One-Shot Hybrid State Redistribution

Eyuri Wakakuwa; Yoshifumi Nakata; Min-Hsiu Hsieh

doi:10.22331/q-2022-05-30-724

One-Shot Hybrid State Redistribution

Eyuri Wakakuwa^1,2, Yoshifumi Nakata^3,4,5, and Min-Hsiu Hsieh^6,7

¹Department of Communication Engineering and Informatics, Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan
²Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
³Yukawa Institute for Theoretical Physics, Kyoto university, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan
⁴Photon Science Center, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
⁵JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
⁶Centre for Quantum Software & Information (UTS:QSI), University of Technology Sydney, Sydney NSW, Australia
⁷Hon Hai (Foxconn) Research Institute, Taipei, Taiwan

Published:	2022-05-30, volume 6, page 724
Eprint:	arXiv:2006.12059v3
Doi:	https://doi.org/10.22331/q-2022-05-30-724
Citation:	Quantum 6, 724 (2022).

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

Abstract

We consider state redistribution of a "hybrid" information source that has both classical and quantum components. The sender transmits classical and quantum information at the same time to the receiver, in the presence of classical and quantum side information both at the sender and at the decoder. The available resources are shared entanglement, and noiseless classical and quantum communication channels. We derive one-shot direct and converse bounds for these three resources, represented in terms of the smooth conditional entropies of the source state. Various coding theorems for two-party source coding problems are systematically obtained by reduction from our results, including the ones that have not been addressed in previous literature.

► BibTeX data

@article{Wakakuwa2022oneshothybridstate,
  doi = {10.22331/q-2022-05-30-724},
  url = {https://doi.org/10.22331/q-2022-05-30-724},
  title = {One-{S}hot {H}ybrid {S}tate {R}edistribution},
  author = {Wakakuwa, Eyuri and Nakata, Yoshifumi and Hsieh, Min-Hsiu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {724},
  month = may,
  year = {2022}
}

► References

[1] Anura Abeyesinghe, Igor Devetak, Patrick Hayden, and Andreas Winter, ``The mother of all protocols: restructuring quantum information’s family tree’’, Proc. Roy. Soc. A 465(2108):2537–2563 (2009).
https://doi.org/10.1098/rspa.2009.0202

[2] Anurag Anshu, Rahul Jain, and Naqueeb Ahmad Warsi, ``A one-shot achievability result for quantum state redistribution’’, IEEE Trans. Inf. Theory 64(3): 1425–1435 (2018).
https://doi.org/10.1109/TIT.2017.2776112

[3] Anurag Anshu, Min-Hsiu Hsieh, and Rahul Jain, ``Noisy quantum state redistribution with promise and the alpha-bit’’, IEEE Trans. Inf. Theory 66(12): 7772–7786 (2020).
https://doi.org/10.1109/TIT.2020.3021289

[4] Zahra Baghali Khanian and Andreas Winter, ``Distributed compression of correlated classicalquantum sources or: The price of ignorance’’, IEEE Trans. Inf. Theory 66(9): 5620–5633 (2020).
https://doi.org/10.1109/TIT.2020.2981322

[5] Mario Berta, Matthias Christandl, Roger Colbeck, Joseph M. Renes, and Renato Renner, ``The uncertainty principle in the presence of quantum memory’’, Nat. Phys. 6(9): 659–662 (2010).
https://doi.org/10.1038/NPHYS1734

[6] Mario Berta, Matthias Christandl, and Renato Renner, ``The quantum reverse shannon theorem based on one-shot information theory’’, Comm. Math. Phys. 306(3):579–615 (2011).
https://doi.org/10.1007/s00220-011-1309-7

[7] Mario Berta, Matthias Christandl, and Dave Touchette, ``Smooth entropy bounds on one-shot quantum state redistribution’’, IEEE Trans. Inf. Theory 62(3):1425–1439 (2016).
https://doi.org/10.1109/TIT.2016.2516006

[8] T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd ed.), Wiley InterScience, 2005.

[9] Nilanjana Datta and Min-Hsiu Hsieh, ``The apex of the family tree of protocols: optimal rates and resource inequalities’’, New J. Phys. 13, 093042 (2011).
https://doi.org/10.1088/1367-2630/13/9/093042

[10] Nilanjana Datta, Min-Hsiu Hsieh, and Jonathan Oppenheim, ``An upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution’’, J. Math. Phys. 57(5), 052203 (2016).
https://doi.org/10.1063/1.4949571

[11] I Devetak and A Winter, ``Classical data compression with quantum side information’’, Phys. Rev. A 68(4), 042301 (2003).
https://doi.org/10.1103/PhysRevA.68.042301

[12] I Devetak, AW Harrow, and A Winter, ``A family of quantum protocols’’, Phys. Rev. Lett. 93(23), 230504 (2004).
https://doi.org/10.1103/PhysRevLett.93.230504

[13] Igor Devetak and Jon Yard, ``Exact cost of redistributing multipartite quantum states’’, Phys. Rev. Lett. 100(23), 230501 (2008).
https://doi.org/10.1103/PhysRevLett.100.230501

[14] Igor Devetak, Aram W. Harrow, and Andreas J. Winter, ``A resource framework for quantum shannon theory’’, IEEE Trans. Inf. Theory 54(10):4587-4618 (2008).
https://doi.org/10.1109/TIT.2008.928980

[15] Frederic Dupuis, Mario Berta, Juerg Wullschleger, and Renato Renner, ``One-shot decoupling’’, Comm. Math. Phys. 328(1):251-284 (2014).
https://doi.org/10.1007/s00220-014-1990-4

[16] M Horodecki, J Oppenheim, and A Winter, ``Partial quantum information’’, Nature 436 (7051): 673–676 (2005).
https://doi.org/10.1038/nature03909

[17] Michal Horodecki, Jonathan Oppenheim, and Andreas Winter, ``Quantum state merging and negative information’’, Comm. Math. Phys. 269(1):107–136 (2007).
https://doi.org/10.1007/s00220-006-0118-x

[18] Min-Hsiu Hsieh and Mark M. Wilde, ``Trading classical communication, quantum communication, and entanglement in quantum shannon theory’’, IEEE Trans. Inf. Theory 56(9): 4705-4730 (2010).
https://doi.org/10.1109/TIT.2010.2054532

[19] Robert Koenig, Renato Renner, and Christian Schaﬀner, ``The operational meaning of min- and max-entropy’’, IEEE Trans. Inf. Theory 55(9): 4337-4347 (2009).
https://doi.org/10.1109/TIT.2009.2025545

[20] T Ogawa and H Nagaoka, ``A new proof of the channel coding theorem via hypothesis testing in quantum information theory’’, In ISIT: 2002 IEEE Int. Symp. Info.Theory, Proceedings, page 73.
https://doi.org/10.1109/ISIT.2002.1023345

[21] Joseph M. Renes and Renato Renner, ``One-shot classical data compression with quantum side information and the distillation of common randomness or secret keys’’, IEEE Trans. Inf. Theory 58(3):1985–1991 (2012).
https://doi.org/10.1109/TIT.2011.2177589

[22] R Renner and S Wolf, ``Simple and tight bounds for information reconciliation and privacy ampliﬁcation’’, In Advances in Cryptology - ASIACRYPT 2005, pp.199-216, Springer, Berlin, Heidelberg.
https://doi.org/10.1007/11593447_11

[23] Renato Renner, ``Security of quantum key distribution’’, Int. J. Quant. Info. 6(1): 1–127, (2008).
https://doi.org/10.1142/S0219749908003256

[24] Benjamin Schumacher, ``Quantum coding’’, Phys. Rev. A 51(4):2738–2747 (1995).
https://doi.org/10.1103/PhysRevA.51.2738

[25] W Forrest Stinespring, ``Positive functions on $c^\ast$ algebras’’, Proc. of the Amer. Math. Soc., 6(2): 211–216 (1955).
https://doi.org/10.2307/2032342

[26] M. Tomamichel, Quantum Information Processing with Finite Resources, Springer Briefs in Mathematical Physics, 2016.

[27] Marco Tomamichel and Masahito Hayashi, ``A hierarchy of information quantities for ﬁnite block length analysis of quantum tasks’’, IEEE Trans. Inf. Theory 59(11):7693–7710 (2013).
https://doi.org/10.1109/TIT.2013.2276628

[28] Marco Tomamichel, Roger Colbeck, and Renato Renner, ``A fully quantum asymptotic equipartition property’’, IEEE Trans. Inf. Theory 55(12):5840–5847 (2009).
https://doi.org/10.1109/TIT.2009.2032797

[29] Marco Tomamichel, Roger Colbeck, and Renato Renner, ``Duality between smooth min-and max-entropies’’, IEEE Trans. Inf. Theory 56(9):46744681 (2010).
https://doi.org/10.1109/TIT.2010.2054130

[30] Armin Uhlmann, ``The “transition probability” in the state space of a $\ast$-algebra’’, Rep. Math. Phys., 9(2):273–279 (1976).
https://doi.org/10.1016/0034-4877(76)90060-4

[31] Alexander Vitanov, Frederic Dupuis, Marco Tomamichel, and Renato Renner, ``Chain rules for smooth min- and max-entropies’’, IEEE Trans. Inf. Theory 59(5):2603–2612 (2013).
https://doi.org/10.1109/TIT.2013.2238656

[32] Eyuri Wakakuwa and Yoshifumi Nakata, ``Randomized partial decoupling uniﬁes one-shot quantum channel capacities’’, arXiv:2004.12593 (2020).
arXiv:2004.12593

[33] Eyuri Wakakuwa and Yoshifumi Nakata, ``One-shot randomized and nonrandomized partial decoupling’’, Comm. Math. Phys. 386(2):589–649 (2021).
https://doi.org/10.1007/s00220021-04136-5

[34] Mark Wilde, Quantum Information Theory, Camb. Univ. Press, 2013.

[35] Jon T. Yard and Igor Devetak, ``Optimal quantum source coding with quantum side information at the encoder and decoder’’, IEEE Trans. Inf. Theory 55 (11):5339–5351 (2009).
https://doi.org/10.1109/TIT.2009.2030494

[36] Ming-Yong Ye, Yan-Kui Bai, and Z. D. Wang, ``Quantum state redistribution based on a generalized decoupling’’, Phys. Rev. A 78(3) (2008).
https://doi.org/10.1103/PhysRevA.78.030302

Cited by

[1] Eyuri Wakakuwa and Yoshifumi Nakata, "One-Shot Triple-Resource Trade-Off in Quantum Channel Coding", IEEE Transactions on Information Theory 69 4, 2400 (2023).

[2] Ian George, Min-Hsiu Hsieh, and Eric Chitambar, 2023 IEEE International Symposium on Information Theory (ISIT) 96 (2023) ISBN:978-1-6654-7554-9.

[3] Xiaomin Liu, Zhengjun Xi, and Heng Fan, "Slepian–Wolf coding with quantum side information", Quantum Information Processing 22 11, 407 (2023).

[4] Manideep Mamindlapally and Andreas Winter, "Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes", IEEE Transactions on Information Theory 69 9, 5857 (2023).

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

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-05 03:46:22) and SAO/NASA ADS (last updated successfully 2024-05-05 03:46:23). 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.