High probability decoupling via approximate unitary designs and efficient relative thermalisation
Aditya Nema, Pranab Sen
备注：
Re-formatted the draft with new title, an easy to read description of the main theorem with explicit values of t.Elucidated comparison of our relative thermalisation theorem with the existing ones and application to Hayden-Preskill toy model for black hole information paradox.Corrected a calculation on moments
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.
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