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## Main Question or Discussion Point

Let suppose I have an observable ##A## with associated projection-valued measure ##\mu_A##

$$A = \int_{a \in \mathbb{R}} a \cdot \textrm{d}\mu_A(a)$$

for a system in the (possibly mixed) state ##\rho##. Let ##S \subset \mathbb{R}## be a measurable subset and let ##Z = \mu_A(S)## be the observable equating 1 if ##A## falls in ##S## and 0 otherwise.

Is this statement meaningful and correct:

Again is this statement meaningful and correct:

$$A = \int_{a \in \mathbb{R}} a \cdot \textrm{d}\mu_A(a)$$

for a system in the (possibly mixed) state ##\rho##. Let ##S \subset \mathbb{R}## be a measurable subset and let ##Z = \mu_A(S)## be the observable equating 1 if ##A## falls in ##S## and 0 otherwise.

Is this statement meaningful and correct:

*If measuring ##A##, with probability ##\textrm{tr}( \rho \cdot \mu_A(S) )## the result will be a point in ##S## (let's call it ##a##) and the system will collapse to state ##\rho' = \mu_A(\{a\})##*

Again is this statement meaningful and correct:

*If measuring ##Z##, with probability ##\textrm{tr}( \rho \cdot \mu_A(S) )## the result will be 1 and the system will collapse to state $$\rho' = \frac{1}{\textrm{tr} (\rho \cdot \mu_A(S))} \cdot \int_{a \in S} \textrm{tr} ( \rho \cdot \mu_A(\{a\}) ) \cdot \textrm{d} \mu_A(a) $$ with any subsequent measurement of ##A## producing a value inside ##S##*