Decoupling with random diagonal unitaries

Yoshifumi Nakata; Christoph Hirche; Ciara Morgan; Andreas Winter

doi:10.22331/q-2017-07-21-18

Decoupling with random diagonal unitaries

Yoshifumi Nakata^1,2,3, Christoph Hirche^1,3, Ciara Morgan^1,4, and Andreas Winter^3,5

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

Published:	2017-07-21, volume 1, page 18
Eprint:	arXiv:1509.05155v5
Doi:	https://doi.org/10.22331/q-2017-07-21-18
Citation:	Quantum 1, 18 (2017).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

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.

Featured image: A decoupling protocol. Alice tries to destroy all correlations with Reference by applying a suitable unitary onto her system before it is sent to Bob via a given quantum channel. We construct an imprecise quantum pseudorandomness that achieves decoupling as strongly as a uniformly distributed random unitary.

Popular summary

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.

► BibTeX data

@article{Nakata2017decouplingrandom,
  doi = {10.22331/q-2017-07-21-18},
  url = {https://doi.org/10.22331/q-2017-07-21-18},
  title = {Decoupling with random diagonal unitaries},
  author = {Nakata, Yoshifumi and Hirche, Christoph and Morgan, Ciara and Winter, Andreas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {18},
  month = jul,
  year = {2017}
}

► References

[1] 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.
https://doi.org/10.1098/rspa.2009.0202

[2] M. Berta. Single-shot Quantum State Merging. arXiv:0912.4495, 2009.
arXiv:0912.4495

[3] 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.
https://doi.org/10.1007/s00220-016-2706-8

[4] W. Brown and O. Fawzi. Decoupling with random quantum circuits. Commun. Math. Phys., 340: 867, 2015. 10.1007/s00220-015-2470-1.
https://doi.org/10.1007/s00220-015-2470-1

[5] 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.
https://doi.org/10.1103/PhysRevA.77.040303

[6] M. D. Choi. Completely positive linear maps on complex matrices. Linear Algebra Appl., 10: 285, 1975. 10.1016/0024-3795(75)90075-0.
https://doi.org/10.1016/0024-3795(75)90075-0

[7] 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.

[8] 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.
https://doi.org/10.1088/1751-8113/40/28/S16

[9] 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.
https://doi.org/10.1103/PhysRevA.80.012304

[10] 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.
https://doi.org/10.1088/1367-2630/13/9/093042

[11] L. del Rio, A. Hutter, R. Renner, and S. Wehner. Relative thermalization. Phys. Rev. E, 94 (2): 022104, 2016. 10.1103/PhysRevE.94.022104.
https://doi.org/10.1103/PhysRevE.94.022104

[12] 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.
https://doi.org/10.1109/TIT.2004.839515

[13] 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.
https://doi.org/10.1103/PhysRevLett.93.080501

[14] 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.
https://doi.org/10.1007/s00220-011-1217-x

[15] D. P. DiVincenzo, D. W. Leung, and B. M. Terhal. Quantum data hiding. IEEE Trans. Inf. Theory, 48: 580, 2002. 10.1109/18.985948.
https://doi.org/10.1109/18.985948

[16] F. Dupuis. The decoupling approach to quantum information theory. PhD thesis, Université de Montréal, 2010. arXiv:1004.1641.
arXiv:1004.1641

[17] F. Dupuis, M. Berta, J. Wullschleger, and R. Renner. One-shot decoupling. Commun. Math. Phys., 328: 251, 2014. 10.1007/s00220-014-1990-4.
https://doi.org/10.1007/s00220-014-1990-4

[18] 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.
https://doi.org/10.1103/PhysRevA.72.032317

[19] 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.
https://doi.org/10.1063/1.2716992

[20] 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.
https://doi.org/10.1007/s00220-009-0873-6

[21] P. Hayden. Decoupling: A building block for quantum information theory. http://qip2011.quantumlah.org/images/QIPtutorial1.pdf, 2012. Accessed: 2017-3-30.
http://qip2011.quantumlah.org/images/QIPtutorial1.pdf

[22] 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.
https://doi.org/10.1088/1126-6708/2007/09/120

[23] 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.
https://doi.org/10.1109/ISIT.2014.6874890

[24] 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.
https://doi.org/10.1007/s00220-006-0118-x

[25] E. Knill. Approximation by Quantum Circuits. arXiv:quant-ph/9508006, 1995.
arXiv:quant-ph/9508006

[26] 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.
https://doi.org/10.1016/0034-4877(72)90011-0

[27] M. Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society Providence, RI, USA, 2001.

[28] R. A. Low. Pseudo-randomness and learning in quantum computation. PhD thesis, University of Bristol, 2010. arXiv:1006.5227.
arXiv:1006.5227

[29] 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.
https://doi.org/10.1142/S0219749913500627

[30] 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.
https://doi.org/10.1140/epjp/i2014-14152-9

[31] 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.
https://doi.org/10.1088/1367-2630/16/5/053043

[32] 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.
https://doi.org/10.1103/PhysRevX.7.021006
arXiv:1609.07021

[33] 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.
https://doi.org/10.1063/1.4983266

[34] 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.
https://doi.org/10.1103/PhysRevLett.98.130502

[35] R. Renner. Security of Quantum Key Distribution. PhD thesis, ETH Zurich, 2005. arXiv:quant-ph/0512258.
arXiv:quant-ph/0512258

[36] 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.
https://doi.org/10.1023/A:1019653202562

[37] 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.
https://doi.org/10.1088/1367-2630/15/5/053022

[38] M. Tomamichel. Quantum Information Processing with Finite Resources. SpringerBriefs in Mathematical Physics, 2016.

[39] 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.
https://doi.org/10.1109/TIT.2009.2032797

[40] 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.
https://doi.org/10.1103/PhysRevA.75.042311

[41] 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.
https://doi.org/10.1103/PhysRevA.78.052332

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", 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] Lin Zhang, Uttam Singh, and Arun Kumar Pati, "Average subentropy, coherence and entanglement of random mixed quantum states", arXiv:1510.08859, (2015).

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

The above citations are from Crossref's cited-by service (last updated successfully 2023-06-09 09:59:04) and SAO/NASA ADS (last updated successfully 2023-06-09 09:59:05). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.