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arXiv:1503.04367 · ANZIAM Journal

On the noise-resolution duality, Heisenberg uncertainty and Shannon's information

Timur E. Gureyev, Frank de Hoog, Yakov I. Nesterets, David M. Paganin

Published Sep 30, 2015• 9 citations
arXivHTMLPDF

Abstract

Several variations of the Heisenberg uncertainty inequality are derived on the basis of `noise-resolution duality' recently proposed by us. The same approach leads to a related inequality that provides an upper limit for the information capacity of imaging systems in terms of the number of imaging quanta (particles) used in the experiment. These results are useful in the context of biomedical imaging constrained by the radiation dose delivered to the sample, or in imaging (e.g., astronomical) problems under low light conditions. References R. Bach, D. Pope, S.-H. Liou and H. Batelaan, Controlled double-slit electron diffraction. New J. Phys. 15:033018, 2013. doi:10.1088/1367-2630/15/3/033018 H. H. Barrett and K. J. Myers, Foundations of image science. Wiley, Hoboken, 2004. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471153001.html H. Bateman and A. Erdelyi, Tables of integral transforms. McGraw–Hill, New York, 1954. http://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353 M. Born and E. Wolf, Principles of optics. Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9781139644181 I. J. Cox and C. J. R. Sheppard, Information capacity and resolution in an optical system. J. Opt. Soc. Am. A 3:1152–1158, 1986. doi:10.1364/JOSAA.3.001152 M. G. Cowling and J. F. Price, Bandwidth versus time concentration: the Heisenberg–Pauli–Weyl inequality. SIAM J. Math. Anal. 15(1):151–165, 1984. doi:10.1137/0515012 F. de Hoog, G. Schmalz and T. E. Gureyev, An uncertainty inequality. Appl. Math. Lett. 38:84–86, 2014. doi:10.1016/j.aml.2014.07.009 I. Dreier, W. Ehm, T. Gneiting and D. Richards, Improved bounds for Laue's constant and multivariate extensions. Math. Nachr. 228:109–122, 2001. doi:10.1002/1522-2616(200108)228:1<109::AID-MANA109>3.0.CO;2-V V. A. Epanechnikov, Non-parametric estimation of a multivariate probability density. Theor. Probab. Appl. 14:153–158, 1969. doi:10.1137/1114019 A. C. Fabian, K. A. Pounds and R. D. Blandford, Frontiers of x-ray astronomy. Cambridge University Press, Cambridge, 2004. P. B. Felgett and E. H. Linfoot, On the assessment of optical images. Phil. T. R. Soc. A 247:369–407, 1955. doi:10.1098/rsta.1955.0001 G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3:207–238, 1997. doi:10.1007/BF02649110 T. E. Gureyev, Ya. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi and G. Tromba, Duality between noise and spatial resolution in linear systems. Opt. Express 22:9087–9094, 2014. doi:10.1364/OE.22.009087 T. E. Gureyev, S. C. Mayo, Ya. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin and G. Tromba, Investigation of imaging quality of synchrotron-based phase-contrast mammographic tomography. J. Phys. D: Appl. Phys. 47:365401, 2014. doi:10.1088/0022-3727/47/36/365401 M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, J. C. H. Spence and D. Starodub, An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy. J. Electron Spectrosc. 170:4–12, 2009. doi:10.1016/j.elspec.2008.10.008 E. Laeng and C. Morpurgo, An uncertainty inequality involving \(L^1\) norms. P. Am. Math. Soc. 127:3565–3572, 1999. http://www.jstor.org/stable/119352 Ya. I. Nesterets and T. E. Gureyev, Young's double-slit experiment: noise-resolution duality. Opt. Express 23:3373–3381, 2015. doi:10.1364/OE.23.003373 C. Shannon: Communication in the presence of noise. P. IRE 37:10–21, 1949. doi:10.1109/JRPROC.1949.232969

DOI
10.21914/anziamj.v56i0.9414

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Several variations of the Heisenberg uncertainty inequality are derived on the basis of `noise-resolution duality' recently proposed by us. The same approach leads to a related inequality that provides an upper limit for the information capacity of imaging systems in terms of the number of imaging quanta (particles) used in the experiment. These results are useful in the context of biomedical imaging constrained by the radiation dose delivered to the sample, or in imaging (e.g., astronomical) problems under low light conditions. References R. Bach, D. Pope, S.-H. Liou and H. Batelaan, Controlled double-slit electron diffraction. New J. Phys. 15:033018, 2013. doi:10.1088/1367-2630/15/3/033018 H. H. Barrett and K. J. Myers, Foundations of image science. Wiley, Hoboken, 2004. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471153001.html H. Bateman and A. Erdelyi, Tables of integral transforms. McGraw–Hill, New York, 1954. http://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353 M. Born and E. Wolf, Principles of optics. Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9781139644181 I. J. Cox and C. J. R. Sheppard, Information capacity and resolution in an optical system. J. Opt. Soc. Am. A 3:1152–1158, 1986. doi:10.1364/JOSAA.3.001152 M. G. Cowling and J. F. Price, Bandwidth versus time concentration: the Heisenberg–Pauli–Weyl inequality. SIAM J. Math. Anal. 15(1):151–165, 1984. doi:10.1137/0515012 F. de Hoog, G. Schmalz and T. E. Gureyev, An uncertainty inequality. Appl. Math. Lett. 38:84–86, 2014. doi:10.1016/j.aml.2014.07.009 I. Dreier, W. Ehm, T. Gneiting and D. Richards, Improved bounds for Laue's constant and multivariate extensions. Math. Nachr. 228:109–122, 2001. doi:10.1002/1522-2616(200108)228:1<109::AID-MANA109>3.0.CO;2-V V. A. Epanechnikov, Non-parametric estimation of a multivariate probability density. Theor. Probab. Appl. 14:153–158, 1969. doi:10.1137/1114019 A. C. Fabian, K. A. Pounds and R. D. Blandford, Frontiers of x-ray astronomy. Cambridge University Press, Cambridge, 2004. P. B. Felgett and E. H. Linfoot, On the assessment of optical images. Phil. T. R. Soc. A 247:369–407, 1955. doi:10.1098/rsta.1955.0001 G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3:207–238, 1997. doi:10.1007/BF02649110 T. E. Gureyev, Ya. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi and G. Tromba, Duality between noise and spatial resolution in linear systems. Opt. Express 22:9087–9094, 2014. doi:10.1364/OE.22.009087 T. E. Gureyev, S. C. Mayo, Ya. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin and G. Tromba, Investigation of imaging quality of synchrotron-based phase-contrast mammographic tomography. J. Phys. D: Appl. Phys. 47:365401, 2014. doi:10.1088/0022-3727/47/36/365401 M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, J. C. H. Spence and D. Starodub, An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy. J. Electron Spectrosc. 170:4–12, 2009. doi:10.1016/j.elspec.2008.10.008 E. Laeng and C. Morpurgo, An uncertainty inequality involving (L^1) norms. P. Am. Math. Soc. 127:3565–3572, 1999. http://www.jstor.org/stable/119352 Ya. I. Nesterets and T. E. Gureyev, Young's double-slit experiment: noise-resolution duality. Opt. Express 23:3373–3381, 2015. doi:10.1364/OE.23.003373 C. Shannon: Communication in the presence of noise. P. IRE 37:10–21, 1949. doi:10.1109/JRPROC.1949.232969
We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.
Statistical aspects of Young's double-slit diffraction experiment are analysed quantitatively. It is shown that the signal-to-noise ratio and the spatial resolution in the detected diffraction pattern satisfy a duality relationship which implies that both of them cannot be improved simultaneously beyond a certain limit if the total number of particles forming the image is fixed. As a consequence of this duality, it is possible to estimate the minimal number of particles that have to be detected in order for two slits separated by a given distance to be resolved with a confidence level corresponding to a pre-defined signal-to-noise ratio, e.g. according to the Rose criterion. These results are related to the recently introduced imaging system quality characteristic which combines the spatial resolution and the noise sensitivity, and allows one to estimate the efficiency with which imaging quanta are utilised in a system to deliver maximal amount of information about the imaged object. The presented results can be useful for applications where the imaging quanta are at a premium or where minimization of the radiation dose is important.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
We initiate a new study on the correspondence between the 20-vertex model and a SOS (Solid-on-Solid) model. In comparison to two previous works of the author which have characterized properties of the transfer and quantum monodromy, matrices of the 20-vertex model from the perspective of the quantum inverse scattering method [T. Ozawa, Commun. Math. Phys. 139, 479 (1991), https://doi.org/10.1007/BF02101876; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], in addition to the structure of nonlocal correlations [H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], the forthcoming approach is an adaptation of a study on the Rational 7-vertex model [P. V. Antonenko and P. A. Valinevich, J. Math. Sci. 284, 582 (2024), https://doi.org/10.1007/s10958-024-07374-9]. For both the 7-vertex and 20-vertex models nonlinearity through the Schrödinger’s equation plays a fundamental role not only in quantitatively describing well-posed initial value problems, but also through the intertwining operations underlying Yang–Baxter equations for the Rational and Universal, R-Matrices. Given a decomposition of the Universal R-Matrix obtained in [H. Boos et al., Rev. Math. Phys. 26, 143005 (2014), https://doi.org/10.1142/S0129055X14300052; H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208] we formalize a novel application of the quantum inverse scattering method which is related to entanglement, and other behaviors of nonlinear Schrödinger’s equations [N. N. Akhmediev, V. M. Eleonskii and N. E. Kulagin, Theor. Math. Phys. 72, 809 (1987), https://doi.org/10.1007/BF01017105; L. D. Carr, C. W. Clark and W. P. Reinhardt, Phys. Rev. A 62, 063611 (2000), https://doi.org/10.1103/PhysRevA.62.063610; D. Keating, N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 391, 1181 (2022), https://doi.org/10.1007/s00220-022-04334-9; K. Nakanishi and T. Ozawa, NoDEA Nonlinear Differ. Equ. Appl. 9, 45 (2002), https://doi.org/10.1007/s00030-002-8118-9; H. Niggemann, A. Klümper and J. Zittartz, Z. Phys. B 104, 103 (1997), https://doi.org/10.1007/s002570050425; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102; N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 356, 535 (2017), https://doi.org/10.1007/s00220-017-2983-x; G. Wang et al., Appl. Math. Lett. 172 (2026), https://doi.org/10.1016/j.aml.2025.109720; J. Yang, Phys. Rev. E 98, 042202 (2018), https://doi.org/10.1103/PhysRevE.98.042202; V. E. Zakharov and S. V. Manakov, Theor. Math. Phys. 19, 551 (1974), https://doi.org/10.1007/BF01035568]. Within the setting of the Rational 7-vertex model, Antonenko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-Matrix of a vertex model into the Boltzmann weight matrix of an SOS model. To further develop perspectives on classes of higher-dimensional SOS models, by leveraging previous computations with L-operators due to the author [L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], and by also manipulating q-exponentials, and other related factors from a factorization of the Universal R-Matrix due to Boos et al. [J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], analogs of intertwining vectors for the 20-vertex model can be obtained. In comparison to the system of equations that have been obtained for the Rational 7-vertex model, the system that one obtains for the 20-vertex model consists of 9 equations for each entry of the R-Matrix. Furthermore, from the fact that the 20-vertex model contains more possible configurations in the sample space than the Rational 7-vertex model does, the Boltzmann weight matrix of the corresponding SOS model asymptotically depends upon several contributions, a few of which include tensor products of basis elements and spectral parameters over the triangular lattice.