We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recently shown to achieve an approximate unitary $2$-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary $2$-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.
Quantum pseudorandomness, approximations of a uniformly distributed random unitary, is one of the random unitaries most commonly used in decoupling. Although it is known that quantum pseudorandomness achieves decoupling if the approximation is sufficiently precise, it has remained open whether such a precision is necessary or not. In this paper, we show for the first time that quantum pseudorandomness with rather imprecise approximations achieves decoupling as strongly as a uniformly distributed random unitary does, opening the possibility to realise decoupling with more efficient random unitaries. As our construction is based on spin-glass-type interactions, our result also has implications that decoupling may be spontaneously achieved in certain types of physically natural many-body systems.
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