override journal defaults
% to display numbers if necessary
%\preprint{}

%Title of paper
\title{Generalized model-independent approach to intrinsic decoherence}

% repeat the \author .. \affiliation  etc. as needed
% \email, \thanks, \homepage, \altaffiliation all apply to the current
% author. Explanatory text should go in the []'s, actual e-mail
% address or url should go in the {}'s for \email and \homepage.
% Please use the appropriate macro foreach each type of information

% \affiliation command applies to all authors since the last
% \affiliation command. The \affiliation command should follow the
% other information
% \affiliation can be followed by \email, \homepage, \thanks as well.
\author{D. Salgado}
\email[]{david.salgado@uam.es}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
\affiliation{Dpto. F\'{\i}sica Te\'{o}rica, Universidad Aut\'{o}noma de Madrid\\
28049 Cantoblanco, Madrid (Spain)}

\author{J.L. S\'{a}nchez-G\'{o}mez}
\email[]{jl.sanchezgomez@uam.es}
%\homepage[]{Your web page}
%\thanks{}
\altaffiliation{Permanent Address}
\affiliation{Dpto. F\'{\i}sica Te\'{o}rica, Universidad Aut\'{o}noma de Madrid\\
28049 Cantoblanco, Madrid (Spain)}


%Collaboration name if desired (requires use of superscriptaddress
%option in \documentclass). \noaffiliation is required (may also be
%used with the \author command).
%\collaboration can be followed by \email, \homepage, \thanks as well.
%\collaboration{}
%\noaffiliation

\date{\today}

\begin{abstract}
A formalism is presented to express decoherence both in the markovian and nonmarkovian regimes and both dissipative and nondissipative in isolated systems. The main physical hypothesis, already contained in the literature \cite{Mil91a,BonOliTomVit00a}, amounts to allowing some internal parameters of the system to evolve in a random fashion. This formalism may also be applicable to open quantum systems. 
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{03.65.Yz,02.50.-r}
% insert suggested keywords - APS authors don't need to do this
\keywords{Decoherence, Markovianity, Dissipation, Randomness}

%\maketitle must follow title, authors, abstract, \pacs, and \keywords
\maketitle

% body of paper here - Use proper section commands
% References should be done using the \cite, \ref, and \label commands
%\section{}
% Put \label in argument of \section for cross-referencing
%\section{\label{}}
%\subsection{}
%\subsubsection{}

% If in two-column mode, this environment will change to single-column
% format so that long equations can be displayed. Use
% sparingly.
%\begin{widetext}
% put long equation here
%\end{widetext}

Decoherence is at the heart of the destruction of quantum superpositions in most physical systems. Its role is basic in questions such as the quantum-classical transition \cite{Zur91a}, the realization of quantum computation or the quantum processing of  information \cite{ChuNie00a}. From a quantitative point of view the effect of decoherence is commonly described by the action of the environment upon the physical system of interest \cite{GiuJooKieKupStaZeh96a,Zur82a}. This view inevitably depends on the chosen modelization of this environment. Recently \cite{BonOliTomVit00a} (see also \cite{Bon99a}) a model-independent approach to decoherence has been initiated based on the hypothesis of randomness of some of the internal parameters of the system, thus reducing the role of the environment. Here we present a triple generalization of this model-independent approach which  allows us to find more general master equations both in the markovian and in the nonmarkovian regimes\footnote{As usual, markovianity is understood in the sense of an evolution which satisfies a semigroup condition.} and both dissipative and nondissipative.


Randomness in an isolated system is introduced by claiming that the evolution time is a stochastic parameter \cite{BonOliTomVit00a,Bon99a} (previous references in this respect are \cite{Mil91a,MoyBuzKimKni93a}). This physical hypothesis is implemented by noticing that the experimental observations are described by the time-averaged density operator, as expressed by the relation 

\begin{equation}
\label{BonDensOp}
\bar{\rho}(t)=\int_{0}^{\infty}dt'P(t,t')\rho(t')
\end{equation}

\noindent where $P(t,t')$ denotes the probability density associated with the random time $t'$ and $\rho(t')$ denotes the usual quantum density operator. To find an expression for $P(t,t')$ two conditions are imposed \cite{BonOliTomVit00a}: i) $\bar{\rho}(t)$ must be a density operator itself; ii) the evolution of $\bar{\rho}(t)$ must be markovian. Under these assumptions and using the integral definition of the Gamma function, one arrives at a fixed probability density $P(t,t')$ (up to two parameters) and thus at a fixed time-averaged density operator $\bar{\rho}(t)$. Here we drop condition ii), thus comprising nonmarkovian evolutions and moreover using different tools we also allow for distinct markovian situations. The nonmarkovian regime is interesting in its own since it is the rigorous type of dynamics (prior to approximations) followed by a physical system coupled to an environment \cite{AlickiLendi87a,GorFriVerKosSud78a}. %We will show how a quantum system can preserve its initial purity when evolving in a nonmarkovian fashion.
Furthermore the experimental advances in mesoscopic scales (for instance, in creating atom lasers with Bose-Einstein condensates \cite{MoyHopSav98a}) also demands a better understanding of this regime. The generality obtained also allows us to embrace dissipative decoherence in contrast to \cite{BonOliTomVit00a,Mil91a}.

The idea is to express the evolution operator $U(t)$ using the spectral decomposition theorem for unitary operators \cite{Kreyszig78a} and then introduce the randomness in the time evolution. Let us recall then that 

\begin{equation}\label{UnitOp}
U(t)=\int_{-\pi}^{\pi}e^{-i\theta t}dE_{\theta}
\end{equation}

\noindent where $E_{\theta}$ is the spectral family on $[-\pi,\pi]$ associated to $U(t)$. The randomness hypothesis can be introduced in different ways. The most na\"{\i}ve way is just to substitute $t$ in expression (\ref{UnitOp}) for a random variable $\chi_{t}$ which expresses the stochastic nature of the evolution. The properties of $\chi_{t}$ (distribution, moments, \dots) will depend on the characteristics of the physical system under study. To grasp this generality it is convenient to consider  $\chi_{t}$ as the solution to an Ito stochastic differential equation \cite{Oksendal98a}:

\begin{equation}
d\chi_{t}=b(t,\chi_{t})dt+\sigma(t,\chi_{t})d\mathcal{B}_{t}
\end{equation} 

\noindent where $b(\cdot,\cdot)$ and $\sigma(\cdot,\cdot)$ are real functions of two real variables and $\mathcal{B}_{t}$ denotes standard real brownian motion. These two functions encode the information of the specific physical system. As a first example we may consider $b(t,\chi_{t})=1$ and $\sigma(t,\chi_{t})=\sigma(t)$ (with clear abuse of notation). This allows us to write $\chi_{t}=t+\int_{0}^{t}\sigma(s)d\mathcal{B}_{s}$, hence the expectation value of $\chi_{t}$ is the standard deterministic time $t$ and its deviation from it is given by $\lambda(t)\equiv\int_{0}^{t}\sigma^{2}(s)ds$. This can be considered a weak-coupling-like limit. With these hypotheses, we may construct a random evolution operator given by

\begin{equation}\label{RandEvolOp}
U_{\text{rand}}(t)=\int_{-\pi}^{\pi}e^{-i\theta t-i\theta\int_{0}^{t}\sigma(s)d\mathcal{B}_{s}}dE_{\theta}
\end{equation}

Following \cite{BonOliTomVit00a} the density operator describing experimental results will be given by

\begin{equation}\label{ExpVal}
\bar{\rho}(t)=\mathbb{E}[\rho(t)]=\mathbb{E}[U_{\text{rand}}(t)\rho(0)U_{\text{rand}}^{\dagger}(t)]
\end{equation}

\noindent where $\mathbb{E}$ denotes expectation value with respect to the probability distribution of $\chi_{t}$. Using Ito calculus \cite{Oksendal98a}, the expectation value $\mathbb{E}[e^{-i(\theta-\theta')\int_{0}^{t}\sigma_{s}d\mathcal{B}_{s}}]=e^{-\frac{\lambda(t)}{2}(\theta-\theta')^2}$ can be readily found and hence one immediately arrives at the following expression for $\bar{\rho}(t)$

\begin{equation}
\bar{\rho}(t)=e^{-it\mathcal{L}_{H}-\frac{\lambda(t)}{2}\mathcal{L}_{H}^{2}}[\rho(0)]
\end{equation}

\noindent where $\mathcal{L}_{H}\equiv[H,\cdot]$ is the commutator with the hamiltonian $H$. Note that this equation automatically satisfies condition i) above, i.e. $\bar{\rho}(t)$ is a density operator (it is selfadjoint, it has trace one and it is positive) provided the initial $\rho(0)$ is a density operator too. This is a first generalization of the results obtained in \cite{BonOliTomVit00a}. If condition ii) is imposed, then the only choice for $\sigma(t)$ will be $\sigma(t)=\gamma^{1/2}$, i.e. a constant. The master equation is the well-known phase-destroying master equation

\begin{equation}\label{PhasDestME}
\dot{\bar{\rho}}(t)=-i[H,\bar{\rho}(t)]-\frac{\gamma}{2}[H,[H,\bar{\rho}(t)]]
\end{equation}

Note that no approximation has been assumed to arrive at this equation, in contrast to \cite{BonOliTomVit00a,Mil91a}. The constant $\gamma$ is thus under this formalism  a measure of the deviation from strictly determinist time evolution. Apart from the evident freedom in the choice of adequately physically motivated choices of functions $b(\cdot,\cdot)$ and $\sigma(\cdot,\cdot)$ (which e.g. may drive us to time-dependent $\gamma(t)$), a higher generalization can be achieved if the previous scheme is slightly changed by substituting the whole phase of the exponential in (\ref{UnitOp}) for a $\theta$-dependent random variable $\chi_{t}(\theta)$. This allows us to write the random evolution operator instead as

\begin{equation}
U_{\text{rand}}(t)=\int_{-\pi}^{\pi}e^{-i\chi_{t}(\theta)}dE_{\theta}
\end{equation}

The former scheme is a particular case of this latter when $\chi_{t}(\theta)=\theta\chi_{t}$. The physical meaning behind this formalism is clear: the random time evolution does not affect on an equal footing to all energy levels of the system. %This is very reasonable. Consider for instance a sustem of harmonic oscillators interacting  with each other. It is unnatural to expect that the oscillator will be equally afected by the environment either if it is in a low-energy level or in a high-energy level.
Following then the same idea as above, we consider $\chi_{t}(t)$ as a solution to an Ito SDE, now $\theta$-dependent

\begin{equation}
d\chi_{t}(\theta)=b(t,\chi_{t}(\theta);\theta)dt+\sigma(t,\chi_{t}(\theta);\theta)d\mathcal{B}_{t}(\theta)
\end{equation}

\noindent i.e., now we have a SDE for each energy level. This must be supplemented with correlation properties among the brownian motions at different $\theta$'s. These will be expressed through the relation

\begin{equation}
d\mathcal{B}_{t}(\theta)d\mathcal{B}_{t}(\theta')=g(t;\theta,\theta')dt
\end{equation}

\noindent where $g(t;\theta,\theta')$ is basically for each $t$ a covariance function expressing the correlations among different $\mathcal{B}_{t}(\theta)$'s. From a physical standpoint it is clear that $g(t;\theta,\theta')=g(t;\theta',\theta)$ and since we are dealing with standard brownian motions $|g(t;\theta,\theta')|\leq 1$ for all $t,\theta,\theta'$.

It is difficult to proceed forward without making some assumptions. One may on one hand discuss on physical grounds particular choices for the different functions involved and find the solution or on the other hand look for general analytical properties which drives us to a sufficiently global situation. As before, as a first example, let us assign simple choices to $b(t,\chi_{t}(\theta);\theta)=h(\theta)$ and  $\sigma(t,\chi_{t}(\theta);\theta)=\sigma(t;\theta)$. The former corresponds to the fact that the unitary part of the dynamics is driven by a renormalized hamiltonian \cite{CalLeg83a}, as the following calculation shows; whereas the second states that the departure from deterministic evolution is caused by an uncorrelated noise with different intensity at different times for each $\theta$. Let us include some details. We have to evaluate the expression:

\begin{widetext}
\begin{equation}\label{RandEvolOpSevEn}
\bar{\rho}(t)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}e^{-i(h(\theta)-h(\theta'))t}\mathbb{E}[e^{-i\left(\int_{0}^{t}\sigma(s;\theta)d\mathcal{B}_{s}(\theta)-\int_{0}^{t}\sigma(s;\theta')d\mathcal{B}_{s}(\theta')\right)}]dE_{\theta}\rho(0)dE_{\theta'}
\end{equation}
\end{widetext}

The expectation value is found using standard techniques. Define the stochastic process $X_{t}(\theta,\theta')$ by the SDE $dX_{t}(\theta,\theta')=\sigma(t;\theta)d\mathcal{B}_{t}(\theta)-\sigma(t;\theta')d\mathcal{B}_{t}(\theta')$; express $\exp(-iX_{t}(\theta,\theta'))$ as a power series; apply the Ito formula (cf. \cite{Oksendal98a}) to $f(X_{t})$ ($f(z)=z^n$); define $\beta_{n}(t;\theta,\theta')\equiv\mathbb{E}[X_{t}^{n}(\theta,\theta')]$ to find

\begin{subequations}
\begin{eqnarray}
\beta_{2n}(t;\theta,\theta')&=&\frac{(2n)!}{2^n n!}\lambda^{n}(t;\theta,\theta')\\
\beta_{2n+1}(t;\theta,\theta')&=&0
\end{eqnarray}
\end{subequations}

\noindent where $\lambda(t;\theta,\theta')\equiv\int_{0}^{t}\eta(s;\theta,\theta')ds$ and we have defined $\eta(t;\theta,\theta')=\sigma^{2}(t;\theta)g(t;\theta,\theta)+\sigma^{2}(t;\theta')g(t;\theta',\theta')-2\sigma(t;\theta)\sigma(t;\theta')g(t;\theta,\theta')$. Finally resum the series and then (\ref{RandEvolOpSevEn}) reduces to

\begin{equation}\label{RenNonMark}
\bar{\rho}(t)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}e^{-it(h(\theta)-h(\theta'))}e^{-\frac{\lambda(t;\theta,\theta')}{2}}dE_{\theta}\rho(0)dE_{\theta'}
\end{equation}

We cannot proceed further without specifying $\sigma(t;\theta)$ and $g(t;\theta,\theta')$. To compare with the previous scheme let us define $\sigma(t;\theta)=\gamma^{1/2}\theta$ and assume a correlation function $g(t;\theta,\theta')=\exp(-\tau^2(\theta-\theta')^2)$. Here $\tau^{-1}$ roughly measures the distance of correlation in some arbitrary units. Then $\lambda(t;\theta,\theta')=\gamma t (\theta^{2}+\theta^{'2}-2\theta\theta'e^{-\tau^2(\theta-\theta')^{2}})$ and the master equation reads

\begin{eqnarray}
\dot{\bar{\rho}}(t)&=&-i[h(H),\bar{\rho}(t)]-\nonumber\\
&-&\frac{\gamma}{2}(H^{2}\bar{\rho}(t)+\bar{\rho}(t)H^{2}-2He^{-\tau^2\mathcal{L}_{H}^{2}}[\bar{\rho}(t)]H)
\end{eqnarray}

This is a CP markovian master equation\footnote{To convince oneself that this denotes a CP evolution, compare with eq. (4.1) in \cite{Lin76}.}  different from (\ref{PhasDestME}), though it reduces to it in the limit $\tau\to 0$, i.e. in the case of infinite-range correlations (all energy levels equally affected by the random evolution). In this way the formalism depicted here generalizes the approach in \cite{BonOliTomVit00a}, even when  restricting ourselves to the markovian regime. Indeed, whenever we may write $\lambda(t;\theta,\theta')=t\tilde{\lambda}(\theta,\theta')$ (which amounts to having $\sigma(t;\theta,\theta')$ and $g(t;\theta,\theta')$ time-independent) we arrive at a markovian evolution, since

\begin{eqnarray}
\bar{\rho}(t)&=&\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}e^{t[-i(h(\theta)-h(\theta'))-\tilde{\lambda}(\theta,\theta')]}dE_{\theta}\rho(0)dE_{\theta'}\nonumber\\
&\equiv&e^{-it\mathcal{L}_{h(H)}+t\mathcal{D}_H}[\rho(0)]
\end{eqnarray}

\noindent where $\mathcal{L}_{h(H)}$ and $\mathcal{D}_H$ denote superoperators.\\
The previous example depicts a typical situation of nondissipative decoherence. In this language and restricting to the previous case (eq. (\ref{RenNonMark})), this is due to the fact that $\lambda(t,\theta,\theta)=0$, as an immediate calculation of $\text{Tr}(\bar{\rho}(t)H)$ shows. Working in the energy eigenstates basis, (\ref{RenNonMark}) leads to $\bar{\rho}(t;E,E')=\exp(-it(h(E)-h(E'))-\lambda(t;E,E')/2)\rho(0;E,E')$, hence 

\begin{eqnarray}
\text{Tr}(\rho(t)H)&=&\int_{\sigma(H)}Ee^{-\frac{\lambda(t;E,E)}{2}}\rho(0;E,E)d\mu(E)=\nonumber\\\label{MeanEn}
&=&\text{Tr}(\rho(0)H)
\end{eqnarray}


\noindent where $\sigma(H)$ denotes the spectrum of $H$ and $\mu(E)$ denotes the measure over this spectrum (a sum if it is discrete and an integral if it is continuous) and where $\lambda(t,\theta,\theta)=0$ has been assumed. From (\ref{MeanEn}) it is clear that if this condition is not satisfied, we have a dissipative decoherence process. This is a third generalization of previous works.\\
Some comments must be made about this formalism. Though it has been presented as an adequate tool to express decoherence in isolated systems, from a mathematical standpoint it can be viewed as an alternative description of environment-induced decoherence provided the stochastic parameter $\chi_{t}$ is properly connected to the surrounding reservoir properties and  the environment-system interaction characteristics. But in our opinion the main benefit from this language is to bring a bit closer the decoherence process and the different models of stochastic state reduction (see \cite{AdlBroBruHug01a} for a review of their common backbone). However this connection demands further mathematical work which is in progress. 

% If you have acknowledgments, this puts in the proper section head.
\begin{acknowledgments}
One of us (D.S.)  acknowledges the support of Madrid Education Council under grant BOCAM 20-08-1999. 
\end{acknowledgments}

% Create the reference section using BibTeX:
%\bibliography{/home/david/Bibliography/Biblio}
\begin{thebibliography}{99}
%\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
%\expandafter\ifx\csname bibnamefont\endcsname\relax
%  \def\bibnamefont#1{#1}\fi
%\expandafter\ifx\csname bibfnamefont\endcsname\relax
%  \def\bibfnamefont#1{#1}\fi
%\expandafter\ifx\csname citenamefont\endcsname\relax
%  \def\citenamefont#1{#1}\fi
%\expandafter\ifx\csname url\endcsname\relax
%  \def\url#1{\texttt{#1}}\fi
%\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
%\providecommand{\bibinfo}[2]{#2}
%\providecommand{\eprint}[2][]{\url{#2}}

\bibitem{Mil91a} G. Milburn, Phys. Rev. A \textbf{44}, 5401 (1991).
%\bibitem[{\citenamefont{Milburn}(1991)}]{Mil91a}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Milburn}},
%  \bibinfo{journal}{Phys. Rev. A}
%  \textbf{\bibinfo{volume}{44}}(\bibinfo{number}{9}) (\bibinfo{year}{1991}).

\bibitem{BonOliTomVit00a} R. Bonifacio, S. Olivares, P. Tombesi, and D. Vitali, Phys. Rev. A \textbf{61}, 053803 (2000).
%\bibitem[{\citenamefont{Bonifacio et~al.}(2000)\citenamefont{Bonifacio,
%  Olivares, Tombesi, and Vitali}}]{BonOliTomVit00a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Bonifacio}},
%  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Olivares}},
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Tombesi}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vitali}},
%  \bibinfo{journal}{Phys. Rev. A} \textbf{\bibinfo{volume}{61}},
%  \bibinfo{pages}{053802} (\bibinfo{year}{2000}).

\bibitem{Zur91a} W.H. Zurek, Phys. Today \textbf{44}(10), 36 (1991). 
%\bibitem[{\citenamefont{Zurek}(1991)}]{Zur91a}
%\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Zurek}},
%  \bibinfo{journal}{Phys. Today}
%  \textbf{\bibinfo{volume}{44}}(\bibinfo{number}{10}), \bibinfo{pages}{36}
%  (\bibinfo{year}{1991}).

\bibitem{ChuNie00a} M.A. Nielsen and I.L. Chuang, \emph{Quantum Computation and Quantum Information} (Cambridge University Press, Cambridge, 2000).
%\bibitem[{\citenamefont{Chuang and Nielsen}(2000)}]{ChuNie00a}
%\bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Chuang}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Nielsen}},
%  \emph{\bibinfo{title}{Quantum Computation and Quantum Information}}
%  (\bibinfo{publisher}{Cambridge University Press},
%  \bibinfo{address}{Cambridge}, \bibinfo{year}{2000}).

\bibitem{GiuJooKieKupStaZeh96a} D. Giulini \emph{et. al.}, \emph{Decoherence and the appearance of a classical world in quantum theory} (Springer-Verlag, Berlin, 1996).
%\bibitem[{\citenamefont{Giulini et~al.}(1996)\citenamefont{Giulini, Joos,
%  Kiefer, Kupsch, Stamatescu, and Zeh}}]{GiuJooKieKupStaZeh96a}
%\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Giulini}},
%  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Joos}},
%  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Kiefer}},
%  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kupsch}},
%  \bibinfo{author}{\bibfnamefont{I.-O.} \bibnamefont{Stamatescu}},
%  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Zeh}},
%  \emph{\bibinfo{title}{Decoherence and the appearance of a classical world in
%  quantum theory}} (\bibinfo{publisher}{Springer-Verlag},
%  \bibinfo{address}{Berlin}, \bibinfo{year}{1996}).

\bibitem{Zur82a} W.H. Zurek, Phys. Rev. D \textbf{26}, 1862 (1982).
%\bibitem[{\citenamefont{Zurek}(1982)}]{Zur82a}
%\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Zurek}},
%  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{26}},
%  \bibinfo{pages}{1862} (\bibinfo{year}{1982}).

\bibitem{Bon99a} R. Bonifacio, Nuo. Cim. B\textbf{114}, 473 (1999).
%\bibitem[{\citenamefont{Bonifacio}(1999)}]{Bon99a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Bonifacio}},
%  \bibinfo{journal}{Nuo. Cim. B} \textbf{\bibinfo{volume}{114}},
%  \bibinfo{pages}{473} (\bibinfo{year}{1999}).

\bibitem{MoyBuzKimKni93a} H. Moya-Cessa, V. Buzek, M. Kim, and P. Knight, Phys. Rev. A \textbf{48}, 3900 (1993).
%\bibitem[{\citenamefont{Moya-Cessa et~al.}(1993)\citenamefont{Moya-Cessa,
%  Buzek, Kim, and Knight}}]{MoyBuzKimKni93a}
%\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Moya-Cessa}},
%  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Buzek}},
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kim}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Knight}},
%  \bibinfo{journal}{Phys. Rev. A}
%  \textbf{\bibinfo{volume}{48}}(\bibinfo{number}{5}), \bibinfo{pages}{3900}
%  (\bibinfo{year}{1993}).

\bibitem{AlickiLendi87a} R. Alicki and K. Lendi, \emph{Quantum dynamical semigroups and applications}, LNP 286 (Springer-Verlag, Berlin, 1987).
%\bibitem[{\citenamefont{Alicki and Lendi}(1987)}]{AlickiLendi87a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Alicki}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Lendi}},
%  \emph{\bibinfo{title}{Quantum dynamical semigroups and applications}}, LNP
%  286 (\bibinfo{publisher}{Springer-Verlag}, \bibinfo{address}{Berlin},
%  \bibinfo{year}{1987}).

\bibitem{GorFriVerKosSud78a} V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan, Rep. Math. Phys. \textbf{13}, 149 (1978).
%\bibitem[{\citenamefont{Gorini et~al.}(1978)\citenamefont{Gorini, Frigerio,
%  Verri, Kossakowski, and Sudarshan}}]{GorFriVerKosSud78a}
%\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Gorini}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Frigerio}},
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Verri}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Kossakowski}},
%  \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Sudarshan}},
%  \bibinfo{journal}{Rep. Math. Phys.} \textbf{\bibinfo{volume}{13}},
%  \bibinfo{pages}{149} (\bibinfo{year}{1978}).

\bibitem{MoyHopSav98a} G. Moy, J. Hope, and C. Savage, quant-ph/9801046 v2 21-Sep-1998.
%\bibitem[{\citenamefont{Moy et~al.}(1998)\citenamefont{Moy, Hope, and
%  Savage}}]{MoyHopSav98a}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Moy}},
%  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Hope}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Savage}},
%  \bibinfo{journal}{quant-ph/9801046 v2 21-Sep-1998}  (\bibinfo{year}{1998}).

\bibitem{Kreyszig78a} E. Kreyszig, \emph{Introductory Functional Analysis with Applications} (J. Wiley \& Sons, New York, 1978).
%\bibitem[{\citenamefont{Kreyszig}(1978)}]{Kreyszig78a}
%\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Kreyszig}},
%  \emph{\bibinfo{title}{Introductory Functional Analysis with Applications}}
%  (\bibinfo{publisher}{John Wiley \& Sons}, \bibinfo{address}{New York},
%  \bibinfo{year}{1978}).

\bibitem{Oksendal98a} B. Oksendal, \emph{Stochastic Differential Equations} (Springer, Berlin, 1998), 5th ed.
%\bibitem[{\citenamefont{Oksendal}(1998)}]{Oksendal98a}
%\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Oksendal}},
%  \emph{\bibinfo{title}{Stochastic Differential Equations}}
%  (\bibinfo{publisher}{Springer}, \bibinfo{address}{Berlin},
%  \bibinfo{year}{1998}), \bibinfo{edition}{5th} ed.

\bibitem{CalLeg83a} A. Caldeira and A. Legget, Physica \textbf{121A}, 587 (1983).
%\bibitem[{\citenamefont{Caldeira and Legget}(1983)}]{CalLeg83a}
%\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Caldeira}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Legget}},
%  \bibinfo{journal}{Physica} \textbf{\bibinfo{volume}{121A}},
%  \bibinfo{pages}{587} (\bibinfo{year}{1983}).

\bibitem{AdlBroBruHug01a} S. Adler, D. Brody, T. Brun, and L. Hughston, J. Math. Phys. A \textbf{34}, 8795 (2001).
%\bibitem[{\citenamefont{Adler et~al.}(2001)\citenamefont{Adler, Brody, Brun,
%  and Hughston}}]{AdlBroBruHug01a}
%\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Adler}},
%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Brody}},
%  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Brun}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Hughston}},
%  \bibinfo{journal}{J. Phys. A} \textbf{\bibinfo{volume}{34}},
%  \bibinfo{pages}{8795} (\bibinfo{year}{2001}).

\bibitem{Lin76} G. Lindblad, Commun. Math. Phys. \textbf{48}, 119 (1976).
%\bibitem[{\citenamefont{Lindblad}(1976)}]{Lin76}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Lindblad}},
%  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{48}},
%  \bibinfo{pages}{119} (\bibinfo{year}{1976}).

\end{thebibliography}




\end{document}
%
% ****** End of file template.aps ******


quant-ph0204141.extracted_bib
\begin{thebibliography}{99}
%\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
%\expandafter\ifx\csname bibnamefont\endcsname\relax
%  \def\bibnamefont#1{#1}\fi
%\expandafter\ifx\csname bibfnamefont\endcsname\relax
%  \def\bibfnamefont#1{#1}\fi
%\expandafter\ifx\csname citenamefont\endcsname\relax
%  \def\citenamefont#1{#1}\fi
%\expandafter\ifx\csname url\endcsname\relax
%  \def\url#1{\texttt{#1}}\fi
%\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
%\providecommand{\bibinfo}[2]{#2}
%\providecommand{\eprint}[2][]{\url{#2}}

\bibitem{Mil91a} G. Milburn, Phys. Rev. A \textbf{44}, 5401 (1991).
%\bibitem[{\citenamefont{Milburn}(1991)}]{Mil91a}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Milburn}},
%  \bibinfo{journal}{Phys. Rev. A}
%  \textbf{\bibinfo{volume}{44}}(\bibinfo{number}{9}) (\bibinfo{year}{1991}).

\bibitem{BonOliTomVit00a} R. Bonifacio, S. Olivares, P. Tombesi, and D. Vitali, Phys. Rev. A \textbf{61}, 053803 (2000).
%\bibitem[{\citenamefont{Bonifacio et~al.}(2000)\citenamefont{Bonifacio,
%  Olivares, Tombesi, and Vitali}}]{BonOliTomVit00a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Bonifacio}},
%  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Olivares}},
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Tombesi}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vitali}},
%  \bibinfo{journal}{Phys. Rev. A} \textbf{\bibinfo{volume}{61}},
%  \bibinfo{pages}{053802} (\bibinfo{year}{2000}).

\bibitem{Zur91a} W.H. Zurek, Phys. Today \textbf{44}(10), 36 (1991). 
%\bibitem[{\citenamefont{Zurek}(1991)}]{Zur91a}
%\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Zurek}},
%  \bibinfo{journal}{Phys. Today}
%  \textbf{\bibinfo{volume}{44}}(\bibinfo{number}{10}), \bibinfo{pages}{36}
%  (\bibinfo{year}{1991}).

\bibitem{ChuNie00a} M.A. Nielsen and I.L. Chuang, \emph{Quantum Computation and Quantum Information} (Cambridge University Press, Cambridge, 2000).
%\bibitem[{\citenamefont{Chuang and Nielsen}(2000)}]{ChuNie00a}
%\bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Chuang}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Nielsen}},
%  \emph{\bibinfo{title}{Quantum Computation and Quantum Information}}
%  (\bibinfo{publisher}{Cambridge University Press},
%  \bibinfo{address}{Cambridge}, \bibinfo{year}{2000}).

\bibitem{GiuJooKieKupStaZeh96a} D. Giulini \emph{et. al.}, \emph{Decoherence and the appearance of a classical world in quantum theory} (Springer-Verlag, Berlin, 1996).
%\bibitem[{\citenamefont{Giulini et~al.}(1996)\citenamefont{Giulini, Joos,
%  Kiefer, Kupsch, Stamatescu, and Zeh}}]{GiuJooKieKupStaZeh96a}
%\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Giulini}},
%  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Joos}},
%  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Kiefer}},
%  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kupsch}},
%  \bibinfo{author}{\bibfnamefont{I.-O.} \bibnamefont{Stamatescu}},
%  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Zeh}},
%  \emph{\bibinfo{title}{Decoherence and the appearance of a classical world in
%  quantum theory}} (\bibinfo{publisher}{Springer-Verlag},
%  \bibinfo{address}{Berlin}, \bibinfo{year}{1996}).

\bibitem{Zur82a} W.H. Zurek, Phys. Rev. D \textbf{26}, 1862 (1982).
%\bibitem[{\citenamefont{Zurek}(1982)}]{Zur82a}
%\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Zurek}},
%  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{26}},
%  \bibinfo{pages}{1862} (\bibinfo{year}{1982}).

\bibitem{Bon99a} R. Bonifacio, Nuo. Cim. B\textbf{114}, 473 (1999).
%\bibitem[{\citenamefont{Bonifacio}(1999)}]{Bon99a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Bonifacio}},
%  \bibinfo{journal}{Nuo. Cim. B} \textbf{\bibinfo{volume}{114}},
%  \bibinfo{pages}{473} (\bibinfo{year}{1999}).

\bibitem{MoyBuzKimKni93a} H. Moya-Cessa, V. Buzek, M. Kim, and P. Knight, Phys. Rev. A \textbf{48}, 3900 (1993).
%\bibitem[{\citenamefont{Moya-Cessa et~al.}(1993)\citenamefont{Moya-Cessa,
%  Buzek, Kim, and Knight}}]{MoyBuzKimKni93a}
%\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Moya-Cessa}},
%  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Buzek}},
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kim}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Knight}},
%  \bibinfo{journal}{Phys. Rev. A}
%  \textbf{\bibinfo{volume}{48}}(\bibinfo{number}{5}), \bibinfo{pages}{3900}
%  (\bibinfo{year}{1993}).

\bibitem{AlickiLendi87a} R. Alicki and K. Lendi, \emph{Quantum dynamical semigroups and applications}, LNP 286 (Springer-Verlag, Berlin, 1987).
%\bibitem[{\citenamefont{Alicki and Lendi}(1987)}]{AlickiLendi87a}
%\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Alicki}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Lendi}},
%  \emph{\bibinfo{title}{Quantum dynamical semigroups and applications}}, LNP
%  286 (\bibinfo{publisher}{Springer-Verlag}, \bibinfo{address}{Berlin},
%  \bibinfo{year}{1987}).

\bibitem{GorFriVerKosSud78a} V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan, Rep. Math. Phys. \textbf{13}, 149 (1978).
%\bibitem[{\citenamefont{Gorini et~al.}(1978)\citenamefont{Gorini, Frigerio,
%  Verri, Kossakowski, and Sudarshan}}]{GorFriVerKosSud78a}
%\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Gorini}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Frigerio}},
%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Verri}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Kossakowski}},
%  \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Sudarshan}},
%  \bibinfo{journal}{Rep. Math. Phys.} \textbf{\bibinfo{volume}{13}},
%  \bibinfo{pages}{149} (\bibinfo{year}{1978}).

\bibitem{MoyHopSav98a} G. Moy, J. Hope, and C. Savage, quant-ph/9801046 v2 21-Sep-1998.
%\bibitem[{\citenamefont{Moy et~al.}(1998)\citenamefont{Moy, Hope, and
%  Savage}}]{MoyHopSav98a}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Moy}},
%  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Hope}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Savage}},
%  \bibinfo{journal}{quant-ph/9801046 v2 21-Sep-1998}  (\bibinfo{year}{1998}).

\bibitem{Kreyszig78a} E. Kreyszig, \emph{Introductory Functional Analysis with Applications} (J. Wiley \& Sons, New York, 1978).
%\bibitem[{\citenamefont{Kreyszig}(1978)}]{Kreyszig78a}
%\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Kreyszig}},
%  \emph{\bibinfo{title}{Introductory Functional Analysis with Applications}}
%  (\bibinfo{publisher}{John Wiley \& Sons}, \bibinfo{address}{New York},
%  \bibinfo{year}{1978}).

\bibitem{Oksendal98a} B. Oksendal, \emph{Stochastic Differential Equations} (Springer, Berlin, 1998), 5th ed.
%\bibitem[{\citenamefont{Oksendal}(1998)}]{Oksendal98a}
%\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Oksendal}},
%  \emph{\bibinfo{title}{Stochastic Differential Equations}}
%  (\bibinfo{publisher}{Springer}, \bibinfo{address}{Berlin},
%  \bibinfo{year}{1998}), \bibinfo{edition}{5th} ed.

\bibitem{CalLeg83a} A. Caldeira and A. Legget, Physica \textbf{121A}, 587 (1983).
%\bibitem[{\citenamefont{Caldeira and Legget}(1983)}]{CalLeg83a}
%\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Caldeira}} \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Legget}},
%  \bibinfo{journal}{Physica} \textbf{\bibinfo{volume}{121A}},
%  \bibinfo{pages}{587} (\bibinfo{year}{1983}).

\bibitem{AdlBroBruHug01a} S. Adler, D. Brody, T. Brun, and L. Hughston, J. Math. Phys. A \textbf{34}, 8795 (2001).
%\bibitem[{\citenamefont{Adler et~al.}(2001)\citenamefont{Adler, Brody, Brun,
%  and Hughston}}]{AdlBroBruHug01a}
%\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Adler}},
%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Brody}},
%  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Brun}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Hughston}},
%  \bibinfo{journal}{J. Phys. A} \textbf{\bibinfo{volume}{34}},
%  \bibinfo{pages}{8795} (\bibinfo{year}{2001}).

\bibitem{Lin76} G. Lindblad, Commun. Math. Phys. \textbf{48}, 119 (1976).
%\bibitem[{\citenamefont{Lindblad}(1976)}]{Lin76}
%\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Lindblad}},
%  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{48}},
%  \bibinfo{pages}{119} (\bibinfo{year}{1976}).

\end{thebibliography}