Decoupling with random diagonal unitaries

TL;DR

This paper demonstrates that using repeated applications of random diagonal unitaries in specific bases can achieve quantum decoupling efficiently, matching the performance of more complex uniform random unitaries, with implications for quantum information protocols.

Contribution

It proves that fewer repetitions of diagonal unitaries suffice for decoupling, showing that approximate unitary 2-designs are adequate, simplifying implementation in quantum protocols.

Findings

Fewer repetitions of diagonal unitaries achieve decoupling at the same rate as uniform unitaries.

Imprecise approximations of unitary 2-designs are sufficient for effective decoupling.

Discusses implications for quantum state merging and thermalisation.

Abstract

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.