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.
 M. Berta. Single-shot Quantum State Merging. arXiv:0912.4495, 2009.
 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. Brown and O. Fawzi. Decoupling with random quantum circuits. Commun. Math. Phys., 340: 867, 2015. 10.1007/s00220-015-2470-1.
 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.
 M. D. Choi. Completely positive linear maps on complex matrices. Linear Algebra Appl., 10: 285, 1975. 10.1016/0024-3795(75)90075-0.
 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.
 L. del Rio, A. Hutter, R. Renner, and S. Wehner. Relative thermalization. Phys. Rev. E, 94 (2): 022104, 2016. 10.1103/PhysRevE.94.022104.
 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.
 D. P. DiVincenzo, D. W. Leung, and B. M. Terhal. Quantum data hiding. IEEE Trans. Inf. Theory, 48: 580, 2002. 10.1109/18.985948.
 F. Dupuis. The decoupling approach to quantum information theory. PhD thesis, Université de Montréal, 2010. arXiv:1004.1641.
 F. Dupuis, M. Berta, J. Wullschleger, and R. Renner. One-shot decoupling. Commun. Math. Phys., 328: 251, 2014. 10.1007/s00220-014-1990-4.
 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.
 A. W. Harrow and R. A. Low. Random quantum circuits are approximate 2-designs. Commun. Math. Phys., 291: 257, 2009. 10.1007/s00220-009-0873-6.
 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.
 E. Knill. Approximation by Quantum Circuits. arXiv:quant-ph/9508006, 1995.
 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.
 R. A. Low. Pseudo-randomness and learning in quantum computation. PhD thesis, University of Bristol, 2010. arXiv:1006.5227.
 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.
 R. Renner. Security of Quantum Key Distribution. PhD thesis, ETH Zurich, 2005. arXiv:quant-ph/0512258.
 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.
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