# 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

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### 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.

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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### Cited by

[1] Hayata Yamasaki and Mio Murao, "Quantum State Merging for Arbitrarily Small-Dimensional Systems", IEEE Transactions on Information Theory 65 6, 3950 (2019).

[2] Pedro Figueroa–Romero, Felix A. Pollock, and Kavan Modi, "Markovianization with approximate unitary designs", Communications Physics 4 1, 127 (2021).

[3] Yoshifumi Nakata, Eyuri Wakakuwa, and Hayata Yamasaki, "One-shot quantum error correction of classical and quantum information", Physical Review A 104 1, 012408 (2021).

[4] Anurag Anshu and Rahul Jain, "Efficient methods for one-shot quantum communication", npj Quantum Information 8 1, 97 (2022).

[5] Eiichi Bannai, Mikio Nakahara, Da Zhao, and Yan Zhu, "On the explicit constructions of certain unitaryt-designs", Journal of Physics A: Mathematical and Theoretical 52 49, 495301 (2019).

[6] Sreeram PG and Vaibhav Madhok, "Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory", Physical Review A 104 3, 032404 (2021).

[7] Yoshifumi Nakata, Christoph Hirche, Masato Koashi, and Andreas Winter, "Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics", Physical Review X 7 2, 021006 (2017).

[8] Uttam Singh, Lin Zhang, and Arun Kumar Pati, "Average coherence and its typicality for random pure states", Physical Review A 93 3, 032125 (2016).

[9] Yoshifumi Nakata, Christoph Hirche, Ciara Morgan, and Andreas Winter, "Unitary 2-designs from random X- and Z-diagonal unitaries", Journal of Mathematical Physics 58 5, 052203 (2017).

[10] Lin Zhang, Uttam Singh, and Arun K. Pati, "Average subentropy, coherence and entanglement of random mixed quantum states", arXiv:1510.08859, Annals of Physics 377, 125 (2017).

[11] Lídia del Rio, Adrian Hutter, Renato Renner, and Stephanie Wehner, "Relative thermalization", Physical Review E 94 2, 022104 (2016).

[12] Eyuri Wakakuwa and Yoshifumi Nakata, "One-Shot Randomized and Nonrandomized Partial Decoupling", Communications in Mathematical Physics 386 2, 589 (2021).

[13] Kerstin Beer and Friederike Anna Dziemba, "Phase-context decomposition of diagonal unitaries for higher-dimensional systems", Physical Review A 93 5, 052333 (2016).

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