Designs via Free Probability

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 relations 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>k</a:mi></a:math> designs through free probability theory. First, by introducing the more general notion of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>k</c:mi></c:math>-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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>k</e:mi></e:math>-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.