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Journal of Signal and Information Processing

Maximal Phase Space Compression

Mihai Octavian Dima, Marian Petre

Published Jan 1, 2013• 0 citations
Publisher

Abstract

The (seldomly quoted) generalised-Heisenberg uncertainty relations are an effect of the quantum correlation coefficient inequalities. The quantum correlation coefficient determines how much a state can be compacted and on what basis. It is shown that how this can be used to best compress a signal (such as a radio wave, or a 2D laser complex field at a focal plane) while at the same time encrypting the signal.

DOI
10.4236/jsip.2013.42024

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The (seldomly quoted) generalised-Heisenberg uncertainty relations are an effect of the quantum correlation coefficient inequalities. The quantum correlation coefficient determines how much a state can be compacted and on what basis. It is shown that how this can be used to best compress a signal (such as a radio wave, or a 2D laser complex field at a focal plane) while at the same time encrypting the signal.
The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.
The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound $3\ensuremath{\hbar}/2$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.
The boundary between classical and quantum correlations is well characterized by linear constraints called Bell inequalities. It is much harder to characterize the boundary of the quantum set itself in the space of no-signaling correlations. For the points on the quantum boundary that violate maximally some Bell inequalities, J. Oppenheim and S. Wehner [Science 330, 1072 (2010)] pointed out a complex property: Alice's optimal measurements steer Bob's local state to the eigenstate of an effective operator corresponding to its maximal eigenvalue. This effective operator is the linear combination of Bob's local operators induced by the coefficients of the Bell inequality, and it can be interpreted as defining a fine-grained uncertainty relation. It is natural to ask whether the same property holds for other points on the quantum boundary, using the Bell expression that defines the tangent hyperplane at each point. We prove that this is indeed the case for a large set of points, including some that were believed to provide counterexamples. The price to pay is to acknowledge that the Oppenheim-Wehner criterion does not respect equivalence under the no-signaling constraint: for each point, one has to look for specific forms of writing the Bell expressions.
Quantum spatial uncertainty is usually introduced as a fundamental property of microscopic systems through the Heisenberg uncertainty relation between position and momentum. In this work, we investigate whether a Heisenberg scale uncertainty bound can emerge from classical higher dimensional geometry without initially assuming wave functions or noncommuting quantum operators. We consider ordinary three dimensional space as a projection of a higher dimensional configuration space containing one compact fourth spatial dimension q 4 with characteristic length L. The compact sector is described by a hidden phase space distribution ρ(q 4 ,p 4 ), where curvature and warping of the compact geometry generate a position- dependent compact momentum p 4 (q 4 ) and therefore a nonzero compact phase space covariance. To connect the hidden compact sector with observable variables, we introduce an explicit compact to three dimensional projection rule. Applying the classical Cauchy-Schwarz inequality to the projected observable variables yields a conditional lower bound on the measurable position-momentum product without any using wave function. The resulting uncertainty scale is determined by the parameter such as compact covariance, the phase overlap between the curvature-induced momentum modulation and the projection coordinate and a geometric compact action scale derived from the higher-dimensional gravitational sector. The compact gravitational action leads to an effective uncertainty scale of the form [Formula: see text] (c is light velocity, L is size of compact dimension and G 4 is gravity constant). The proposed framework provides a compact-geometric mechanism capable of generating a Heisenberg-scale projected uncertainty bound from higher dimensional classical geometry.