QUANTUM ESTIMATION THEORY AND OPTICAL DETECTION

In quantum mechanics we call “observable” any physical quantity that can be represented by numbers. An observable is associated in a one-to-one way with a selfadjoint operator \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} \) acting on the Hilbert space H S of the quantum system S, and the spectrum of \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} \) represents the set of all possible readings from the measurement. Let us consider, for example, an observable with spectrum equal to whole real line R, and with spectral decomposition $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X} = \int {xd\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{E} (x)} $$ (1) .

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