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arXiv:1811.12592 · Scientific Reports

Noise-resolution uncertainty principle in classical and quantum systems

Timur E. Gureyev, Alexander Kozlov, David M. Paganin, Yakov I. Nesterets, Harry M. Quiney

Published May 12, 2020• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41598-020-64539-7

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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.
An uncertainty inequality is presented that establishes a lower limit for the product of the variance of the time-averaged intensity of a mode of a quantized electromagnetic field and the degree of its spatial localization. The lower limit is determined by the vacuum fluctuations within the volume corresponding to the width of the mode. This result also leads to a generalized form of the Heisenberg uncertainty principle for boson fields in which the lower limit for the product of uncertainties in the spatial and momentum localization of a mode is equal to the product of Planck's constant and a dimensionless functional which reflects the joint signal-to-noise ratio of the position and momentum of vacuum fluctuations in the region of the phase space occupied by the mode. Experimental X-ray synchrotron measurements provide an initial verification of the proposed theory in the case of Poisson statistics.
In (1), which deals with entropic uncertainty relationships, it is pointed out that x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values. We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.
One of the most surprising consequences of quantum mechanics is the entanglement of two or more distant particles. Even though there are still open questions regarding some fundamental issues related to entangled systems, quantum entanglement has started to play important roles in practical applications. Quantum imaging is one of the hot topics. Quantum imaging has many interesting features which are useful for different applications. For example, quantum imaging can be nonlocal, which is useful for secure two-dimensional information transfer. Quantum imaging can reach a much higher spatial resolution compared with classical imaging, even beyond the diffraction limit. This is useful for lithography and other microsystems fabrication technology. The super-resolution does not represent a violation of the uncertainty principle, it is just a quantum multi-particle phenomenon. Can quantum imaging be simulated classically? This question is closely related to the concerns of Einstein-Podolsky-Rosen of 1935. An attempted answer is given based on the analysis of a recent experiment of biphoton imaging-interference/diffraction.
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 L1L^1L1 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
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements, whereas the long standing mathematical formulation does not allow such an interpretation. Recently, a new relation was found by the present author to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing. Here, we discuss the state-dependent notion of precise measurements of a given observable, and consider a perfect distance such that zero distance implies the perfect correlation. Then, we shall show that even in the perfect error notions both the position measuring noise and the momentum disturbance can be arbitrarily small. We also show that it is possible to generalize the new noise-disturbance uncertainty relation to perfect error distances.