Stochastic quantization of the dynamics of interacting qubit systems
Sergio De Filippo\cite{byline}
The dynamics of all states of a qubit system with arbitrary, even time dependent, one qubit Hamiltonians and two qubit interactions is realized as one and the same diffusion process for systems with time dependent statistical weight in a space whose dimension grows only linearly with the number of qubits. The ensuing Fokker-Planck equation for the corresponding nonpositive probability density is equivalent to the von Neumann equation for the quantum state. Presumably the effectiveness of the stochastic process as a numerical quantization method decreases as the number of qubits grows.
quant-ph0002018.tex

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\begin{document}
\draft
\author{Sergio De Filippo\cite{byline}}
\address{Dipartimento di Scienze Fisiche, Universit\`a di Salerno\\
Via Allende I-84081 Baronissi (SA) ITALY\\
and \\
Unit\`{a} INFM Salerno}
\date{\today}
\title{Stochastic quantization of the dynamics of interacting qubit systems}
\maketitle

\begin{abstract}
The dynamics of all states of a qubit system with arbitrary, even time
dependent, one qubit Hamiltonians and two qubit interactions is realized as
one and the same diffusion process for systems with time dependent
statistical weight in a space whose dimension grows only linearly with the
number of qubits. The ensuing Fokker-Planck equation for the corresponding
nonpositive probability density is equivalent to the von Neumann equation
for the quantum state. Presumably the effectiveness of the stochastic
process as a numerical quantization method decreases as the number of qubits
grows.
\end{abstract}

\pacs{03.65.-w, 03.65.Bz, 03.67.Lx,  05.40.-a}

In the past few years, theoretical and experimental research on quantum
computation, mainly sparked by the discovery of the quantum factorization
algorithm by Shor\cite{Shor}, led to a growing interest in quantum models
with finite dimensional state spaces.\cite{Steane,Preskil} This result
confirmed a former foresight of Feynman\cite{Feynman1} who had anticipated
that amazing computational capabilities were inherent in quantum systems due
to the exponential growth of their state space dimension as a function of
their physical dimension.\cite{Jozsa} Nevertheless, in spite of a general
feeling, there is no formal proof that classical stochastic systems have a
less effective scaling of their computational power versus physical
dimensions\cite{Steane}. Apart from the relevance for the foundations of
quantum mechanics, this issue makes the quest for classical stochastic
models of finite dimensional quantum dynamics particularly appealing. It was
still Feynman\cite{Feynman2} who suggested that if one wants to get rid of
quantum amplitudes and construct a quantum theory only in terms of
probabilities, one has to use generalized non-positive probabilities. In
fact a local traditional probabilistic theory always obeys Bell's
inequalities\cite{Bell}, contrary to what happens for quantum mechanical
states. The aim of this paper is to prove that quantum evolution of qubit
systems, namely arrays of two state quantum systems, with two qubit
interactions, is equivalent to a suitable classical stochastic dynamics of
particular non-positive probability densities in a phase space whose
dimension is proportional to the number of qubits, which is the crucial
feature of a classical state space.

To fit the issue in the due context, it is worthwhile citing that Moyal \cite
{Moyal} and Wigner\cite{Wigner} showed long ago how quantum states of point
particles can be represented as nonpositive real functions on classical
phase spaces. Their dynamics, which in the classical limit obeys the
Liouville equation for the probability density, is nonlocal in space, i.e.
their evolution cannot be described in terms of a partial differential
equation of finite order. As a consequence its description as some sort of
generalized classical diffusion process is not viable.

If instead one considers, to begin with, a system of two interacting qubits,
their generic (mixed) state can be expressed as an at most quadratic
function of the Pauli generators of the corresponding $sl(2,C)$ algebras. If
these generators are replaced by real variables, these states can be
interpreted as nonpositive probability densities giving rise to expectation
values coinciding with the quantum ones. On the other hand their quadratic
character allows for the description of their time evolution by means of a
second order partial differential equation. It is shown that this equation
can be read as the master equation of a suitably generalized diffusion
process. As to the generalization involved, if one wants to define
statistical ensembles for non-positive probability densities, one has to
introduce for each element in the ensemble a real statistical weight, which,
as it is only natural to expect, will itself evolve in time according to a
suitable differential equation. Although presented for notational simplicity
with reference to a two qubit system, the result is general and the phase
space is always a direct product of single qubit phase spaces.

Stated in these terms the result seems to prove that quantum computational
systems can be effectively simulated by classical stochastic ones. Actually,
since, as explained below, effectiveness seems to be achieved only for short
computational times, the problem is subtler and deserves further
investigation. A still deeper distinction between true quantum computation
and stochastic simulation in its present form resides in the lack within the
latter of a mechanism of wave function collapse. This of course does not
hold for NMR computing\cite{CoryFahmy,Gershenfeld} where no wave function
collapse occurs.

Before proceeding to the construction of the quantization scheme it is
worthwhile remarking that it has nothing to do with Parisi-Wu quantization 
\cite{Parisi}, which is based on the addition of stochastic terms to a
fictitious dissipative dynamics which has the ground state as a fixed point.
On the other hand, while Nelson's approach\cite{Nelson} deals with real time
evolution, it is tailored for point particle dynamics and it does not follow
Feynman's suggestion about nonpositive probabilities and Moyal-Wigner
viewpoint about treating pure and mixed states on the same footing.

To be specific, consider a generic two qubit system formally identified with
a two spin $1/2$ system, whose Hamiltonian, apart from an additive constant,
has the traceless form 
\begin{equation}
H=-B_{i}^{\alpha }\hat{S}_{i}^{\alpha }-\hat{S}_{i}^{1}J_{ij}\;\hat{S}%
_{j}^{2},  \label{H}
\end{equation}
where summation over repeated indices and apices is implied, $\hat{S}%
_{i}^{1},\hat{S}_{i}^{2}$ denote the two spin $1/2$ vector components
defining the usual Clifford algebra 
\begin{equation}
\left[ \hat{S}_{i}^{\alpha },\hat{S}_{j}^{\beta }\right] =i\delta _{\alpha
\beta }\varepsilon _{ijk}\hat{S}_{k}^{\alpha },\;\;\hat{S}_{i}^{\alpha }%
\hat{S}_{j}^{\beta }=\frac{1}{4}\delta _{\alpha \beta }\delta _{ij},
\label{Clifford}
\end{equation}
while $\vec{B}^{1},\vec{B}^{2}$ and $\hat{J}\;$\ denote possibly time
dependent c-number vectors and a $3\times 3$ matrix.

The most general state, i.e. unit trace non negative Hermitian operator,
reads 
\begin{equation}
\rho =\frac{1}{4}\left( {\bf 1}+4\left\langle \hat{S}_{i}^{\alpha
}\right\rangle \hat{S}_{i}^{\alpha }+16\left\langle \hat{S}_{i}^{1}\hat{S}%
_{j}^{2}\right\rangle \hat{S}_{i}^{1}\hat{S}_{j}^{2}\right) ,  \label{rho}
\end{equation}
where ${\bf 1}$ is the unit operator in the four dimensional Hilbert space.
Dynamical evolution is completely characterized in terms of the expectation
values appearing in Eq. (\ref{rho}), i.e. by 
\begin{eqnarray}
\frac{d\left\langle \hat{S}_{i}^{1}\right\rangle }{dt} &=&-i\left\langle %
\left[ \hat{S}_{i}^{1},H\right] \right\rangle =-\varepsilon _{ijk}\left(
B_{j}^{1}\left\langle \hat{S}_{k}^{1}\right\rangle -J_{jl}\left\langle 
\hat{S}_{k}^{1}\hat{S}_{l}^{2}\right\rangle \right)   \nonumber \\
\frac{d\left\langle \hat{S}_{i}^{2}\right\rangle }{dt} &=&-i\left\langle %
\left[ \hat{S}_{i}^{2},H\right] \right\rangle =-\varepsilon _{ijk}\left(
B_{j}^{2}\left\langle \hat{S}_{k}^{2}\right\rangle -J_{lj}\left\langle 
\hat{S}_{l}^{1}\hat{S}_{k}^{2}\right\rangle \right)   \label{Heisenberg} \\
\frac{d\left\langle \hat{S}_{i}^{1}\hat{S}_{j}^{2}\right\rangle }{dt}
&=&-\varepsilon _{ikl}B_{k}^{1}\left\langle \hat{S}_{l}^{1}\hat{S}%
_{j}^{2}\right\rangle -\varepsilon _{jkl}B_{k}^{2}\left\langle \hat{S}%
_{i}^{1}\hat{S}_{l}^{2}\right\rangle -\frac{\varepsilon _{imk}}{4}%
J_{mj}\left\langle \hat{S}_{k}^{1}\right\rangle -\frac{\varepsilon _{jmk}}{4}%
J_{im}\left\langle \hat{S}_{k}^{2}\right\rangle ,  \nonumber
\end{eqnarray}
where in the last equation anticommutation properties of the Clifford
algebra (\ref{Clifford}) were used.

If in Eq. (\ref{rho}) spin components outside expectation values are
promoted to the components $S_{i}^{1},S_{i}^{2}$ of ordinary real vectors $%
\vec{S}^{1},\vec{S}^{2}$, then for the generic traceless observable one gets 
\begin{equation}
\left\langle c_{i}^{\alpha }\hat{S}_{i}^{\alpha }+c_{ij}\hat{S}_{i}^{1}%
\hat{S}_{j}^{2}\right\rangle =\frac{\int\limits_{r_{1}\leq \left| \vec{S}%
^{1}\right| \leq R_{1}}d\vec{S}^{1}\int\limits_{r_{2}\leq \left| \vec{S}%
^{1}\right| \leq R_{2}}d\vec{S}^{2}\left( \frac{3c_{i}^{\alpha
}S_{i}^{\alpha }}{4\left| \vec{S}^{\alpha }\right| ^{2}}+\frac{%
9c_{ij}S_{i}^{1}S_{j}^{2}}{16}\right) \rho \left( \vec{S}^{1},\vec{S}%
^{2}\right) }{\int\limits_{r_{1}\leq \left| \vec{S}^{1}\right| \leq R_{1}}d%
\vec{S}^{1}\int\limits_{r_{2}\leq \left| \vec{S}^{1}\right| \leq R_{2}}d\vec{%
S}^{2}\rho \left( \vec{S}^{1},\vec{S}^{2}\right) }
\end{equation}
for arbitrary values of $r_{1},R_{1},r_{2},R_{2}$, by which $\rho $ can be
interpreted as an unnormalizable non-positive probability density evolving
according to the master equation 
\begin{equation}
\frac{\partial \rho }{\partial t}=\frac{d\left\langle \hat{S}_{i}^{\alpha
}\right\rangle }{dt}S_{i}^{\alpha }+4\frac{d\left\langle \hat{S}_{i}^{1}%
\hat{S}_{j}^{2}\right\rangle }{dt}S_{i}^{1}S_{j}^{2}.  \label{master0}
\end{equation}
Here and henceforth only polynomial probability densities obtained from Eq. (%
\ref{rho}) are considered and called quantum states. Using Eq.s (\ref
{Heisenberg}), Eq. (\ref{master0}) is seen to be equivalent to 
\begin{eqnarray}
\frac{\partial \rho }{\partial t} &=&\left( \vec{B}^{\alpha }\wedge \vec{S}%
^{\alpha }\right) \cdot \frac{\partial \rho }{\partial \vec{S}^{\alpha }}%
-\left( \vec{S}^{1}\wedge \hat{J}\vec{S}^{2}\right) \cdot \frac{\partial
\rho }{\partial \vec{S}^{1}}-\left( \vec{S}^{2}\wedge \hat{J}^{T}\vec{S}%
^{1}\right) \cdot \frac{\partial \rho }{\partial \vec{S}^{2}}  \nonumber \\
&&+\frac{1}{4}\vec{S}^{1}\wedge \frac{\partial }{\partial \vec{S}^{1}}\cdot 
\hat{J}\frac{\partial \rho }{\partial \vec{S}^{2}}+\frac{1}{4}\vec{S}%
^{2}\wedge \frac{\partial }{\partial \vec{S}^{2}}\cdot \hat{J}^{T}\frac{%
\partial \rho }{\partial \vec{S}^{1}}  \label{master} \\
&&+\left( \vec{S}^{2}\wedge \hat{J}^{T}\vec{S}^{1}\right) \cdot \frac{%
\partial }{\partial \vec{S}^{2}}\vec{S}^{1}\cdot \frac{\partial \rho }{%
\partial \vec{S}^{1}}+\left( \vec{S}^{1}\wedge \hat{J}\vec{S}^{2}\right)
\cdot \frac{\partial }{\partial \vec{S}^{1}}\vec{S}^{2}\cdot \frac{\partial
\rho }{\partial \vec{S}^{2}},  \nonumber
\end{eqnarray}
where standard vector notation was used. While the first three terms above
give the Liouville equation for two interacting classical tops, the
following ones cannot be interpreted as resulting from a deterministic
dynamics in the space of $\vec{S}^{1}$ and $\vec{S}^{2}$ since they contain
second partial derivatives. In the following it is shown that the above
equation can be interpreted as the master equation of a generalized
diffusion process in this space.

The generalization implied consists in considering a statistical ensemble of
equivalent systems each one endowed with a nonpositive real statistical
weight $w$ obeying an ODE 
\begin{equation}
dw=h(\vec{z},t)wdt
\end{equation}
and whose states are represented by coordinate vectors $\vec{z}$ obeying a
Langevin equation 
\begin{equation}
d\vec{z}=\vec{f}(\vec{z},t)dt+\sum_{j}\vec{g}_{j}(\vec{z},t)d\omega _{j},
\end{equation}
where the gaussian random variables $d\omega _{i}$ have zero mean $\overline{%
d\omega _{i}}=0$ and covariances defined by $\overline{\dot{\omega}_{i}(t)%
\dot{\omega}_{j}(t^{\prime })}$ $=\delta _{ij}\delta (t-t^{\prime })$, or
less naively by 
\[
\overline{\left[ \omega _{i}(t+dt)-\omega _{i}(t)\right] \left[ \omega
_{j}(t^{\prime }+dt^{\prime })-\omega _{j}(t^{\prime })\right] }%
=\int_{t}^{t+dt}d\tilde{t}\,\int_{t^{\prime }}^{t^{\prime }+dt^{\prime }}d%
\tilde{t}^{\prime }\;\delta (\tilde{t}-\tilde{t}^{\prime }). 
\]

A statistical ensemble $(\vec{z},w_{i})_{i=1,2,..N}$ defines a nonpositive
probability density $\rho (\vec{z},t)$ whose averages coincide with the
weighted ensemble averages 
\begin{equation}
\rho (\vec{z},t)=\frac{1}{N}\sum_{i=1}^{N}w_{i}\delta \left( \vec{z}-\vec{z}%
_{i}(t)\right) \Rightarrow \int d\vec{z}\rho k(\vec{z})=\frac{1}{N}%
\sum_{i=1}^{N}w_{i}k(\vec{z}_{i})
\end{equation}
and whose time evolution contains one term more than usual: 
\begin{equation}
d\rho =-\frac{1}{N}\sum_{i=1}^{N}w_{i}d\vec{z}_{i}\cdot \frac{\partial }{%
\partial \vec{z}}\delta \left( \vec{z}-\vec{z}_{i}(t)\right) +\frac{1}{N}%
\sum_{i=1}^{N}\delta \left( \vec{z}-\vec{z}_{i}(t)\right) dw_{i}=-\frac{%
\partial }{\partial \vec{z}}\cdot \left( d\vec{z}\rho \right) +h\rho dt.
\end{equation}

Starting from this equation one can obtain the generalized Fokker-Planck
(F-P) equation: 
\begin{equation}
\frac{\partial \rho }{\partial t}=-\frac{\partial }{\partial \vec{z}}\cdot
\left( \vec{f}(\vec{z},t)\rho \right) +\frac{1}{2}\sum_{j}\frac{\partial }{%
\partial \vec{z}}\otimes \frac{\partial }{\partial \vec{z}}\cdot \left[ \vec{%
g}_{j}(\vec{z},t)\otimes \vec{g}_{j}(\vec{z},t)\rho \right] +h(\vec{z})\rho .
\label{genFP}
\end{equation}

Of course one could even adopt a more traditional, though less insightful
viewpoint, by treating $w$ as a further dynamical variable and introducing a
positive probability density 
\begin{equation}
\rho _{P}(\vec{z},w,t)=\frac{1}{N}\sum_{i=1}^{N}\delta \left( \vec{z}-\vec{z}%
_{i}(t)\right) \delta \left( w-w_{i}(t)\right) \Rightarrow \rho (\vec{z}%
,t)=\int_{-\infty }^{+\infty }w\rho _{P}(\vec{z},w,t)dw,
\end{equation}
in which case the additional term in the generalized F-P equation is simply
the transcription of a term in the ordinary F-P equation for $\rho _{P}(\vec{%
z},w,t)$: 
\begin{equation}
-\int_{-\infty }^{+\infty }w\frac{\partial }{\partial w}\left[ h(\vec{z}%
,t)w\rho _{P}(\vec{z},w,t)\right] dw=\int_{-\infty }^{+\infty }h(\vec{z}%
,t)w\rho _{P}(\vec{z},w,t)dw=h(\vec{z},t)\rho (\vec{z},t),
\end{equation}
where it was assumed that $w^{2}\rho _{P}(\vec{z},w,t)\rightarrow 0$ as $%
w\rightarrow \pm \infty $.

While this alternative setting leads to the usual form of the F-P equation,
the price to pay consists, when evaluating averages, in replacing the
generic dynamical variable $k(\vec{z})$ with the product $wk(\vec{z})$,
which is of course the remnant of the nonpositive character of the original
probability density and then the source of the possible violation of Bell's
inequalities.

To be specific, if one defines the vector fields 
\begin{equation}
\vec{F}^{1}\equiv \left( \vec{S}^{1}\wedge \hat{J}\frac{\vec{S}^{2}}{\left| 
\vec{S}^{2}\right| }\right) \wedge \hat{J}\frac{\vec{S}^{2}}{\left| \vec{S}%
^{2}\right| },\;\vec{F}^{2}\equiv \left( \vec{S}^{2}\wedge \hat{J}^{T}\frac{%
\vec{S}^{1}}{\left| \vec{S}^{1}\right| }\right) \wedge \hat{J}^{T}\frac{\vec{%
S}^{1}}{\left| \vec{S}^{1}\right| },
\end{equation}
one can show that Eq.(\ref{master}) is the F-P equation of the system of
stochastic differential equations 
\begin{eqnarray}
d\vec{S}^{1} &=&\left[ -\vec{B}^{1}\wedge \vec{S}^{1}+5\vec{S}^{1}\wedge 
\hat{J}\vec{S}^{2}+\vec{F}^{1}+6\left| \vec{S}^{1}\right| ^{2}\vec{S}^{1}-%
\frac{1}{2}\vec{S}^{1}\right] dt  \nonumber \\
&&+\frac{d\vec{\eta}^{1}}{2}\wedge \vec{S}^{1}+\hat{J}\frac{d\vec{\eta}^{2}}{%
2}+d\xi _{1}\vec{S}^{1}\wedge \hat{J}\frac{\vec{S}^{2}}{\left| \vec{S}%
^{2}\right| }+d\xi _{2}\left| \vec{S}^{1}\right| \vec{S}^{1}  \label{st1} \\
d\vec{S}^{2} &=&\left[ -\vec{B}^{2}\wedge \vec{S}^{2}+5\vec{S}^{2}\wedge 
\hat{J}^{T}\vec{S}^{1}+\vec{F}^{2}+6\left| \vec{S}^{2}\right| ^{2}\vec{S}%
^{2}-\frac{1}{2}\vec{S}^{2}\right] dt  \nonumber \\
&&+\frac{d\vec{\eta}^{2}}{2}\wedge \vec{S}^{2}+\hat{J}^{T}\frac{d\vec{\eta}%
^{1}}{2}+d\xi _{2}\vec{S}^{2}\wedge \hat{J}^{T}\frac{\vec{S}^{1}}{\left| 
\vec{S}^{1}\right| }+d\xi _{1}\left| \vec{S}^{2}\right| \vec{S}^{2}
\label{st2} \\
dw &=&w\left[ \frac{1}{2}\left( \frac{\partial }{\partial \vec{S}^{1}}\cdot 
\vec{F}^{1}+\frac{\partial }{\partial \vec{S}^{2}}\cdot \vec{F}^{2}\right)
+15\left( \left| \vec{S}^{1}\right| ^{2}+\left| \vec{S}^{2}\right|
^{2}\right) -\frac{3}{2}\right] dt,  \label{stw}
\end{eqnarray}
where  $\vec{\eta}^{1},\vec{\eta}^{2},\xi _{1},\xi _{2}$ are zero mean
independent gaussian $\delta $-correlated random variables: 
\begin{equation}
\;\;\overline{\dot{\eta}_{i}^{\alpha }(t)\dot{\eta}_{j}^{\beta }(t^{\prime })%
}=\delta _{\alpha \beta }\delta _{ij}\delta (t-t^{\prime }),\;\overline{\dot{%
\xi}_{\alpha }(t)\dot{\xi}_{\beta }(t^{\prime })}=\delta _{\alpha \beta
}\delta (t-t^{\prime }),\;\overline{\dot{\eta}_{i}^{\alpha }(t)\dot{\xi}%
_{\beta }(t^{\prime })}=0
\end{equation}
The contribution to the F-P equation due to terms in $\vec{\eta}^{1}$, apart
from operators vanishing on quantum states and before averaging on $\vec{\eta%
}^{1}$, reads 
\[
\frac{1}{dt}\left[ \frac{1}{8}\frac{\partial }{\partial \vec{S}^{1}}\cdot
\left( d\vec{\eta}^{1}\wedge \vec{S}^{1}\right) \frac{\leftarrow \partial
\rightarrow }{\partial \vec{S}^{1}}\cdot \left( d\vec{\eta}^{1}\wedge \vec{S}%
^{1}\right) +\frac{1}{4}\frac{\partial }{\partial \vec{S}^{1}}\cdot \left( d%
\vec{\eta}^{1}\wedge \vec{S}^{1}\right) d\vec{\eta}^{1}\cdot \hat{J}\frac{%
\partial }{\partial \vec{S}^{2}}\right] .
\]
where here and henceforth the symbol $\leftarrow \partial \rightarrow $
denotes that the corresponding derivative operator applies \ even to the
left, and after averaging it reads 
\[
-\frac{1}{4}\vec{S}^{1}\cdot \frac{\partial }{\partial \vec{S}^{1}}-\frac{1}{%
4}\frac{\partial }{\partial \vec{S}^{1}}\cdot \vec{S}^{1}+\frac{1}{4}\left( 
\vec{S}^{1}\wedge \frac{\partial }{\partial \vec{S}^{1}}\right) \cdot \hat{J}%
\frac{\partial }{\partial \vec{S}^{2}}.
\]
Of course analogous relations hold true for terms in $\vec{\eta}^{2}$.

As to the terms in $\xi ^{1}$ and $\xi ^{2}$, after averaging they give rise
respectively to 
\begin{eqnarray*}
&&\frac{\partial }{\partial \vec{S}^{1}}\cdot \left( \vec{S}^{1}\wedge \hat{J%
}\frac{\vec{S}^{2}}{\left| \vec{S}^{2}\right| }\right) \frac{\leftarrow
\partial \rightarrow }{\partial \vec{S}^{2}}\cdot \left| \vec{S}^{2}\right| 
\vec{S}^{2}+\frac{1}{2}\frac{\partial }{\partial \vec{S}^{2}}\cdot \left| 
\vec{S}^{2}\right| \vec{S}^{2}\frac{\leftarrow \partial \rightarrow }{%
\partial \vec{S}^{2}}\cdot \left| \vec{S}^{2}\right| \vec{S}^{2} \\
&&+\frac{1}{2}\frac{\partial }{\partial \vec{S}^{1}}\cdot \left( \vec{S}%
^{1}\wedge \hat{J}\frac{\vec{S}^{2}}{\left| \vec{S}^{2}\right| }\right) 
\frac{\leftarrow \partial \rightarrow }{\partial \vec{S}^{1}}\cdot \left( 
\vec{S}^{1}\wedge \hat{J}\frac{\vec{S}^{2}}{\left| \vec{S}^{2}\right| }%
\right) \\
&=&4\frac{\partial }{\partial \vec{S}^{1}}\cdot \left( \vec{S}^{1}\wedge 
\hat{J}\vec{S}^{2}\right) +\left( \vec{S}^{1}\wedge \hat{J}\vec{S}%
^{2}\right) \cdot \frac{\partial }{\partial \vec{S}^{1}}\vec{S}^{2}\cdot 
\frac{\partial }{\partial \vec{S}^{2}}+ \\
&&+\frac{\partial }{\partial \vec{S}^{1}}\cdot \frac{\vec{F}^{1}}{2}+\frac{%
\vec{F}^{1}}{2}\cdot \frac{\partial }{\partial \vec{S}^{1}}+3\frac{\partial 
}{\partial \vec{S}^{2}}\cdot \left| \vec{S}^{2}\right| ^{2}\vec{S}%
^{2}+3\left| \vec{S}^{2}\right| ^{2}\vec{S}^{2}\cdot \frac{\partial }{%
\partial \vec{S}^{2}}
\end{eqnarray*}
for $\xi ^{1}$ and an analogous expression for $\xi ^{2}$.

Adding these terms to the Liouville operator of the deterministic terms in
Eq.s (\ref{st1},\ref{st2}) and to the ones generated by Eq. (\ref{stw}) for
the statistical weight according to Eq. (\ref{genFP}), one gets the proof of
the equivalence between the stochastic differential equations (\ref{st1},\ref
{st2},\ref{stw}) and the F-P Eq. (\ref{master}).

Of course the present result and its proof hold basically unchanged for
systems with more than two qubits, provided that there are only pair
interactions. To be specific, consider, in its proof, what turns out when
taking account of the time evolution due to a particular two qubit
interaction. The crucial point is that, if $\rho $ is an arbitrary
polynomial at most linear in every spin vector, the $\hat{J}$ dependent part
of the right hand side of Eq. (\ref{master}) coincides with what is got
considering spin components as operators and taking the corresponding
commutator appearing in the von Neumann equation. Then in general to obtain
the master equation one simply has to introduce operators of the same form
as in Eq. (\ref{master}) for each pair interaction, which obviously leads to
an immediate generalization of equations (\ref{st1},\ref{st2},\ref{stw}).

Some comments are in order to the numerical implementation of the
quantization method. Once fixed an initial quantum state, one takes an
ensemble sampling the corresponding probability density for $\left| \vec{S}%
^{\alpha }\right| \leq M$ where $M$ is properly chosen. Than the numerical
simulation will work for $\left| \vec{S}^{\alpha }\right| \leq m<M$ \ until
the effect of the lacking sampling for $\left| \vec{S}^{\alpha }\right| >M$
will be felt at $\left| \vec{S}^{\alpha }\right| \leq m$. Of course the
reliable simulation time will be the longer the greater $M$ and the lower $m$
are chosen. In other words finite statistical ensembles undergo an aging
effect. So that if one wants to simulate for longer times, one has to reset
them periodically. In doing this at each step a fraction of the statistical
information encoded is lost.

A thorough numerical analysis is needed to assess the real power of the
method as the number of qubits grows. A way out of the seeming contradiction
between the linear growth of the simulation space in the present
quantization method and the exponential growth, with the number of qubits,
of the computational resources needed for simulation of quantum systems\cite
{Feynman1} in the author opinion is to be found precisely in this aging
effect. An indication in this direction comes from the observation that the
logarithmic derivative of the statistical weight is proportional to the
number of interaction terms, which is to be interpreted as a growing
compensating effect for the depopulation and overpopulation effect of time
evolution in different regions of the $\vec{S}^{\alpha }$ space as the
number of pair interactions grows. This would produce an aging effect even
for an hypothetical infinite statistical ensemble sampling the whole space,
as can be guessed for instance for the totally disordered state, whose
density matrix is proportional to the identity matrix. While the probability
density is obviously constant for this state, this is not the case for the
statistical ensemble. In particular the ultimate fate of every stochastic
trajectory is to run away to infinity, due to the cubic deterministic term.
While this term can be eliminated with alternative forms of the stochastic
differential equations, \ probably the present one is the most convenient as
to the behavior close to the origin.

It is relevant to remark that the aging of the statistical ensembles
reconciles another apparent contradiction inherent in the use of a typically
irreversible diffusion process to describe reversible unitary quantum
evolution.

It should also be stressed that, although spin notation in this paper played
in general only a formal role, the presented result applies to spin systems
like arbitrarily disordered anisotropic Heisenberg systems. In the case of
long range interactions of course the quantization method is numerically
less effective due to the quadratic growth of the number of stochastic terms.

Finally it should be remarked that the present setting has a natural
generalization to quantum open systems in the Markoff approximation in terms
of additional stochastic terms. One can naturally introduce a fluctuating
part in the $\vec{B}^{\alpha }$ vectors, which simply amounts to the
addition of further terms in the F-P equation.

Acknowledgments -- This paper was supported by M.U.R.S.T., Italy and
I.N.F.M., Salerno.

\begin{references}
\bibitem[*]{byline}  Phone: +39 89 965229; FAX: +39 89 965275; electronic
address: defilippo@sa.infn.it

\bibitem{Shor}  P.W. Shor, in {\it Proceedings of the 35th Annual Symposium
on Foundations of Computer Science}, edited by S. Goldwasser (IEEE Computer
Society, Los Alamitos, CA, 1994), p. 124.

\bibitem{Steane}  A.M. Steane, Rep. Prog. Theor. Phys. {\bf 61}, 117 (1998),
references therein.

\bibitem{Preskil}  J. Preskill, {\it A Course on Q. C.},
http://www.theory.caltech.edu/\symbol{126}preskill/ph229

\bibitem{Feynman1}  R. Feynman, Int. J. Theor. Phys. {\bf 21}, 467 (1982).

\bibitem{Jozsa}  R. Jozsa, Lect. Notes in Comp. Science {\bf 1509}, 103
(1999).

\bibitem{Feynman2}  R. Feynman, in {\it Quantum Implications}, edited by
B.J. Hiley and F.D. Peats (Routledge \& Kegan, London 1987), p. 235.

\bibitem{Bell}  J.S. Bell, Physics {\bf 1},195 (1964).

\bibitem{Moyal}  J.E. Moyal, Proc. Camb. Phyl. Soc. {\bf 45},\ 99 (1949).

\bibitem{Wigner}  E. Wigner, Phys. Rev. {\bf 40}, 749 (1932).

\bibitem{CoryFahmy}  D.G. Cory, A.F. Fahmy, T.F. Havel, Proc. Natl. Acad.
Sci. {\bf 94},1634 (1997).

\bibitem{Gershenfeld}  N.A. Gershenfeld, I.L. Chuang, Science {\bf 275}, 350
(1997).

\bibitem{Parisi}  G. Parisi and Wu Yong Shi, Sci. Sin. {\bf 24}, 483 (1981).

\bibitem{Nelson}  E. Nelson, {\it Quantum Fluctuations}, Princeton Univ.
Press, Princeton NJ (1985).\newpage 
\end{references}

\end{document}

quant-ph0002018.extracted_bib
\bibitem[*]{byline}  Phone: +39 89 965229; FAX: +39 89 965275; electronic
address: defilippo@sa.infn.it


\bibitem{Shor}  P.W. Shor, in {\it Proceedings of the 35th Annual Symposium
on Foundations of Computer Science}, edited by S. Goldwasser (IEEE Computer
Society, Los Alamitos, CA, 1994), p. 124.


\bibitem{Steane}  A.M. Steane, Rep. Prog. Theor. Phys. {\bf 61}, 117 (1998),
references therein.


\bibitem{Preskil}  J. Preskill, {\it A Course on Q. C.},
http://www.theory.caltech.edu/\symbol{126}preskill/ph229


\bibitem{Feynman1}  R. Feynman, Int. J. Theor. Phys. {\bf 21}, 467 (1982).


\bibitem{Jozsa}  R. Jozsa, Lect. Notes in Comp. Science {\bf 1509}, 103
(1999).


\bibitem{Feynman2}  R. Feynman, in {\it Quantum Implications}, edited by
B.J. Hiley and F.D. Peats (Routledge \& Kegan, London 1987), p. 235.


\bibitem{Bell}  J.S. Bell, Physics {\bf 1},195 (1964).


\bibitem{Moyal}  J.E. Moyal, Proc. Camb. Phyl. Soc. {\bf 45},\ 99 (1949).


\bibitem{Wigner}  E. Wigner, Phys. Rev. {\bf 40}, 749 (1932).


\bibitem{CoryFahmy}  D.G. Cory, A.F. Fahmy, T.F. Havel, Proc. Natl. Acad.
Sci. {\bf 94},1634 (1997).


\bibitem{Gershenfeld}  N.A. Gershenfeld, I.L. Chuang, Science {\bf 275}, 350
(1997).


\bibitem{Parisi}  G. Parisi and Wu Yong Shi, Sci. Sin. {\bf 24}, 483 (1981).


\bibitem{Nelson}  E. Nelson, {\it Quantum Fluctuations}, Princeton Univ.
Press, Princeton NJ (1985).\newpage 
