Skip to main content
SpringerLink
Log in

Quantum Information Geometry and Quantum Estimation

  • Chapter
  • First Online:
Quantum Information Theory

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

In Chap. 3 we examined the discrimination of two unknown quantum states. This chapter will consider the estimation of a parameter \(\theta \), which labels an unknown state parameterized by a continuous variable \(\theta \). It is a remarkable property of quantum mechanics that a measurement inevitably leads to the state reduction. Therefore, when one performs a measurement for state estimation, it is necessary to choose the measurement that extracts as much information as possible. This problem is called quantum estimation, and the optimization of the measurement is an important topic in quantum information theory. In the classical theory of estimation (of probability distributions) discussed in Sect. 2.2, we saw that the estimation is intimately related to geometrical structures such as the inner product. We can expect that such geometrical structures will also play an important role in the quantum case. The study of geometrical structures in the space of quantum states is called quantum information geometry and is an important field in quantum information theory. This chapter will examine the geometrical structure of quantum systems and discuss its applications to estimation theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The Bogoljubov inner product is also called the canonical correlation in statistical mechanics. In linear response theory, it is often used to give an approximate correlation between two different physical quantities.

References

  1. D. Petz, G. Toth, Lett. Math. Phys. 27, 205 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  2. C.W. Helstrom, Minimum mean-square error estimation in quantum statistics. Phys. Lett. 25A, 101–102 (1976)

    ADS  Google Scholar 

  3. D. Petz, Monotone metrics on matrix spaces. Lin. Alg. Appl. 224, 81–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Petz, C. Sudár, Extending the Fisher metric to density matrices, eds. by O.E. Barndorff-Nielsen, E.B.V. Jensenin. Geometry in Present Day Science, vol. 21 (World Scientific, Singapore, 1998)

    Google Scholar 

  5. H. Nagaoka, The world of quantum information geometry. IEICE Trans. J88-A(8), 874–885 (2005) (in Japanese)

    Google Scholar 

  6. C.W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976)

    MATH  Google Scholar 

  7. A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982); originally published in Russian (1980)

    Google Scholar 

  8. S. Amari, H. Nagaoka, Methods of Information Geometry (AMS & Oxford University Press, Oxford, 2000)

    MATH  Google Scholar 

  9. A. Uhlmann, Density operators as an arena for differential geometry. Rep. Math. Phys. 33, 253–263 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. K. Matsumoto, Geometry of a Quantum State, Master’s Thesis (Graduate School of Engineering, University of Tokyo, Japan, Department of Mathematical Engineering and Information Physics, 1995). (in Japanese)

    Google Scholar 

  11. H. Nagaoka, An asymptotically efficient estimator for a one-dimensional parametric model of quantum statistical operators, Proceedings 1988 IEEE International Symposium on Information Theory, vol. 198 (1988)

    Google Scholar 

  12. H. Nagaoka, On the relation between Kullback divergence and Fisher information—from classical systems to quantum systems, in Proceedings Joint Mini-Workshop for Data Compression Theory and Fundamental Open Problems in Information Theory (1992), pp. 63–72. (Originally in Japanese; also appeared as Chap. 27 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  13. H. Nagaoka, On the parameter estimation problem for quantum statistical models, in Proceedings 12th Symposium on Information Theory and Its Applications (SITA) (1989), pp. 577–582. (Also appeared as Chap. 10 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  14. H. Nagaoka, Differential geometrical aspects of quantum state estimation and relative entropy, eds. by V.P. Belavkin, O. Hirota, R.L. Hudson. Quantum Communications and Measurement (Plenum, New York, 1995), pp. 449–452

    Google Scholar 

  15. F. Hiai, D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99–114 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. A. Fujiwara, Statistical Estimation Theory for Quantum States, Master’s Thesis (Graduate School of Engineering, University of Tokyo, Japan, Department of Mathematical Engineering and Information Physics, 1993). (in Japanese)

    Google Scholar 

  17. A. Fujiwara, Private communication to H. Nagaoka (1996)

    Google Scholar 

  18. H. Nagaoka, Private communication to A. Fujiwara (1991)

    Google Scholar 

  19. H.L. van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968)

    MATH  Google Scholar 

  20. R.D. Gill, B.Y. Levit, Applications of the van Tree inequality: a Bayesian Cramér-Rao bound. Bernoulli 1, 59–79 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. R.D. Gill, Conciliation of Bayes and pointwise quantum state estimation: asymptotic information bounds in quantum statistics, eds. by V.P. Belavkin, M. Guta. Quantum Stochastics & Information: Statistics, Filtering & Control (World Scientific, 2008), pp. 239–261

    Google Scholar 

  22. H. Nagaoka, On fisher information of quantum statistical models, in Proceedings 10th Symposium on Information Theory and Its Applications (SITA), Enoshima, Kanagawa, Japan, 19–21 November 1987, pp. 241–246. (Originally in Japanese; also appeared as Chap. 9 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  23. T.Y. Young, Asymptotically efficient approaches to quantum-mechanical parameter estimation. Inf. Sci. 9, 25–42 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Gill, S. Massar, State estimation for large ensembles. Phys. Rev. A 61, 042312 (2000)

    Article  ADS  Google Scholar 

  25. M. Hayashi, K. Matsumoto, Statistical model with measurement degree of freedom and quantum physics, in RIMS koukyuroku Kyoto University, vol. 1055 (1998), pp. 96–110 (in Japanese). (Also appeared as Chap. 13 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  26. M. Hayashi, Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation. J. Phys. A Math. Gen. 35, 7689–7727 (2002); quant-ph/0202003 (2002). (Also appeared as Chap. 28 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  27. M. Hayashi, Second-order asymptotics in fixed-length source coding and intrinsic randomness. IEEE Trans. Inf. Theory 54, 4619–4637 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Matsumoto, Uhlmann’s parallelism in quantum estimation theory. quant-ph/9711027 (1997)

    Google Scholar 

  29. K. Matsumoto, A Geometrical Approach to Quantum Estimation Theory, Ph.D. Thesis (Graduate School of Mathematical Sciences, University of Tokyo, 1997)

    Google Scholar 

  30. K. Matsumoto, The asymptotic efficiency of the consistent estimator, Berry-Uhlmann’ curvature and quantum information geometry, eds. by P. Kumar, G.M. D’ariano, O. Hirotain. Quantum Communication, Computing, and Measurement 2 (Plenum, New York, 2000), pp. 105–110

    Google Scholar 

  31. M. Hayashi, K. Matsumoto, Asymptotic performance of optimal state estimation in qubit system. J. Math. Phys. 49, 102101 (2008); quant-ph/0411073 (2004)

    Google Scholar 

  32. M. Hayashi, Quantum estimation and quantum central limit theorem. Sugaku 55, 4, 368–391 (2003) (in Japanese); English translation is in Selected Papers on Probability and Statistics (American Mathematical Society Translations Series 2) vol. 277 (2009), pp 95–123

    Google Scholar 

  33. K. Matsumoto, Seminar notes (1999)

    Google Scholar 

  34. N. Giri, W. von Waldenfels, An algebraic version of the central limit theorem. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 42, 129–134 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Petz, An Invitation to the Algebra of Canonical Commutation Relations. Leuven Notes in Mathematical and Theoretical Physics, vol. 2 (1990)

    Google Scholar 

  36. M. Hayashi, Asymptotic quantum estimation theory for the thermal states family, eds. by P. Kumar, G.M. D’ariano, O. Hirota. Quantum Communication, Computing, and Measurement 2 (Plenum, New York, 2000) pp. 99–104; quant-ph/9809002 (1998). (Also appeared as Chap. 14 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  37. H.P. Yuen, M. Lax, Multiple-parameter quantum estimation and measurement of non-selfadjoint observables. IEEE Trans. Inf. Theory 19, 740 (1973)

    Article  MATH  Google Scholar 

  38. M. Hayashi, Asymptotic large deviation evaluation in quantum estimation theory, in Proceedings Symposium on Statistical Inference and Its Information-Theoretical Aspect (1998), pp. 53–82 (in Japanese)

    Google Scholar 

  39. A.S. Holevo, Some statistical problems for quantum Gaussian states. IEEE Trans. Inf. Theory 21, 533–543 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  40. D. Bruß, A. Ekert, C. Machiavello, Optimal universal quantum cloning and state estimation. Phys. Rev. Lett. 81, 2598–2601 (1998). (also appeared as Chap. 24 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  41. R.F. Werner, Optimal cloning of pure states. Phys. Rev. A 58, 1827 (1998)

    Article  ADS  Google Scholar 

  42. M. Keyl, R.F. Werner, Optimal cloning of pure states, judging single clones. J. Math. Phys. 40, 3283–3299 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. H. Fan, K. Matsumoto, M. Wadati, Quantum cloning machines of a d-level system. Phys. Rev. A 64, 064301 (2001)

    Article  ADS  Google Scholar 

  44. H. Fan, K. Matsumoto, X. Wang, M. Wadati, Quantum cloning machines for equatorial qubits. Phys. Rev. A 65, 012304 (2002)

    Article  ADS  Google Scholar 

  45. K. Usami, Y. Nambu, Y. Tsuda, K. Matsumoto, K. Nakamura, Accuracy of quantum-state estimation utilizing Akaike’s information criterion. Phys. Rev. A 68, 022314 (2003)

    Article  ADS  Google Scholar 

  46. M. Hayashi, Minimization of deviation under quantum local unbiased measurements, Master’s Thesis (Graduate School of Science, Kyoto University, Japan, Department of Mathematics, 1996)

    Google Scholar 

  47. M. Hayashi, A linear programming approach to attainable Cramer–Rao type bound, eds. by O. Hirota, A.S. Holevo, C.M. Cavesin. Quantum Communication, Computing, and Measurement (Plenum, New York, 1997), pp. 99–108. (Also appeared as Chap. 12 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  48. M. Hayashi, A linear programming approach to attainable cramer-rao type bound and randomness conditions. Kyoto-Math 97–08; quant-ph/9704044 (1997)

    Google Scholar 

  49. A. Fujiwara, H. Nagaoka, Coherency in view of quantum estimation theory, eds. by K. Fujikawa, Y.A. Ono. Quantum Coherence and Decoherence (Elsevier, Amsterdam, 1996), pp. 303–306

    Google Scholar 

  50. A. Fujiwara, H. Nagaoka, Quantum Fisher metric and estimation for pure state models. Phys. Lett. 201A, 119–124 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. A. Fujiwara, H. Nagaoka, An estimation theoretical characterization of coherent states. J. Math. Phys. 40, 4227–4239 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. K. Matsumoto, A new approach to the Cramér-Rao type bound of the pure state model. J. Phys. A Math. Gen. 35, 3111–3123 (2002)

    Article  ADS  MATH  Google Scholar 

  53. D. Petz, Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105, 123–131 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. D. Petz, Monotonicity of quantum relative entropy revisited. Rev. Math. Phys. 15, 29–91 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  56. V.P. Belavkin, Generalized uncertainty relations and efficient measurements in quantum systems. Teor. Mat. Fiz. 26(3), 316–329 (1976); quant-ph/0412030 (2004)

    Google Scholar 

  57. H. Nagaoka, A new approach to Cramér–Rao bound for quantum state estimation. IEICE Tech. Rep. IT 89-42(228), 9–14 (1989)

    Google Scholar 

  58. H. Nagaoka, A generalization of the simultaneous diagonalization of Hermitian matrices and its relation to quantum estimation theory. Trans. Jpn. Soc. Ind. Appl. Math. 1, 43–56 (1991) (Originally in Japanese; also appeared as Chap. 11 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  59. Y. Tsuda, K. Matsumoto, Quantum estimation for non-differentiable models. J. Phys. A Math. Gen. 38(7), 1593–1613 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. D.C. Brody, L.P. Hughston, R. Soc, Lond. Proc. A 454, 2445–2475 (1998)

    Article  ADS  Google Scholar 

  61. Y. Tsuda, Estimation of Polynomial of Complex Amplitude of Quantum Gaussian States, in Annual Meeting of the Japan Statistical Society, Proceedings, 2005. (in Japanese)

    Google Scholar 

  62. G.M. D’Ariano, Homodyning as universal detection, eds. by O. Hirota, A.S. Holevo, C.A. Caves. Quantum Communication, Computing, and Measurement (Plenum, New York, 1997), pp. 253–264; quant-ph/9701011 (1997). (also appeared as Chap. 33 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  63. L.M. Artiles, R.D. Gill, M.I. Guţă, An invitation to quantum tomography (II). J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67, 109–134 (2005)

    Google Scholar 

  64. C.W. Helstrom, Int. J. Theor. Phys. 11, 357 (1974)

    Article  Google Scholar 

  65. A.S. Holevo, Covariant measurements and uncertainty relations. Rep. Math. Phys. 16, 385–400 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. M. Ozawa, On the noncommutative theory of statistical decisions. Res. Reports Inform. Sci. A-74 (1980)

    Google Scholar 

  67. N.A. Bogomolov, Minimax measurements in a general statistical decision theory. Teor. Veroyatnost. Primenen. 26, 798–807 (1981); (English translation: Theory Probab. Appl., 26, 4, 787–795 (1981))

    Google Scholar 

  68. A.S. Holevo, Covariant Measurements and Optimality, in Chap. IV of Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982). Originally published in Russian (1980): "Bounds for generalized uncertainty of the shift parameter. Lecture Notes in Mathematics 1021, 243–251 (1983)

    Google Scholar 

  69. S. Massar, S. Popescu, Optimal extraction of information from finite quantum ensembles. Phys. Rev. Lett. 74, 1259 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. M. Hayashi, Asymptotic estimation theory for a finite dimensional pure state model. J. Phys. A Math. Gen. 31, 4633–4655 (1998). (Also appeared as Chap. 23 of Asymptotic Theory of Quantum Statistical Inference, M. Hayashi eds.)

    Google Scholar 

  71. M. Hayashi, On the second order asymptotics for pure states family. IEICE Trans. E88-JA, 903–916 (2005) (in Japanese)

    Google Scholar 

  72. E. Bagan, M.A. Ballester, R.D. Gill, A. Monras, R. Muñoz-Tapia, O. Romero-Isart, Optimal full estimation of qubit mixed states. Phys. Rev. A 73, 032301 (2006)

    Google Scholar 

  73. A. Fujiwara, Quantum channel identification problem. Phys. Rev. A 63, 042304 (2001)

    Article  ADS  Google Scholar 

  74. M. Sasaki, M. Ban, S.M. Barnett, Optimal parameter estimation of a depolarizing channel. Phys. Rev. A 66, 022308 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  75. D.G. Fischer, H. Mack, M.A. Cirone, M. Freyberger, Enhanced estimation of a noisy quantum channel using entanglement. Phys. Rev. A 64, 022309 (2001)

    Article  ADS  Google Scholar 

  76. A. Fujiwara, H. Imai, Quantum parameter estimation of a generalized Pauli channel. J. Phys. A Math. Gen. 36, 8093–8103 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. A. Fujiwara, Estimation of a generalized amplitude-damping channel. Phys. Rev. A 70, 012317 (2004)

    Article  ADS  Google Scholar 

  78. F. Tanaka, Investigation on fisher metric on the classical statistical models and the quantum ones, Master’s thesis (Graduate School of Information Science and Technology, University of Tokyo, Japan, Department of Mathematical Informatics, 2004). (in Japanese)

    Google Scholar 

  79. F. De Martini, A. Mazzei, M. Ricci, G.M. D’Ariano, Pauli tomography: complete characterization of a single qubit device. Fortschr. Phys. 51, 342–348 (2003)

    Article  MATH  Google Scholar 

  80. V. Bužek, R. Derka, S. Massar, Optimal quantum clocks. Phys. Rev. Lett. 82, 2207 (1999)

    Article  ADS  Google Scholar 

  81. A. Acín, E. Jané, G. Vidal, Optimal estimation of quantum dynamics. Phys. Rev. A 64, 050302(R) (2001)

    Article  ADS  Google Scholar 

  82. A. Fujiwara, Estimation of SU(2) operation and dense coding: an information geometric approach. Phys. Rev. A 65, 012316 (2002)

    Article  ADS  Google Scholar 

  83. M.A. Ballester, Estimation of unitary quantum operations. Phys. Rev. A 69, 022303 (2004)

    Article  ADS  Google Scholar 

  84. E. Bagan, M. Baig, R. Muñoz-Tapia, Entanglement-assisted alignment of reference frames using a dense covariant coding. Phys. Rev. A 69, 050303(R) (2004)

    Article  ADS  Google Scholar 

  85. M. Hayashi, Parallel treatment of estimation of SU(2) and phase estimation. Phys. Lett. A 354, 183–189 (2006)

    Article  ADS  Google Scholar 

  86. M. Hayashi, Estimation of SU(2) action by using entanglement, in Proceedings 9th Quantum Information Technology Symposium (QIT9), NTT Basic Research Laboratories, Atsugi, Kangawa, Japan, 11–12 December 2003, pp. 9–13 (in Japanese)

    Google Scholar 

  87. G. Chiribella, G.M. D’Ariano, P. Perinotti, M.F. Sacchi, Efficient use of quantum resources for the transmission of a reference frame. Phys. Rev. Lett. 93, 180503 (2004)

    Article  ADS  Google Scholar 

  88. E. Bagan, M. Baig, R. Muñoz-Tapia, Quantum reverse-engineering and reference-frame alignment without nonlocal correlations. Phys. Rev. A 70, 030301(R) (2004)

    Article  ADS  Google Scholar 

  89. G. Chiribella, G.M. D’Ariano, P. Perinotti, M.F. Sacchi, Covariant quantum measurements which maximize the likelihood. Phys. Rev. A 70, 061205 (2004)

    Article  ADS  Google Scholar 

  90. G. Chiribella, G.M. D’Ariano, P. Perinotti, M.F. Sacchi, Maximum likelihood estimation for a group of physical transformations. Int. J. Quantum Inform. 4, 453 (2006)

    Article  MATH  Google Scholar 

  91. M. Hayashi, S. Vinjanampathy, L.C. Kwek, Unattainable & attainable bounds for quantum sensors (2016). arXiv:1602.07131

  92. G. Chiribella, G.M. D’Ariano, M.F. Sacchi, Optimal estimation of group transformations using entanglement. Phys. Rev. A 72, 042338 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  93. M. Hayashi, Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation. Commun. Math. Phys. 304(3), 689–709 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  94. M.A. Morozowa, N.N. Chentsov, Itogi Nauki i Tekhniki 36, 289–304 (1990). (in Russian)

    Google Scholar 

  95. M. Hayashi, Characterization of several kinds of quantum analogues of relative entropy. Quant. Inf. Comput. 6, 583–596 (2006)

    MathSciNet  MATH  Google Scholar 

  96. K. Matsumoto, Reverse estimation theory, complementarity between RLD and SLD, and monotone distances. quant-ph/0511170 (2005); Proceedings 9th Quantum Information Technology Symposium (QIT13), Touhoku University, Sendai, Miyagi, Japan, 24–25 November 2005, pp. 81–86

    Google Scholar 

  97. A. Fujiwara, A Geometrical Study in Quantum Information Systems, Ph.D. Thesis (Department of Mathematical Engineering and Information Physics, Graduate School of Engineering, University of Tokyo, Japan, 1995)

    Google Scholar 

  98. M. Ohya, D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Nagoya, Aichi, Japan

    Masahito Hayashi

Authors
  1. Masahito Hayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masahito Hayashi .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hayashi, M. (2017). Quantum Information Geometry and Quantum Estimation. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-49725-8_6

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-49723-4

  • Online ISBN: 978-3-662-49725-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation