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Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit

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Abstract

We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them; replacing classical bits by zero-bits makes teleportation asymptotically reversible. This decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applications to state merging, the quantum channel capacity, entanglement distillation, quantum identification and remote state preparation. The source of these results is the theory of approximate quantum error correction. The action of a quantum channel is reversible if and only if no information is leaked to the environment, a characterization that is useful even in approximate form. However, different notions of approximation lead to qualitatively different forms of quantum error correction in the limit of large dimension. We study the effect of a constraint on the dimension of the reference system when considering information leakage. While the resulting condition fails to ensure that the entire input can be corrected, it does ensure that all subspaces of dimension matching that of the reference are correctable. The size of the reference can be characterized by a parameter \(\alpha \); we call the associated resource an \(\alpha \)-bit. Changing \(\alpha \) interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of \(\alpha \)-bits, including the applications above, and determine the \(\alpha \)-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants.

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Notes

  1. An ebit is another term for a Bell pair of two qubits.

  2. The authors are ashamed to note that they were unaware of the existence of co-cobits until six months after this paper had first appeared on arXiv.

  3. Technically, our construction requires the use of shared randomness to achieve this rate but it can be eliminated by block coding.

  4. As proved in Lemma 12, the \(\alpha \)-bit capacity is a continuous function of \(\alpha \), as is the entanglement-assisted \(\alpha \)-bit capacity. However, the amortised \(\alpha \)-bit capacity has a discontinuity at \(\alpha = 1\). As a result, it remains an open question to find and prove amortised universal subspace error correction capacities for subspace dimensions that grow sublinearly but faster than \(d^\alpha \) for any \(\alpha < 1\).

References

  1. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)

    Google Scholar 

  2. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69(20), 2881–2884 (1992)

    Google Scholar 

  3. Harrow, A.: Coherent communication of classical messages. Phys. Rev. Lett. 92(9), 097902 (2004)

    Google Scholar 

  4. Wilde, M.M., Krovi, H., Brun, T.A.: Coherent communication with continuous quantum variables. Phys. Rev. A 75(6), 060303(R) (2007). arXiv:quant-ph/0612170

    Google Scholar 

  5. Wilde, M.M.: Quantum Information Theory, 1st edn. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  6. Harrow, A.W., Shor, P.W.: Time reversal and exchange symmetries of unitary gate capacities. arXiv preprint arXiv:quant-ph/0511219 (2005)

  7. Schumacher, B., Westmoreland, M.D.: Approximate quantum error correction. Quantum Inf. Process. 1(1), 5–12 (2002)

    Google Scholar 

  8. Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900 (1997)

    Google Scholar 

  9. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54(5), 3824 (1996)

    Google Scholar 

  10. Leung, D.W., Nielsen, M.A., Chuang, I.L., Yamamoto, Y.: Approximate quantum error correction can lead to better codes. Phys. Rev. A 56(4), 2567 (1997)

    Google Scholar 

  11. Crépeau, C., Gottesman, D., Smith, A.: Approximate quantum error-correcting codes and secret sharing schemes. In: Annual International Conference on the Theory and Applications of Cryptographic Techniques. Springer, pp. 285–301 (2005)

  12. Schumacher, B., Nielsen, M.A.: Quantum data processing and error correction. Phys. Rev. A 54(4), 2629 (1996)

    Google Scholar 

  13. Duan, R., Severini, S., Winter, A.: Zero-error communication via quantum channels, noncommutative graphs, and a quantum lovász number. IEEE Trans. Inf. Theory 59(2), 1164–1174 (2013)

    Google Scholar 

  14. Kretschmann, D., Werner, R.F.: Tema con variazioni: quantum channel capacity. New J. Phys. 6(1), 26 (2004)

    Google Scholar 

  15. Winter, A.: Quantum and classical message identification via quantum channels. In: Hirota, O. (ed.) Festschrift “A. S. Holevo 60”, pp. 171–188. Rinton Press, Princeton (2004)

    Google Scholar 

  16. Hayden, P., Winter, A.: Weak decoupling duality and quantum identification. IEEE Trans. Inf. Theory 58(7), 4914–4929 (2012)

    Google Scholar 

  17. Winter, A.: Identification via quantum channels in the presence of prior correlation and feedback. In: Ahlswede, R. (ed.) Information Transfer and Combinatorics. Lecture Notes in Computer Science, vol. 4123. Springer, Berlin (2006)

    Google Scholar 

  18. Bennett, C.H., Shor, P.W.: Quantum information theory. IEEE Trans. Inf. Theory 44(6), 2724–2742 (1998)

    Google Scholar 

  19. Lloyd, S.: Capacity of the noisy quantum channel. Phys. Rev. A 55(3), 1613 (1997)

    Google Scholar 

  20. Shor, P.W.: The quantum channel capacity and coherent information. In: Lecture Notes, MSRI Workshop on Quantum Computation (2002)

  21. Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)

    Google Scholar 

  22. Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)

    Google Scholar 

  23. Nielsen, M.A., Caves, C.M., Schumacher, B., Barnum, H.: Information-theoretic approach to quantum error correction and reversible measurement. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 454(1969), 277–304 (1998)

    Google Scholar 

  24. Kretschmann, D., Schlingemann, D., Werner, R.F.: The information-disturbance tradeoff and the continuity of stinespring’s representation. IEEE Trans. Inf. Theory 54(4), 1708–1717 (2008)

    Google Scholar 

  25. Hayden, P., Horodecki, M., Winter, A., Yard, J.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15(01), 7–19 (2008)

    Google Scholar 

  26. Paulsen, V.: Completely Bounded Maps and Operator Algebras, vol. 78. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  27. Cerf, N.J., Adami, C.: Negative entropy and information in quantum mechanics. Phys. Rev. Lett. 79(26), 5194 (1997)

    Google Scholar 

  28. Bennett, C.H., 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(10), 2637–2655 (2002). arXiv:quant-ph/0106052

    Google Scholar 

  29. Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1998). arXiv:quant-ph/9712042

    Google Scholar 

  30. Uhlmann, A.: The “transition probability” in the state space of a *-algebra. Rep. Math. Phys. 9(2), 273–279 (1976)

    Google Scholar 

  31. Devetak, I., Harrow, A.W., Winter, A.J.: A resource framework for quantum shannon theory. IEEE Trans. Inf. Theory 54(10), 4587–4618 (2008)

    Google Scholar 

  32. Fawzi, O., Hayden, P., Sen, P.: From low-distortion norm embeddings to explicit uncertainty relations and efficient information locking. J. ACM (JACM) 60(6), 44 (2013)

    Google Scholar 

  33. Devetak, I., Harrow, A.W., Winter, A.: A family of quantum protocols. Phys. Rev. Lett. 93(23), 230504 (2004)

    Google Scholar 

  34. Devetak, I., Winter, A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93(8), 080501 (2004)

    Google Scholar 

  35. Horodecki, M., Oppenheim, J., Winter, A.: Quantum state merging and negative information. Commun. Math. Phys. 269(1), 107–136 (2007)

    Google Scholar 

  36. Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: restructuring quantum information’s family tree. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, p. 20090202. The Royal Society (2009)

  37. Bennett, C.H., 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(5), 2926–2959 (2014). arXiv:0912.5537

    Google Scholar 

  38. Berta, M., Christandl, M., Renner, R.: The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306(3), 579–615 (2011). arXiv:0912.3805

    Google Scholar 

  39. Bennett, C.H., Hayden, P., Leung, D.W., Shor, P.W., Winter, A.: Remote preparation of quantum states. IEEE Trans. Inf. Theory 51(1), 56–74 (2005)

    Google Scholar 

  40. Harrow, A., Hayden, P., Leung, D.: Superdense coding of quantum states. Phys. Rev. Lett. 92(18), 187901 (2004)

    Google Scholar 

  41. Hayden, P., Leung, D.W., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)

    Google Scholar 

  42. Abeyesinghe, A., Hayden, P., Smith, G., Winter, A.J.: Optimal superdense coding of entangled states. IEEE Trans. Inf. Theory 52(8), 3635–3641 (2006)

    Google Scholar 

  43. Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Problems Inf. Transm. 9, 177–183 (1973)

    Google Scholar 

  44. Dupuis, F.: The decoupling approach to quantum information theory. PhD thesis, University of Montreal (2010). arXiv:1004.1641

  45. Dupuis, F., Berta, M., Wullschleger, J., Renner, R.: One-shot decoupling. Commun. Math. Phys. 328(1), 251–284 (2014). arXiv:1012.6044

    Google Scholar 

  46. Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phys. 2(11), 754–758 (2006)

    Google Scholar 

  47. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998). arXiv:quant-ph/9608006

    Google Scholar 

  48. Wilson, R.: The Finite Simple Groups, vol. 251. Springer, Berlin (2009)

    Google Scholar 

  49. Fannes, M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31(4), 291–294 (1973)

    Google Scholar 

  50. Yard, J., Hayden, P., Devetak, I.: Capacity theorems for quantum multiple-access channels: classical-quantum and quantum–quantum capacity regions. IEEE Trans. Inf. Theory 54(7), 3091–3113 (2008)

    Google Scholar 

  51. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A.: Capacities of quantum erasure channels. Phys. Rev. Lett. 78(16), 3217–3220 (1997). arXiv:quant-ph/9701015

    Google Scholar 

  52. Devetak, I., Shor, P.W.: The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256(2), 287–303 (2005)

    Google Scholar 

  53. Hsieh, M.-H., Wilde, M.M.: Trading classical communication, quantum communication, and entanglement in quantum shannon theory. IEEE Trans. Inf. Theory 56(9), 4705–4730 (2010)

    Google Scholar 

  54. Ledoux, M.: The Concentration of Measure Phenomenon. American Mathematical Society, Providence (2005)

    Google Scholar 

  55. Smith, G., Yard, J.: Quantum communication with zero-capacity channels. Science 321(5897), 1812–1815 (2008)

    Google Scholar 

  56. Fawzi, O., Renner, R.: Quantum conditional mutual information and approximate markov chains. arXiv preprint arXiv:1410.0664 (2014)

  57. Junge, M., Renner, R., Sutter, D., Wilde, M.M., Winter, A.: Universal recovery from a decrease of quantum relative entropy. arXiv preprint arXiv:1509.07127 (2015)

  58. Pastawski, F., Eisert, J., Wilming, H.: Quantum source-channel codes. arXiv preprint arXiv:1611.07528 (2016)

  59. Kato, K., Furrer, F., Murao, M.: Information-theoretical analysis of topological entanglement entropy and multipartite correlations. Phys. Rev. A 93(2), 022317 (2016)

    Google Scholar 

  60. Swingle, B., McGreevy, J.: Mixed s-sourcery: building many-body states using bubbles of nothing. Phys. Rev. B 94(15), 155125 (2016)

    Google Scholar 

  61. Cotler, J., Hayden, P., Salton, G., Swingle, B., Walter, M.: Entanglement wedge reconstruction via universal recovery channels. arXiv preprint arXiv:1704.05839 (2017)

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Acknowledgements

We thank David Ding, Aram Harrow, Michael Walter, and Andreas Winter for valuable discussions. This work was supported by AFOSR (FA9550-16-1- 0082), CIFAR and the Simons Foundation.

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  1. Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, USA

    Patrick Hayden & Geoffrey Penington

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Correspondence to Geoffrey Penington.

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Communicated by M. M. Wolf

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Hayden, P., Penington, G. Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit. Commun. Math. Phys. 374, 369–432 (2020). https://doi.org/10.1007/s00220-020-03689-1

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