5. Transition probabilities

Next: 3. The Five Platonic Up: 2. The algorithm Previous: 4. Jumps are implemented

5. Transition probabilities

Given the actual state ${\bf r}$ of the quantum system, and the configuration of the detectors $\{{\bf n}[i], i=1,2,\ldots ,N\}$ we compute probabilities $p[i], \sum_{i=1}^N p[i]=1$ for the

detector to flip. Then we select randomly, with the calculated probability distribution, the flipping detector, and we implement the jump by changing ${\bf r}$ to ${\bf r}^\prime$ according to the formula (

), with ${\bf n}={\bf n}[i]$ . The probabilities

are computed using the theory of piecewise deterministic Markov processes applied to the case of quantum measurements - as developed within EEQT.According to EEQT the probabilities

are given by the formula:

$\begin{displaymath} p[i]=const\cdot\mbox{Tr}\left(P({\bf r})P({\bf n}[i],\alpha)^2P({\bf r})\right) \end{displaymath}$

(16)

where

is the normalizing constant. Using cyclic permutation under the trace, as well as the fact that $P({\bf r} )^2=P({\bf r})$ we find, taking trace of both sides of the formula (

), that

are proportional to $\lambda(\alpha,{\bf n}[i],{\bf r})$ given by (

), thus

$\begin{displaymath} p[i]=\frac{1+\alpha^2+2\alpha ({\bf n}[i]\cdot {\bf r})}{N(1+\alpha^2)}. \end{displaymath}$

(17)

Note that, owing to the fact that $\sum_{k=1}^N {\bf n}[k]={\bf0}$ we have $\sum_{i=1}^N p[i]=1$ , as it should be.

Next: 3. The Five Platonic Up: 2. The algorithm Previous: 4. Jumps are implemented

2002-04-11