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We prove a new concentration result for non-catalytic decoupling by showingthat, for suitably large $t$, applying a unitary chosen uniformly at randomfrom an approximate $t$-design on a quantum system followed by a fixed quantumoperation almost decouples, with high probability, the given system fromanother reference system to which it may initially have been correlated.Earlier works either did not obtain high decoupling probability, or usedprovably inefficient unitaries, or required catalytic entanglement fordecoupling. In contrast, our approximate unitary designs always guaranteedecoupling 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 unitarydesigns achieve relative thermalisation in quantum thermodynamics withexponentially high probability. We also show the scrambling property of blackhole, when the black hole evolution is according to pseudorandom approximateunitary $t$-design, as opposed to the Haar random evolution considered earlierby Hayden-Preskill.

High probability decoupling via approximate unitary designs and efficient relative thermalisation