From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Deterministic classical dynamical systems have an ergodic hierarchy, from through mixing, to Bernoulli that are "as random as a coin-toss". Dual-unitary circuits been recently introduced solvable models of many-body nonintegrable quantum chaotic having hierarchy properties. We extend this include the apex putative which is Bernoulli, in sense correlations single and two-particle observables vanish at space-time separated points. derive condition based on entangling power $e_p(U)$ basic unitary building block, $U$, circuit, guarantees when maximized, corresponds circuits. Additionally we show, both analytically numerically, how local-averaging over realizations single-particle unitaries, $u_i$ $v_i$ such block $U^\prime = (u_1 \otimes u_2 ) U (v_1 v_2 )$ leads identification average mixing rate being determined predominantly by $e_p(U)$. Finally provide several, analytical numerical, ways construct dual-unitary operators covering entire possible range power. coupled cat map for all local dimensions 2-unitary or perfect tensor odd dimensions, can be used build