Full expectation-value statistics for randomly sampled pure states in high-dimensional quantum systems

Area of Science:

• Quantum mechanics
• High-dimensional Hilbert spaces
• Statistical mechanics

Background:

• Understanding the distribution of expectation values is key in quantum mechanics.
• Random sampling from Hilbert spaces is a fundamental concept.

Purpose of the Study:

• To analyze the distribution of expectation values of arbitrary observables.
• To investigate deviations from Gaussian behavior in high dimensions.

Main Methods:

• Analytical calculations of expectation value distributions.
• Sampling normalized vectors from high-dimensional Hilbert spaces.
• Analysis of eigenvalue distributions satisfying Wigner's semicircle law.

Main Results:

• The distribution shows a narrow, approximately Gaussian peak.
• Significant deviations from Gaussian behavior are observed in the tails.
• For Wigner's semicircle law eigenvalues, a large deviation function reveals symmetric nonanalyticities.

Conclusions:

• The study provides a detailed characterization of expectation value distributions in quantum systems.
• The findings offer insights analogous to critical phenomena in thermodynamics.
• This work advances the understanding of random quantum states and observables.