Lorenzo Campos Venuti, Paolo Zanardi
We consider the quantum expectation value $\mathcal{A}=<\psi|A|\psi>$ of an
observable A over the state $|\psi>$. We derive the exact probability
distribution of $\mathcal{A}$ seen as a random variable when $|\psi>$ varies
over the set of all pure states equipped with the Haar-induced measure. The
probability density is obtained with elementary means by computing its
characteristic function, both for non-degenerate and degenerate observables. To
illustrate our results we compare the exact predictions for one-dimensional
projectors with the concentration bounds obtained using Levy's lemma. Finally
we comment on the relevance of the central limit theorem.
View original:
http://arxiv.org/abs/1202.4810
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