3. Fuzzy projections

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3. Fuzzy projections

For each ${\bf n}$ let $P({\bf n},\alpha )$ be the fuzzy projection opearator defined by the formula:

$\begin{displaymath} P({\bf n},\alpha )=\frac{1}{2}(I+\alpha\sigma ({\bf n})) \end{displaymath}$

(11)

where $0\leq\alpha\leq 1$ ), or better: $1-\alpha ,$ is a parameter that measures the "fuzziness." The extreme cases are not the very interesting ones: for $\alpha=0$ we get the identity operator - maximal fuzziness and no information whatsoever, while for $\alpha=1$ we get the sharp projection $P({\bf n})= P({\bf n},\alpha=1 ).$

We restrict the range of the parameter $\alpha$ to the interval because only in this range $P({\bf n},\alpha )$ is a positive operator. It is easy to see that this is so. Indeed, a Hermitian matrix is positive when its eigenvalues are positive, and the eigenvalues of $P({\bf n},\alpha )$ are $(1\pm \alpha)/2 ,$ thus $-1\leq \alpha\leq 1 .$ On the other hand negative $\alpha$ for ${\bf n}$ is the same as positive $\alpha$ for $-{\bf n},$ thus we restrict the range of $\alpha$ to

It is the operators $P({\bf n},\alpha )$ that will act on quantum states to implement "quantum jumps" whenever detectors "flip."

The overall coefficient in the definition (), chosen to be $\frac{1}{2}$ here, is not important because in applications each of the operators $P({\bf n},\alpha )$ is multiplied by a coupling constant, and, in our case, when we are not interested in timing of the jumps, the value of the coupling constant plays no role.

Next: 4. Jumps are implemented Up: 2. The algorithm Previous: 2. Spin directions

2002-04-11