Decoupling with random diagonal unitaries

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.

26 pages, 3 figures. v2: 19 pages, 3 figures, both results and presentations are improved. One conjecture in the previous version was proven. v3: 16 pages, 1 figure. v4: doi links are added, published version

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• Efficient methods for one-shot quantum communication
• On the explicit constructions of certain unitary $t$-designs
• Decoupling by local random unitaries without simultaneous smoothing, and applications to multi-user quantum information tasks
• Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory
• Quantum State Merging for Arbitrarily Small-Dimensional Systems
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Decoding general error correcting codes and the role of complementarity
• Optimized synthesis of circuits for diagonal unitary matrices with reflection symmetry