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Optimizing Quantum Models of Classical Channels: The Reverse Holevo Problem

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Abstract

Given a classical channel—a stochastic map from inputs to outputs—the input can often be transformed into an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation. This inverts Holevo’s original question of quantifying the capacity of quantum channels with classical resources: We determine the lowest-capacity quantum channel required to simulate a classical channel. We also show that this problem is equivalent to another, involving the local generation of a distribution from common entanglement.

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Article Open access 05 August 2022

Roman Rietsche, Christian Dremel, … Jan-Marco Leimeister

Notes

  1. The mapping g need not be explicitly deterministic, but if \(f_1\) and \(f_2\) are deterministic, then this will require g to be so as well.

References

  1. Meyer, D.A.: Quantum computing classical physics. Philos. Trans. R. Soc. Lond. A 360(1792), 395–405 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  2. Yung, M.H., Nagaj, D., Whitfield, J.D., Aspuru-Guzik, A.: Simulation of classical thermal states on a quantum computer: a transfer-matrix approach. Phys. Rev. A 82(6), 060302 (2010)

    Article  ADS  Google Scholar 

  3. Yepez, J.: Quantum lattice-gas model for computational fluid dynamics. Phys. Rev. E 63(4), 046702 (2001)

    Article  ADS  Google Scholar 

  4. Yepez, J.: Quantum computation of fluid dynamics. In: Quantum Computing and Quantum Communications, pp. 34–60. Springer, Berlin (1999)

  5. Sinha, S., Russer, P.: Quantum computing algorithm for electromagnetic field simulation. Quant. Inf. Process. 9(3), 385–404 (2010)

    Article  MathSciNet  Google Scholar 

  6. Yepez, J.: Quantum lattice-gas model for the diffusion equation. Int. J. Mod. Phys. C 12(09), 1285–1303 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  7. Berman, G.P., Ezhov, A.A., Kamenev, D.I., Yepez, J.: Simulation of the diffusion equation on a type-II quantum computer. Phys. Rev. A 66(1), 012310 (2002)

    Article  ADS  Google Scholar 

  8. Yepez, J.: Quantum lattice-gas model for the Burgers equation. J. Stat. Phys. 107(1–2), 203–224 (2002)

    Article  Google Scholar 

  9. Harris, S.A., Kendon, V.M.: Quantum-assisted biomolecular modelling. Philos. Trans. R. Soc. Lond. A 368(1924), 3581–3592 (2010)

    ADS  Google Scholar 

  10. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  11. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219

  12. Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150502 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  13. Crutchfield, J.P., Young, K.: Inferring statistical complexity. Phys. Rev. Lett. 63, 105–108 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  14. Shalizi, C.R., Crutchfield, J.P.: Computational mechanics: pattern and prediction, structure and simplicity. J. Stat. Phys. 104, 817–879 (2001)

    Article  MathSciNet  Google Scholar 

  15. Crutchfield, J.P.: Between order and chaos. Nat. Phys. 8, 17–24 (2012)

    Article  MathSciNet  Google Scholar 

  16. Gu, M., Wiesner, K., Rieper, E., Vedral, V.: Quantum mechanics can reduce the complexity of classical models. Nat. Commun. 3, 762 (2012)

    Article  ADS  Google Scholar 

  17. Tan, R., Terno, D.R., Thompson, J., Vedral, V., Gu, M.: Towards quantifying complexity with quantum mechanics. Eur. J. Phys. Plus 129, 191 (2014)

    Article  Google Scholar 

  18. Mahoney, J.R., Aghamohammadi, C., Crutchfield, J.P.: Occam’s quantum strop: synchronizing and compressing classical cryptic processes via a quantum channel. Sci. Rep. 6, 20495 (2016)

    Article  ADS  Google Scholar 

  19. Riechers, P.M., Mahoney, J.R., Aghamohammadi, C., Crutchfield, J.P.: A closed-form shave from Occam’s quantum razor: exact results for quantum compression. Phys. Lett. A 93, 052317 (2016)

    Google Scholar 

  20. Binder, F.C., Thompson, J., Gu, M.: A practical, unitary simulator for non-Markovian complex processes. Phys. Rev. Lett. 120, 240502 (2017)

    Article  MathSciNet  Google Scholar 

  21. Loomis, S.P., Crutchfield, J.P.: Strong and weak optimizations in classical and quantum models of stochastic processes. J. Stat. Phys. 176, 1317–1342 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  22. Liu, Q., Elliot, T.J.,Binder, F.C., Franco, C.D., Gu, M.: Optimal stochastic modelling with unitary quantum dynamics. arXiv:1810.09668

  23. Bennett, C., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83, 3081 (1999)

    Article  ADS  Google Scholar 

  24. Bennett, C., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48, 2637–2655 (2002)

    Article  MathSciNet  Google Scholar 

  25. Bennett, C., Devetak, I., Harrow, A.W., Shor, P.W., Winter, A.: The quantum reverse Shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Trans. Inf. Theory 60, 2926–2959 (2014)

    Article  MathSciNet  Google Scholar 

  26. Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inf. Transm. 9, 177 (1973)

    Google Scholar 

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

    Article  ADS  Google Scholar 

  28. Holevo, A.S.: Coding theorems for quantum channels. Russ. Math. Surveys 53, 1295–1331 (1996)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Wyner, A.D.: The common information of two dependent random variables. IEEE Trans. Inf. Theory 21, 163–179 (1975)

    Article  MathSciNet  Google Scholar 

  31. Kumar, G.R., Li, C.T., El Gamal, A.: Exact common information. In: 2014 IEEE ISIT, pp. 161–165. IEEE (2014)

  32. Boyd, A.B., Mandal, D., Crutchfield, J.P.: Thermodynamics of modularity: structural costs beyond the Landauer bound. Phys. Rev. X 8(3), 031036 (2018)

    Google Scholar 

  33. Ahlswede, R.A., Winter, A.: Strong converse for identification via quantum channels. IEEE Trans. Inf. Theory 48, 346 (2002). Addendum: IEEE Trans. Inf. Theory, 49(1), 346, 2003

    MathSciNet  MATH  Google Scholar 

  34. Winter, A.: Secret, public and quantum correlation cost of triples of random variables. In: 2005 IEEE ISIT. IEEE (2005)

  35. Ash, R.B.: Information Theory. Dover, New York (1990)

    MATH  Google Scholar 

  36. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50(2), 221 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  37. Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183(1), 14–18 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  38. Horodecki, M., Oppenheim, J., Winter, A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  39. Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications, 3rd edn. Springer, New York (2011)

    Book  Google Scholar 

  40. Ghirardi, G., Bassi, A.: A general scheme for ensemble purification. Phys. Lett. A 309, 24 (2003)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

We thank Ryan G. James and Fabio Anza for useful conversations. As a faculty member, James P. Crutchfield thanks the Santa Fe Institute and the Telluride Science Research Center for their hospitality during visits. This material is based upon work supported by, or in part by, the John Templeton Foundation Grant 52095, Foundational Questions Institute Grant FQXi-RFP-1609, and U. S. Army Research Laboratory and the U. S. Army Research Office under Contract W911NF-13-1-0390 and Grant W911NF-18-1-0028.

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Authors and Affiliations

  1. Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA, 95616, USA

    Samuel P. Loomis, John R. Mahoney, Cina Aghamohammadi & James P. Crutchfield

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  1. Samuel P. Loomis
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  2. John R. Mahoney
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  3. Cina Aghamohammadi
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  4. James P. Crutchfield
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Correspondence to James P. Crutchfield.

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Communicated by Christian Maes.

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Loomis, S.P., Mahoney, J.R., Aghamohammadi, C. et al. Optimizing Quantum Models of Classical Channels: The Reverse Holevo Problem. J Stat Phys 181, 1966–1985 (2020). https://doi.org/10.1007/s10955-020-02649-2

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