Observable-Projected Ensembles

Deep thermalisation is a novel paradigm which refines the notion of thermal equilibrium. We investigate the observable-projected ensemble, an ensemble of subsystem states conditioned on measuring an extensive bath observable, in order to study deep thermalisation. While the conventional projected ensemble forces us to perform a full projection onto the set of states in the bath, which are exponentially many in the bath size, the observable-projected ensemble provides an advantage by featuring the number of measurement outcomes only linear in the bath size.

We focus on two cases: a random Haar initial state and energy eigenstates of many-body quantum systems. We compute the k-th moment of this ensemble of mixed states and compare it to the Scrooge ensemble which often describes the conventional projected ensembles. We find that the k-th moment clusters around its expectation value, but that the expectation value can deviate from the Scrooge answer. We make use of canonical typicality and the many-body Berry conjecture (MBBC) to perform analytical computations and perform numerics to confirm our use of the MBBC.