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.
 A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter. The mother of all protocols : Restructuring quantum information's family tree. Proc. R. Soc. A, 465: 2537, 2009. 10.1098/rspa.2009.0202.
 F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki. Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys., 346 (2): 397–434, September 2016. 10.1007/s00220-016-2706-8.
 W. G. Brown, Y. S. Weinstein, and L. Viola. Quantum pseudorandomness from cluster-state quantum computation. Phys. Rev. A, 77 (4): 040303(R), 2008. 10.1103/PhysRevA.77.040303.
 R. Cleve, D. Leung, L. Liu, and C. Wang. Near-linear constructions of exact unitary 2-designs. Quant. Info. & Comp., 16 (9 & 10): 0721–0756, 2016.
 O. C. O. Dahlsten, R. Oliveira, and M. B. Plenio. The emergence of typical entanglement in two-party random processes. J. Phys. A: Math. Theor., 40 (28): 8081, 2007. 10.1088/1751-8113/40/28/S16.
 C. Dankert, R. Cleve, J. Emerson, and E. Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A, 80: 012304, 2009. 10.1103/PhysRevA.80.012304.
 N. Datta and M.-H. Hsieh. The apex of the family tree of protocols : optimal rates and resource inequalities. New J. Phys., 13: 093042, 2011. 10.1088/1367-2630/13/9/093042.
 I. Devetak. The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory, 51 (1): 44–55, 2005. 10.1109/TIT.2004.839515.
 I. Devetak and A. Winter. Relating Quantum Privacy and Quantum Coherence: An Operational Approach. Phys. Rev. Lett., 93 (8): 080501, 2004. 10.1103/PhysRevLett.93.080501.
 I. T. Diniz and D. Jonathan. Comment on ``Random quantum circuits are approximate 2-designs". Commun. Math. Phys., 304: 281, 2011. 10.1007/s00220-011-1217-x.
 B. Groisman, S. Popescu, and A. Winter. Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A, 72 (3): 032317, 2005. 10.1103/PhysRevA.72.032317.
 D. Gross, K. Audenaert, and J. Eisert. Evenly distributed unitaries: On the structure of unitary designs. J. of Math. Phys., 48 (5): 052104, 2007. 10.1063/1.2716992.
 P. Hayden. Decoupling: A building block for quantum information theory. http://qip2011.quantumlah.org/images/QIPtutorial1.pdf, 2012. Accessed: 2017-3-30.
 P. Hayden and J. Preskill. Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys., 2007 (09): 120, 2007. 10.1088/1126-6708/2007/09/120.
 C. Hirche and C. Morgan. Efficient achievability for quantum protocols using decoupling theorems. In Proc. 2014 IEEE Int. Symp. Info. Theory, page 536, 2014. 10.1109/ISIT.2014.6874890.
 M. Horodecki, J. Oppenheim, and A. Winter. Quantum state merging and negative information. Comms. Math. Phys., 269: 107–136, 2007. 10.1007/s00220-006-0118-x.
 A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys., 3: 275, 1972. 10.1016/0034-4877(72)90011-0.
 M. Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society Providence, RI, USA, 2001.
 Y. Nakata and M. Murao. Diagonal-unitary 2-designs and their implementations by quantum circuits. Int. J. Quant. Inf., 11: 1350062, 2013. 10.1142/S0219749913500627.
 Y. Nakata and M. Murao. Diagonal quantum circuits: their computational power and applications. Eur. Phys. J. Plus, 129: 152, 2014. 10.1140/epjp/i2014-14152-9.
 Y. Nakata, M. Koashi, and M. Murao. Generating a state t-design by diagonal quantum circuits. New J. Phys., 16: 053043, 2014. 10.1088/1367-2630/16/5/053043.
 Y. Nakata, C. Hirche, M. Koashi, and A. Winter. Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics. Phys. Rev. X, 7 (2): 021006, 2017a. 10.1103/PhysRevX.7.021006. see also arXiv:1609.07021.
 Y. Nakata, C. Hirche, C. Morgan, and A. Winter. Implementing unitary 2-designs using random diagonal-unitary matrices. J. Math. Phys., 58 (5): 052203, 2017b. 10.1063/1.4983266.
 R. Oliveira, O. C. O. Dahlsten, and M. B. Plenio. Generic Entanglement Can Be Generated Efficiently. Phys. Rev. Lett., 98 (13): 130502, 2007. 10.1103/PhysRevLett.98.130502.
 B. Schumacher and M. D. Westmoreland. Approximate Quantum Error Correction. Quant. Info. Proc., 1 (1-2): 5–12, 2002. ISSN 1570-0755, 1573-1332. 10.1023/A:1019653202562.
 O. Szehr, F. Dupuis, M. Tomamichel, and R. Renner. Decoupling with unitary approximate two-designs. New J. Phys., 15: 053022, 2013. 10.1088/1367-2630/15/5/053022.
 M. Tomamichel. Quantum Information Processing with Finite Resources. SpringerBriefs in Mathematical Physics, 2016.
 M. Tomamichel, R. Colbeck, and R. Renner. A Fully Quantum Asymptotic Equipartition Property. IEEE Trans. Inf. Theory, 55 (12): 5840, 2009. 10.1109/TIT.2009.2032797.
 G. Tóth and J. J. García-Ripoll. Efficient algorithm for multiqudit twirling for ensemble quantum computation. Phys. Rev. A, 75 (4): 042311, 2007. 10.1103/PhysRevA.75.042311.
 Y. S. Weinstein, W. G. Brown, and L. Viola. Parameters of pseudorandom quantum circuits. Phys. Rev. A, 78 (5): 052332, 2008. 10.1103/PhysRevA.78.052332.
 Hayata Yamasaki and Mio Murao, "Quantum State Merging for Arbitrarily Small-Dimensional Systems", IEEE Transactions on Information Theory 65 6, 3950 (2019).
 Pedro Figueroa–Romero, Felix A. Pollock, and Kavan Modi, "Markovianization with approximate unitary designs", Communications Physics 4 1, 127 (2021).
 Eiichi Bannai, Mikio Nakahara, Da Zhao, and Yan Zhu, "On the explicit constructions of certain unitary t-designs", Journal of Physics A: Mathematical and Theoretical 52 49, 495301 (2019).
 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).
 Uttam Singh, Lin Zhang, and Arun Kumar Pati, "Average coherence and its typicality for random pure states", Physical Review A 93 3, 032125 (2016).
 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).
 Lídia del Rio, Adrian Hutter, Renato Renner, and Stephanie Wehner, "Relative thermalization", Physical Review E 94 2, 022104 (2016).
 Eyuri Wakakuwa and Yoshifumi Nakata, "One-Shot Randomized and Nonrandomized Partial Decoupling", arXiv:1903.05796.
 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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