Decoupling with random diagonal unitaries

Yoshifumi Nakata1,2,3, Christoph Hirche1,3, Ciara Morgan1,4, and Andreas Winter3,5

1Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany.
2Photon Science Center, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan.
3Departament de Física: Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.
4School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4. Ireland.
5ICREA–Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, ES-08010 Barcelona, Spain

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.

One of the most fundamental protocols in quantum information science is decoupling, aiming to destroy all correlations between two quantum systems by applying a unitary on one of the systems. The importance of decoupling lies not only in the fact that it is the key primitive in the mother protocol of quantum Shannon theory, implying that many other information processing protocols can be obtained by decoupling method, but also in its connection to fundamental physics of black holes and quantum thermodynamics.

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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