Designs via Free Probability

Abstract

Unitary Designs have become a vital tool for investigating pseudorandomness since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relation to quantum chaotic evolution and in particular to the Eigenstate Thermalization Hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and $k$-designs through Free Probability theory. First, by introducing the more general notion of $k$-freeness, we show that it can be used as an alternative probe to designs. In turn, free probability theory comes with several tools, useful for instance for the calculation of mixed moments or the so-called $k$-fold quantum channels. Our second result is the connection to quantum dynamics. Quantum ergodicity, and correspondingly ETH, apply to a restricted class of physical observables, as already discussed in the literature. In this spirit, we show that unitary evolution with generic Hamiltonians always leads to freeness at sufficiently long times, but only when the operators considered are restricted within the ETH class. Our results provide a direct link between unitary designs, quantum chaos and the Eigenstate Thermalization Hypothesis, and shed new light on the universality of late-time quantum dynamics.

17 pages, 4 figures; expanded discussion on the k-fold Haar channel, its decomposition in terms of free cumulants and ensembles of states

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• Quantum Algorithms for Inverse Participation Ratio Estimation in multi-qubit and multi-qudit systems
• Fermionic Magic Resources of Quantum Many-Body Systems
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