Research Article
BibTex RIS Cite

A Rosetta Stone for information theory and differential equations

Year 2018, Volume: 1 Issue: 1, 45 - 64, 30.09.2018

Alessandro Selvitella

https://doi.org/10.33434/cams.448407

Abstract

In this paper, we propose a dictionary between Partial Differential Equations and Information Theory. As a model case, we will discuss in detail the example of the Schrödinger Equation and Shannon Information Theory. Comments will be made in both the continuous and discrete case and in both the noiseless and noisy case.

Keywords

Information Theory PDEs Nyquist Bit Rate Shannon Capacity Strichartz Estimates Strichartz Estimates

References

  • [1] S. AMARI AND H. NAGAOKA, Methods of Information Geometry, Translations of Mathematical Monographs Vol. 191 Am. Math. Soc. (2000).
  • [2] C. ATKINSON AND A. F. S. MITCHELL, Rao’s Distance Measure, Samkhy˜a-The Indian Journal of Statistics 43 (1981) 345-365.
  • [3] J. BENNETT, N. BEZ, A. CARBERY AND D. HUNDERTMARK, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 no. 2 (2009) 147-158.
  • [4] P. BILLINGSLEY, Probability and Measure, (2012) Wiley.
  • [5] S. BOBKOV, I. GENTIL AND M. LEDOUX, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. 80 (2001) 669-696.
  • [6] A. BURCHARD, A Short Course on Rearrangement Inequalities, http://www.math.toronto.edu/almut/ rearrange.pdf
  • [7] M. CHRIST AND Q. REN´E , Gaussians rarely extremize adjoint Fourier restriction inequalities for paraboloids, Proc. Amer. Math. Soc. 142 no. 3 (2014) 887-896.
  • [8] K. CONRAD, Probability Distributions and Maximum Entropy, http://www.math.uconn.edu/˜kconrad/ blurbs/analysis/entropypost.pdf
  • [9] T. M. COVER AND J. A. THOMAS, Elements of Information Theory, (1991) Wiley.
  • [10] A. DE BOUARD AND A. DEBUSSCHE, The nonlinear Schr¨odinger equation with white noise dispersion, J. Funct. Anal. 259 no. 5 (2010) 1300-1321.
  • [11] A. DEMBO, T. M. COVER AND J. A. THOMAS, Information Theoretic Inequalities, IEEE Transactions on Information Theory Vol. 37 no. 6 (1991) 1501-1518.
  • [12] J. DOLBEAULT, An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation, Discrete Contin. Dyn. Syst. 8 no. 2 (2002) 361-380.
  • [13] S. FLEGO, A. PLASTINO AND A. R. PLASTINO, Information Theory Consequences of the Scale-Invariance of Schr¨odinger’s Equation, Entropy 13 (2011) 2049-2058.
  • [14] D. FOSCHI, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 no. 1 (2005) 1-24.
  • [15] D. FOSCHI, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 no. 4 (2007) 739-774.
  • [16] B. R. FRIEDEN, Fisher Information as the basis for the Schr¨odinger wave equation, Am. J. Phys. 57 (1989) 1004-1008.
  • [17] B. R. FRIEDEN, Physics from Fisher Information: A Unification, 1st Ed. (1998) Cambridge University Press.
  • [18] B. R. FRIEDEN, Science from Fisher Information: A Unification, 2nd Ed. (2004) Cambridge University Press.
  • [19] B. R. FRIEDEN AND R. A. GATENBY, Principle of maximum Fisher information from Hardy’s axioms applied to statistical systems, Phys. Rev. E 88, 042144 (2013).
  • [20] B. R. FRIEDEN, A. PLASTINO, A. R. PLASTINO AND B. H. SOFFER, Schr¨odinger link between nonequilibrium thermodynamics and Fisher information, Physical Review E 66 046128 (2002)
  • [21] P. GARBACZEWSKI, Differential entropy and time Review, Entropy 7 (2005) 253-299.
  • [22] P. GIBILISCO, F. HIAI AND D. PETZ, Quantum covariance, quantum Fisher information, and the uncertainty relations, IEEE Trans. Inform. Theory 55 no. 1 (2009) 439-443.
  • [23] M. HAYASHI, S. ISHIZAKA, A. KAWACHI, G. KIMURA, GEN AND T. OGAWA, Introduction to quantum information science, (2015) Springer.
  • [24] M. KEEL, T. TAO, Endpoint Strichartz estimates, Amer. J. Math. 120 no. 5 (1998) 955-980.
  • [25] M. KUNZE, On the existence of a maximizer for the Strichartz inequality, Comm. Math. Phys. 243 no. 1 (2003) 137-162.
  • [26] R. LANDAUER, Information is Physical, Proc. Workshop on Physics and Computation PhysComp’92 IEEE Comp. Sci. Press (1993) 1-4.
  • [27] S. LUO, On Covariance and Quantum Fisher Information, Theory Prob. Appl. Vol. 53 No.2 (2009) 329-334.
  • [28] S. LUO, Quantum Fisher Information and Uncertainty Relations, Lett. Math. Phys. Volume 53 Issue 3 (2000) 243-251.
  • [29] S. LUO AND Q. ZHANG, On skew information, IEEE Trans. Inform. Theory 50 (2004) no. 8 1778-1782.
  • [30] S. LUO AND Q. ZHANG, Correction to ”On skew information”, IEEE Trans. Inform. Theory 51 no. 12 (2005) 4432.
  • [31] S. LUO AND Q. ZHANG, An informational characterization of Schr¨odinger’s uncertainty relations, J. Statist. Phys. 114 no. 5-6 (2004) 1557-1576.
  • [32] E. LUTWAK, S. LV, D. YANG AND G. ZHANG, Extensions of Fisher information and Stam’s inequality IEEE Trans. Inform. Theory 58 no. 3 (2012) 1319-1327.
  • [33] D. J. C. MACKAY, Information Theory, Inference, and Learning Algorithms, (2003) Cambridge University Press.
  • [34] K. MARDIA AND P. E. JUPP, Directional Statistics, (1999) Wiley.
  • [35] K. MARTON, An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces, Journal of Functional Analysis 264 (2013) 34-61.
  • [36] H. NAGAOKA, On Fisher information of quantum statistical models, Proc. 10th Symposium on Information Theory and Its Applications (1987) 241-246 (in Japanese).
  • [37] D. PETZ, Covariance and Fisher information in quantum mechanics, J. Phys. A 35 no. 4 (2002) 929-939.
  • [38] D. PETZ AND C. GHINEA Introduction to quantum Fisher information, Quantum probability and related topics QP–PQ: Quantum Probab. White Noise Anal. 27 (2011) 261-281.
  • [39] A. SELVITELLA, Remarks on the sharp constant for the Schrodinger Strichartz estimate and applications, Electronic Journal of Differential Equations Vol. 2015 No. 270 (2015) 1-19.
  • [40] C. E. SHANNON, A Mathematical Theory of Communication, The Bell System Technical Journal, Vol. 27 379-423/623-656 (July/October 1948).
  • [41] A. STEFANOV AND P. KEVREKIDIS, Asymptotic behaviour of small solutions for the discrete nonlinear Schr¨odinger and Klein-Gordon equations, Nonlinearity 18 no. 4 (2005) 1841-1857.
  • [42] R.S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 no. 3 (1977) 705-714.
  • [43] T.TAO, Nonlinear Dispersive Equations: Local and Global Analysis, (2006) CBMS Number 106.
  • [44] F. OTTO AND C. VILLANI, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000) 361-400.
  • [45] C. VILLANI, A short proof of the ”concavity of entropy power”, IEEE Trans. Inform. Theory 46 no.4 (2000) 1695-1696.
  • [46] A. WEHRL, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221-260.
  • [47] E. WEINSTEIN, Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application, https://www.youtube.com/watch?v=h5gnATQMtPg
  • [48] E. WEINSTEIN, Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application, http://pirsa.org/pdf/files/7c58ac48-fe90-425d-96c0-42fedcde51b7.pdf
  • [49] T. T. WU AND C. N. YANG, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D (3) 12 no. 12 (1975) 3845-3857.
  • [50] J. ZHOU, Derivatives in Mathematics and Physics,http://faculty.math.tsinghua.edu.cn/˜jzhou/ Connection04.pdf

Year 2018, Volume: 1 Issue: 1, 45 - 64, 30.09.2018

Alessandro Selvitella

https://doi.org/10.33434/cams.448407

Abstract

References

  • [1] S. AMARI AND H. NAGAOKA, Methods of Information Geometry, Translations of Mathematical Monographs Vol. 191 Am. Math. Soc. (2000).
  • [2] C. ATKINSON AND A. F. S. MITCHELL, Rao’s Distance Measure, Samkhy˜a-The Indian Journal of Statistics 43 (1981) 345-365.
  • [3] J. BENNETT, N. BEZ, A. CARBERY AND D. HUNDERTMARK, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 no. 2 (2009) 147-158.
  • [4] P. BILLINGSLEY, Probability and Measure, (2012) Wiley.
  • [5] S. BOBKOV, I. GENTIL AND M. LEDOUX, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. 80 (2001) 669-696.
  • [6] A. BURCHARD, A Short Course on Rearrangement Inequalities, http://www.math.toronto.edu/almut/ rearrange.pdf
  • [7] M. CHRIST AND Q. REN´E , Gaussians rarely extremize adjoint Fourier restriction inequalities for paraboloids, Proc. Amer. Math. Soc. 142 no. 3 (2014) 887-896.
  • [8] K. CONRAD, Probability Distributions and Maximum Entropy, http://www.math.uconn.edu/˜kconrad/ blurbs/analysis/entropypost.pdf
  • [9] T. M. COVER AND J. A. THOMAS, Elements of Information Theory, (1991) Wiley.
  • [10] A. DE BOUARD AND A. DEBUSSCHE, The nonlinear Schr¨odinger equation with white noise dispersion, J. Funct. Anal. 259 no. 5 (2010) 1300-1321.
  • [11] A. DEMBO, T. M. COVER AND J. A. THOMAS, Information Theoretic Inequalities, IEEE Transactions on Information Theory Vol. 37 no. 6 (1991) 1501-1518.
  • [12] J. DOLBEAULT, An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation, Discrete Contin. Dyn. Syst. 8 no. 2 (2002) 361-380.
  • [13] S. FLEGO, A. PLASTINO AND A. R. PLASTINO, Information Theory Consequences of the Scale-Invariance of Schr¨odinger’s Equation, Entropy 13 (2011) 2049-2058.
  • [14] D. FOSCHI, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 no. 1 (2005) 1-24.
  • [15] D. FOSCHI, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 no. 4 (2007) 739-774.
  • [16] B. R. FRIEDEN, Fisher Information as the basis for the Schr¨odinger wave equation, Am. J. Phys. 57 (1989) 1004-1008.
  • [17] B. R. FRIEDEN, Physics from Fisher Information: A Unification, 1st Ed. (1998) Cambridge University Press.
  • [18] B. R. FRIEDEN, Science from Fisher Information: A Unification, 2nd Ed. (2004) Cambridge University Press.
  • [19] B. R. FRIEDEN AND R. A. GATENBY, Principle of maximum Fisher information from Hardy’s axioms applied to statistical systems, Phys. Rev. E 88, 042144 (2013).
  • [20] B. R. FRIEDEN, A. PLASTINO, A. R. PLASTINO AND B. H. SOFFER, Schr¨odinger link between nonequilibrium thermodynamics and Fisher information, Physical Review E 66 046128 (2002)
  • [21] P. GARBACZEWSKI, Differential entropy and time Review, Entropy 7 (2005) 253-299.
  • [22] P. GIBILISCO, F. HIAI AND D. PETZ, Quantum covariance, quantum Fisher information, and the uncertainty relations, IEEE Trans. Inform. Theory 55 no. 1 (2009) 439-443.
  • [23] M. HAYASHI, S. ISHIZAKA, A. KAWACHI, G. KIMURA, GEN AND T. OGAWA, Introduction to quantum information science, (2015) Springer.
  • [24] M. KEEL, T. TAO, Endpoint Strichartz estimates, Amer. J. Math. 120 no. 5 (1998) 955-980.
  • [25] M. KUNZE, On the existence of a maximizer for the Strichartz inequality, Comm. Math. Phys. 243 no. 1 (2003) 137-162.
  • [26] R. LANDAUER, Information is Physical, Proc. Workshop on Physics and Computation PhysComp’92 IEEE Comp. Sci. Press (1993) 1-4.
  • [27] S. LUO, On Covariance and Quantum Fisher Information, Theory Prob. Appl. Vol. 53 No.2 (2009) 329-334.
  • [28] S. LUO, Quantum Fisher Information and Uncertainty Relations, Lett. Math. Phys. Volume 53 Issue 3 (2000) 243-251.
  • [29] S. LUO AND Q. ZHANG, On skew information, IEEE Trans. Inform. Theory 50 (2004) no. 8 1778-1782.
  • [30] S. LUO AND Q. ZHANG, Correction to ”On skew information”, IEEE Trans. Inform. Theory 51 no. 12 (2005) 4432.
  • [31] S. LUO AND Q. ZHANG, An informational characterization of Schr¨odinger’s uncertainty relations, J. Statist. Phys. 114 no. 5-6 (2004) 1557-1576.
  • [32] E. LUTWAK, S. LV, D. YANG AND G. ZHANG, Extensions of Fisher information and Stam’s inequality IEEE Trans. Inform. Theory 58 no. 3 (2012) 1319-1327.
  • [33] D. J. C. MACKAY, Information Theory, Inference, and Learning Algorithms, (2003) Cambridge University Press.
  • [34] K. MARDIA AND P. E. JUPP, Directional Statistics, (1999) Wiley.
  • [35] K. MARTON, An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces, Journal of Functional Analysis 264 (2013) 34-61.
  • [36] H. NAGAOKA, On Fisher information of quantum statistical models, Proc. 10th Symposium on Information Theory and Its Applications (1987) 241-246 (in Japanese).
  • [37] D. PETZ, Covariance and Fisher information in quantum mechanics, J. Phys. A 35 no. 4 (2002) 929-939.
  • [38] D. PETZ AND C. GHINEA Introduction to quantum Fisher information, Quantum probability and related topics QP–PQ: Quantum Probab. White Noise Anal. 27 (2011) 261-281.
  • [39] A. SELVITELLA, Remarks on the sharp constant for the Schrodinger Strichartz estimate and applications, Electronic Journal of Differential Equations Vol. 2015 No. 270 (2015) 1-19.
  • [40] C. E. SHANNON, A Mathematical Theory of Communication, The Bell System Technical Journal, Vol. 27 379-423/623-656 (July/October 1948).
  • [41] A. STEFANOV AND P. KEVREKIDIS, Asymptotic behaviour of small solutions for the discrete nonlinear Schr¨odinger and Klein-Gordon equations, Nonlinearity 18 no. 4 (2005) 1841-1857.
  • [42] R.S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 no. 3 (1977) 705-714.
  • [43] T.TAO, Nonlinear Dispersive Equations: Local and Global Analysis, (2006) CBMS Number 106.
  • [44] F. OTTO AND C. VILLANI, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000) 361-400.
  • [45] C. VILLANI, A short proof of the ”concavity of entropy power”, IEEE Trans. Inform. Theory 46 no.4 (2000) 1695-1696.
  • [46] A. WEHRL, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221-260.
  • [47] E. WEINSTEIN, Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application, https://www.youtube.com/watch?v=h5gnATQMtPg
  • [48] E. WEINSTEIN, Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application, http://pirsa.org/pdf/files/7c58ac48-fe90-425d-96c0-42fedcde51b7.pdf
  • [49] T. T. WU AND C. N. YANG, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D (3) 12 no. 12 (1975) 3845-3857.
  • [50] J. ZHOU, Derivatives in Mathematics and Physics,http://faculty.math.tsinghua.edu.cn/˜jzhou/ Connection04.pdf
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Alessandro Selvitella

Publication Date September 30, 2018
Submission Date July 27, 2018
Acceptance Date September 16, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Selvitella, A. (2018). A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences, 1(1), 45-64. https://doi.org/10.33434/cams.448407
AMA Selvitella A. A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences. September 2018;1(1):45-64. doi:10.33434/cams.448407
Chicago Selvitella, Alessandro. “A Rosetta Stone for Information Theory and Differential Equations”. Communications in Advanced Mathematical Sciences 1, no. 1 (September 2018): 45-64. https://doi.org/10.33434/cams.448407.
EndNote Selvitella A (September 1, 2018) A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences 1 1 45–64.
IEEE A. Selvitella, “A Rosetta Stone for information theory and differential equations”, Communications in Advanced Mathematical Sciences, vol. 1, no. 1, pp. 45–64, 2018, doi: 10.33434/cams.448407.
ISNAD Selvitella, Alessandro. “A Rosetta Stone for Information Theory and Differential Equations”. Communications in Advanced Mathematical Sciences 1/1 (September 2018), 45-64. https://doi.org/10.33434/cams.448407.
JAMA Selvitella A. A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences. 2018;1:45–64.
MLA Selvitella, Alessandro. “A Rosetta Stone for Information Theory and Differential Equations”. Communications in Advanced Mathematical Sciences, vol. 1, no. 1, 2018, pp. 45-64, doi:10.33434/cams.448407.
Vancouver Selvitella A. A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences. 2018;1(1):45-64.
 Download Cover Image

Creative Commons License   The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..