I Choosing the eigenvector stochastically

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Mentz114

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The equation ##\hat{O}|\psi\rangle \rightarrow \alpha_n|\mathbf{e}_n\rangle## where ##|\mathbf{e}_n\rangle## is an eigenvector of the operator and ##\alpha_n## is its eigenvalue, is central in the QT formalism. This is as much as we can get from quantum theory but an ideal instrument should enable us to estimate the ##p_n\equiv |\alpha_n|^2## from the observed frequencies of different outcomes.

When there are 2 eigenstates it seems straightforward to define a stochastic evolution operator thus ##\hat{D}=diag( e^{\lambda(p_0-s)t}, e^{-\lambda(p_0-s)t})##
where ##s## is a random variate (nearly) uniformly distributed in ##(0,1)##. ##\hat{D}## selects which eigenstate to grow and which to shrink with the proportions ##p## and ##1-p##. ##\hat{D}## evolves with time to having only one non-null eigenvector.
##\hat{D}## has to be considered part of the apparatus and the acquisition of ##s## could be from a random phase or from another special interaction perhaps with an internal state.

With more than 2 dimensions ##\hat{D} ## has the form

\begin{align}
\hat{D} &= \left[ \begin{array}{cccc}
e^{\lambda(p_0-s)t} & 0 & 0 & 0 \\\
0 & e^{-\lambda(p_0-s)(p_0+p_1-s)t} & 0 & 0 \\\
0 & 0 & e^{-\lambda(p_0+p_1-s)(p_0+p_1+p_2-s)t} & 0 \\\
0 & 0 & 0 & e^{\lambda(s-(p_0+p_2+p_3))t}
\end{array} \right]
\end{align}


For any ##s## only one exponent is positive.
 
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