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Quantum Ruzsa Divergence to Quantify Magic (2401.14385v2)

Published 25 Jan 2024 in quant-ph, cs.IT, math-ph, math.IT, math.MP, and math.PR

Abstract: In this work, we investigate the behavior of quantum entropy under quantum convolution and its application in quantifying magic. We first establish an entropic, quantum central limit theorem (q-CLT), where the rate of convergence is bounded by the magic gap. We also introduce a new quantum divergence based on quantum convolution, called the quantum Ruzsa divergence, to study the stabilizer structure of quantum states. We conjecture a ``convolutional strong subadditivity'' inequality, which leads to the triangle inequality for the quantum Ruzsa divergence. In addition, we propose two new magic measures, the quantum Ruzsa divergence of magic and quantum-doubling constant, to quantify the amount of magic in quantum states. Finally, by using the quantum convolution, we extend the classical, inverse sumset theory to the quantum case. These results shed new insight into the study of the stabilizer and magic states in quantum information theory.

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References (67)
  1. Quantum entropy and central limit theorem. Proceedings of the National Academy of Sciences, 120(25):e2304589120, 2023.
  2. Discrete quantum Gaussians and central limit theorem. arXiv:2302.08423, 2023b.
  3. Ju. V. Linnik. An information-theoretic proof of the central limit theorem with Lindeberg conditions. Theory of Probability & Its Applications, 4(3):288–299, 1959.
  4. Andrew Barron. Entropy and the central limit theorem. Ann. Probab., 14(1):336–342, Sep 1986.
  5. Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc., 17(4):975–982, 2004.
  6. Solution of Shannon’s problem on the monotonicity of entropy. Probability Theory and Related Fields, 129(3):381–390, 2004.
  7. Entropy and the central limit theorem. Probability Theory and Related Fields, 129(3):391–409, Sep 2004.
  8. The entropic central limit theorem for discrete random variables. In 2022 IEEE International Symposium on Information Theory (ISIT), pages 708–713, 2022.
  9. C. Cushen and R. Hudson. A quantum-mechanical central limit theorem. J. Appl. Probab., 8(3):454–469, Feb 1971.
  10. K. Hepp and E.H. Lieb. Phase-transitions in reservoir-driven open systems with applications to lasers and superconductors. Helv. Phys. Acta, 46(5):573–603, Feb 1973.
  11. K. Hepp and E.H. Lieb. On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model. Ann. Phys., 76(2):360–404, Feb 1973.
  12. N. Giri and W. von Waldenfels. An algebraic version of the central limit theorem. Probab. Theory Relat. Fields, 42(2):129–134, June 1978.
  13. D. Goderis and P. Vets. Central limit theorem for mixing quantum systems and the ccr-algebra of fluctuations. Commun. Math. Phys., 122(2):249–265, June 1978.
  14. T. Matsui. Bosonic central limit theorem for the one-dimensional xy model. Rev. Math. Phys., 14(07n08):675–700, June 2002.
  15. M. Cramer and J. Eisert. A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states. New J. Phys., 12(5):055020, May 2010.
  16. Central limit theorem for locally interacting Fermi gas. Commun. Math. Phys., 285(1):175–217, May 2009.
  17. A central limit theorem in many-body quantum dynamics. Commun. Math. Phys., 321(2):371–417, July 2013.
  18. Central limit theorems for the large-spin asymptotics of quantum spins. Probab. Theory Relat. Fields, 130(4):493–517, Dec 2004.
  19. Non-commutative central limits. Probab. Theory Relat. Fields, 82(4):527–544, Aug 1989.
  20. A quantum central limit theorem for sums of independent identically distributed random variables. J. Math. Phys., 51(1):015208, 2010.
  21. L. Accardi and Y. G. Lu. Quantum central limit theorems for weakly dependent maps. ii. Acta Math. Hung., 63(3):249–282, Sep 1994.
  22. Zhengwei Liu. Exchange relation planar algebras of small rank. Transactions of the American Mathematical Society, 368(12):8303–8348, Mar 2016.
  23. Block maps and Fourier analysis. Science China Mathematics, 62(8):1585–1614, Aug 2019.
  24. M. Hayashi. Quantum estimation and the quantum central limit theorem. Am. Math. Soc. Trans. Ser., 2(227):95–123, Sep 2009.
  25. Continuous-variable entanglement distillation and noncommutative central limit theorems. Phys. Rev. A, 87:042330, Apr 2013.
  26. Convergence rates for the quantum central limit theorem. Commun. Math. Phys., 383(1):223–279, Apr 2021.
  27. On a generalized central limit theorem and large deviations for homogeneous open quantum walks. Journal of Statistical Physics, 188(1):8, 2022.
  28. Towards optimal convergence rates for the quantum central limit theorem. arXiv preprint arXiv:2310.09812, 2023.
  29. Stabilizer testing and magic entropy. arXiv:2306.09292, 2023c.
  30. D. Gottesman. Stabilizer codes and quantum error correction. arXiv:quant-ph/9705052, 1997.
  31. Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R2493–R2496, Oct 1995.
  32. Alexei Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2–30, Jun 2003.
  33. D Gottesman. The Heisenberg representation of quantum computers. In Proc. XXII International Colloquium on Group Theoretical Methods in Physics, 1998, pages 32–43, 1998.
  34. Improved classical simulation of quantum circuits dominated by Clifford gates. Phys. Rev. Lett., 116:250501, Jun 2016.
  35. Trading classical and quantum computational resources. Phys. Rev. X, 6:021043, Jun 2016.
  36. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3:181, September 2019.
  37. Lower bounds on the non-Clifford resources for quantum computations. Quantum Sci. Technol., 5(3):035009, May 2020.
  38. Quantifying quantum speedups: Improved classical simulation from tighter magic monotones. PRX Quantum, 2:010345, Mar 2021.
  39. Classical simulation of quantum circuits by half Gauss sums. Commun. Math. Phys., 390:471–500, Mar 2022.
  40. Efficient classical simulation of noisy quantum computation. arXiv:1810.03176, 2018.
  41. Efficient classical simulation of Clifford circuits with nonstabilizer input states. Phys. Rev. Lett., 123:170502, Oct 2019.
  42. A polynomial-time classical algorithm for noisy random circuit sampling. STOC, page 945–957, Jun 2023.
  43. Dax Enshan Koh. Further extensions of Clifford circuits and their classical simulation complexities. Quantum Information & Computation, 17(3&4):0262–0282, 2017.
  44. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71:022316, Feb 2005.
  45. Magic can enhance the quantum capacity of channels. arXiv:2401.12105, 2024.
  46. Entropy power inequalities for qudits. J. Math. Phys., 57(5):052202, 2016.
  47. On a quantum entropy power inequality of Audenaert, Datta, and Ozols. J. Math. Phys., 57(6):062203, 2016.
  48. Free probability and operator algebras. European Mathematical Society, 2016.
  49. Additive Combinatorics, volume 105. Cambridge University Press, 2006.
  50. Terence Tao. Sumset and inverse sumset theory for Shannon entropy. Combinatorics, Probability and Computing, 19(4):603–639, 2010.
  51. Sumset and inverse sumset inequalities for differential entropy and mutual information. IEEE Transactions on Information Theory, 60(8):4503–4514, 2014.
  52. Sumsets and entropy revisited. arXiv preprint arXiv:2306.13403, 2023.
  53. On a conjecture of Marton. arXiv preprint arXiv:2311.05762, 2023.
  54. Entropy bounds on abelian groups and the Ruzsa divergence. IEEE Transactions on Information Theory, 64(1):77–92, 2018.
  55. Imre Z. Ruzsa. Sumsets and entropy. Random Structures & Algorithms, 34(1):1–10, 2009.
  56. Mokshay Madiman. On the entropy of sums. In 2008 IEEE Information Theory Workshop, pages 303–307, 2008.
  57. Entropy and set cardinality inequalities for partition-determined functions. Random Structures & Algorithms, 40(4):399–424, 2012.
  58. Thomas M Cover. Elements of information theory. John Wiley & Sons, 1999.
  59. Proof of the strong subadditivity of quantum‐mechanical entropy. Journal of Mathematical Physics, 14(12):1938–1941, 11 1973.
  60. D. Gross. Hudson’s theorem for finite-dimensional quantum systems. J. Math. Phys., 47(12):122107, 2006.
  61. Quantum Boolean functions. Chicago Journal of Theoretical Computer Science, 2010(1), January 2010.
  62. Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A, 54:1862–1868, Sep 1996.
  63. Mean entropy of states in classical statistical mechanics. Communications in Mathematical Physics, 5(4):288–300, 1967.
  64. Mean entropy of states in quantum‐statistical mechanics. Journal of Mathematical Physics, 9(7):1120–1125, 10 1968.
  65. Limits on classical communication from quantum entropy power inequalities. Nature Photon, 7(2):142–146, 2013.
  66. The entropy power inequality for quantum systems. IEEE Trans. Inform. Theory, 60(3):1536–1548, 2014.
  67. A generalization of the entropy power inequality to bosonic quantum systems. Nature Photon, 8(3):958–964, 2014.
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