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Source coding with quantum side information at several decoders

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Abstract

We consider the task of the classical source coding with quantum side information at several decoders. This is a quantum generalization of classical Sgarro’s three correlated information sources. We focus on classical–quantum sources, which involve the classical coding part, using the quantum part as side information at the decoder. We consider two models: non-entangled and entangled side information. To obtain optimal coding rate, we develop a quantum version of maximal simultaneous codes. The achievable rate is found to be determined by the maximally quantum conditional entropy between the classical source and side information. In particular, our result shows that the more the size of the quantum part, the smaller the coding rate.

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References

  1. Shannon, C.E.: A mathematical theory of communication. Bell. Syst. Tech. J. 27(3), 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  2. Slepian, D., Wolf, J.: Noiseless coding of correlated information sources. IEEE Trans. Inf. Theory 19(4), 471–480 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wyner, A.: On source coding with side information at the decoder. IEEE Trans. Inf. Theory 21(3), 294–300 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahlswede, R., Körner, J.: Source coding with side information and a converse for degraded broadcast channels. IEEE Trans. Inf. Theory 21(6), 629–637 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Tsinghua University Press, Beijing (2006)

    MATH  Google Scholar 

  6. Gamal, A.E., Kim, Y.H.: Network Information Theory. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  7. Sgarro, A.: Source coding with side information at several decoders. IEEE Trans. Inf. Theory 23(2), 179–182 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jozsa, R., Schumacher, B.: A new proof of the quantum noiseless coding theorem. Int. J. Opt. 41(12), 2343–2349 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Schumacher, B.: Quantum coding. Phys. Rev. A 51(4), 2738–2747 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  10. Ahn, C., Doherty, A.C., Hayden, P., Winter, A.J.: On the distributed compression of quantum information. IEEE Trans. Inf. Theory 52(10), 4349–4357 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: restructuring quantum information’s family tree. Proc. R. Soc. A. 465, 2537–2563 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Devetak, I., Winter, A.: Classical data compression with quantum side information. Phys. Rev. A 68(4), 042301 (2003)

    Article  ADS  Google Scholar 

  13. Hsieh, M.-H., Watanabe, S.: Channel simulation and coded source compression. IEEE Trans. Inf. Theory 62(11), 6609–6619 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Baghali Khanian, Z., Winter, A.: Distributed compression of correlated classical quantum sources or: the price of ignorance. IEEE Trans. Inf. Theory 66(9), 5620–5633 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cheng, H.-C., Hanson, E.P., Datta, N., Hsieh, M.-H.: Duality between source coding with quantum side information and classical-quantum channel coding. IEEE Trans. Inf. Theory 1–1 (2022)

  16. Yard, J.T., Devetak, I.: Optimal quantum source coding with quantum side information at the encoder and decoder. IEEE Trans. Inf. Theory 55(11), 5339–5351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, B., Huang, X., Zhou, R., Wei, Y., Wan, Q., Pang, C.: A theoretical framework for quantum image representation and data loading scheme. Sci. China Inf. Sci. 57(3), 1–11 (2014)

    Article  MATH  Google Scholar 

  18. Mosonyi, M., Ogawa, T.: Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions. IEEE Trans. Inf. Theory 67(3), 1668–1698 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cheng, H.-C., Hanson, E.P., Datta, N., Hsieh, M.-H.: Non-asymptotic classical data compression with quantum side information. IEEE Trans. Inf. Theory 67(2), 902–930 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Takeuchi, D., Watanabe, S.: The achievable rate region of Wyner-Ahlswede-Korner coding problem for mixed sources. In: IEEE Information Theory Workshop (ITW). pp. 1–6 (2021)

  21. Cheng, H.-C., Datta, N., Rouzé, C.: Strong converse bounds in quantum network information theory. IEEE Trans. Inf. Theory 67(4), 2269–2292 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Merhav, N.: Finite-state source-channel coding for individual source sequences with source side information at the decoder. IEEE Trans. Inf. Theory 68(3), 1532–1544 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sohail, M.A., Anwar Atif, T., Pradhan, S.S.: Unified approach for computing sum of sources over CQ-MAC. In: IEEE International Symposium on Information Theory (ISIT). pp. 1868–1873 (2022)

  24. Korzekwa, K., Puchała, Z., Tomamichel, M., Życzkowski, K.: Encoding classical information into quantum resouces. IEEE Trans. Inf. Theory 68(7), 4518–4530 (2022)

    Article  MATH  Google Scholar 

  25. Lee, Y., Takagi, R., Yamasaki, H., Adesso, G., Lee, S.: State exchange with quantum side information. Phys. Rev. Lett. 122(1), 010502 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  26. Lee, Y., Lee, S.: State transfer with quantum side information. Quan. Inf. Proc. 17, 268 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heegard, C., Berger, T.: Rate distortion when side information may be absent. IEEE Trans. Inf. Theory 31(6), 727–734 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Timo, R., Chan, T., Grant, A.: Rate-distortion with side-information at many decoders. IEEE Trans. Inf. Theory 57(8), 5240–5257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Timo, R., Oechtering, T.J., Wigger, M.: Source coding problems with conditionally less noisy side information. IEEE Trans. Inf. Theory 60(9), 5516–5532 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Unal, S., Wagner, A.B.: A rate-distortion approach to index coding. IEEE Trans. Inf. Theory 62(11), 6359–6378 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sun, H.: Secure groupcast with shared keys. IEEE Trans. Inf. Theory 68(7), 4681–4699 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ye, F., Dupraz, E., Mheich, Z., Amis, K.: Optimized rate-adaptive protograph-based LDPC codes for source coding with side information. IEEE Trans. Inf. Theory 67(6), 3879–3889 (2019)

    Google Scholar 

  33. Parthasarathy, K.R.: Coding theorems of classical and quantum information theory. Hindustan Book Agency (2013)

  34. Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44(1), 269–273 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)

    Article  ADS  Google Scholar 

  36. Winter, A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Theory 45(7), 2481–2485 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wyner, A., Ziv, J.: The rate-distortion function for source coding with side information at the decoder. IEEE Trans. Inf. Theory 22(1), 1–10 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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  1. College of Computer Science, Shaanxi Normal University, Xi’an, 710062, Shaanxi, China

    Xu Han, Xiaomin Liu & Zhengjun Xi

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  1. Xu Han
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  2. Xiaomin Liu
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  3. Zhengjun Xi
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Correspondence to Zhengjun Xi.

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Han, X., Liu, X. & Xi, Z. Source coding with quantum side information at several decoders. Quantum Inf Process 22, 212 (2023). https://doi.org/10.1007/s11128-023-03936-z

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