Unitary designs for continuous variable systems

The study of information scrambling in many-body systems has sharpened our understanding of quantum chaos. In discrete variable (DV) systems (finite-dimensional, e.g. spins), the scrambling ‘strength’ of a unitary is often measured by its closeness to a Haar random unitary. This leads to a hierarchy of increasingly fine-grained measures of scrambling known as ‘unitary k-designs’. Here, we extend the notion of unitary designs to continuous variable (CV) systems (infinite-dimensional, e.g. photons). Although there is no generalization of Haar randomness to CV systems, we show that averages of physical quantities over Haar random unitaries remain well-defined in the CV limit, and use this to define CV unitary designs. Surprisingly, Gaussian unitaries, despite being non-interacting, form a CV 2-design and can therefore `quasi-scramble' information. Extending further, we show that unitary 4-designs maximize the phase space volume of generic time-evolved operators.