  We prove a new concentration result for non-catalytic decoupling by showing
that, for suitably large $t$, applying a unitary chosen uniformly at random
from an approximate $t$-design on a quantum system followed by a fixed quantum
operation almost decouples, with high probability, the given system from
another reference system to which it may initially have been correlated.
Earlier works either did not obtain high decoupling probability, or used
provably inefficient unitaries, or required catalytic entanglement for
decoupling. In contrast, our approximate unitary designs always guarantee
decoupling with exponentially high probability and, under certain conditions,
lead to computationally efficient unitaries. As a result we conclude that,
under suitable conditions, efficiently implementable approximate unitary
designs achieve relative thermalisation in quantum thermodynamics with
exponentially high probability. We also show the scrambling property of black
hole, when the black hole evolution is according to pseudorandom approximate
unitary $t$-design, as opposed to the Haar random evolution considered earlier
by Hayden-Preskill.
