Abstract
We explore how the expectation values of a largely arbitrary observable are distributed when normalized vectors are randomly sampled from a high-dimensional Hilbert space. Our analytical results predict that the distribution exhibits a very narrow peak of approximately Gaussian shape, while the tails significantly deviate from a Gaussian behavior. In the important special case that the eigenvalues of satisfy Wigner's semicircle law, the expectation-value distribution for asymptotically large dimensions is explicitly obtained in terms of a large deviation function, which exhibits two symmetric nonanalyticities akin to critical points in thermodynamics.
- Received 23 May 2018
- Revised 5 September 2018
DOI:https://doi.org/10.1103/PhysRevE.99.012126
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