Uncertainty Inequality Research Articles

Heisenberg uncertainty inequality for certain Lie groups

We establish analogues of Heisenberg uncertainty inequality for some classes of Lie groups, such as connected and simply connected nilpotent Lie groups, diamond Lie groups and Heisenberg motion groups.

Uncertainty principles for the q-bessel fourier transform

ABSTRACTThe aim of this paper is to continue the study of the uncertainty principle to the q-Bessel Fourier transform , that began in [Dhaouadi L. Heisenberg uncertainty principle for the q-Bessel Fourier transform. Preprint; 2007; Dhaouadi L. On the q-Bessel Fourier transform. Bull Math Anal Appl. 2013;5(2):42–60; Hleili M, Nefzi B, Bsaissa A. A variation on uncertainty principles for the generalized q-Bessel Fourier transform. J Math Anal Appl. 2016;440:823–832]. More precisely, we show an extension of Faris's local uncertainty principle and a general form of Heisenberg-type uncertainty inequality for on the space . We also prove new uncertainty principles for on the subspaces of functions that are essentially concentrated on S and bandlimited on Σ, or functions that are essentially timelimited on S and bandlimited on Σ, where S and Σ are general subsets of finite measure.

The octonionic Fourier transform: Uncertainty relations and convolution

The octonionic Fourier transform: Uncertainty relations and convolution

Uncertainty equality with quantum memory and its experimental verification

As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.

Uncertainty relations for a non-canonical phase-space noncommutative algebra

We consider a non-canonical phase-space deformation of the Heisenberg–Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg–Pauli–Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.

Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix

Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix

Uncertainties of genuinely incompatible triple measurements based on statistical distance

We investigate the measurement uncertainties of a triple of positive-operator-valued measures based on statistical distance and formulate state-independent tight uncertainty inequalities satisfied by the three measurements in terms of triplewise joint measurability. In particular, uncertainty inequalities for three unbiased qubit measurements are presented with analytical lower bounds which relates to the necessary and sufficient condition of the triplewise joint measurability of the given triple. We show that the measurement uncertainties for a triple measurement are essentially different from the ones obtained by pairwise measurement uncertainties by comparing the lower bounds of different measurement uncertainties.

An Note on Uncertainty Inequalities for Deformed Harmonic Oscillators

The aim of this paper is to prove some uncertainty inequalities for a class of integral operators associated to deformed harmonic oscillators.

Minimum-uncertainty states and completeness of non-negative quasiprobability of finite-dimensional quantum systems

We construct minimum-uncertainty states and a non-negative quasi probability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relationof the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasi probability distribution. Its marginal distributions are smeared out. However, we show that this quasi probability is optimal in the sense that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasi probability is complete, i.e., it contains full information of the state. We show that the obtained quasi probability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.

Heisenberg–Pauli–Weyl inequality for connected nilpotent Lie groups

The purpose of this paper is to formulate and prove an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality for connected nilpotent Lie groups with noncompact center. Representation theory and a localized Plancherel formula play an important role in the proof.

Quaternionic shearlet transform

Quaternionic shearlet transform

Uncertainty principle for the quaternion Fourier transform

Uncertainty principle for the quaternion Fourier transform

Egalitarianism under Severe Uncertainty

In the spring of 2009, a novel strain of the H1N1 influenza virus, containing a never before witnessed combination of gene segments from human influenza, two forms of swine influenza, and avian influenza, 1 was declared a global pandemic. The UK Government had to decide whether to undertake, at a cost of £1.2 billion (USD 1.9 billion at the time, equivalent to 1 percent of that year’s health budget), an extensive set of preparatory measures, including the purchase of both antiviral medication and a novel vaccine in quantities sufficient to cover the entire UK population, or whether instead to take substantially less costly measures, which would involve having only a limited supply of these medicines and vaccines at hand.2 The possible.

Uncertainty Inequalities for the Linear Canonical Hilbert Transform

The Heisenberg uncertainty principle plays an important role in signal processing, applied mathematical and physics community. The extensions of the traditional uncertainty principle to the novel transforms have received considerable attention recently. In this paper, the uncertainty inequalities under the linear canonical Hilbert transform (LCHT) are considered. First, the uncertainty principle for the complex signals associated with the one-dimensional LCHT is investigated. Subsequently, the uncertainty inequality for the complex signals derived by the two-dimensional LCHT is proposed. It is shown that the uncertainty bounds for the complex signals derived by LCHT are different from the general complex signals in the LCT domain. In addition, the example and potential applications are also presented to show the correctness and useful of the derived results. Therefore, the uncertainty principles presented here not only can enrich the ensemble of the classical uncertainty principle, but can have many real applications too.

Uncertainty Relation Based on Wigner-Yanase-Dyson Skew Information with Quantum Memory.

We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states.

Norm Estimates and Uncertainty Principles Associated with the Laguerre Integral Transform

In this paper we present some results concerning the action of the Laguerre integral transform over the class of polynomials and over the class of infinitely differentiable functions. We prove several uncertainty inequalities of Heisenberg-type, and we also indicate an explicit boundedness for two classes of generalized Laguerre integral transforms which include the classical Riemann Liouville fractional integral.

Formula omitted]-coherent states associated with the noncommutative complex plane [formula omitted] for the Biedenharn–Macfarlane [formula omitted]-oscillator

formula omitted]-coherent states associated with the noncommutative complex plane [formula omitted] for the Biedenharn–Macfarlane [formula omitted]-oscillator

Windowed Bessel Fourier transform in quantum calculus and applications

This paper deals firstly with some q-harmonic analysis properties for the q-windowed Bessel Fourier transform related to the q-Bessel function of the third kind as Plancherel formula, inversion formula in $$\mathcal {L}_{q,2,\nu }$$ . Secondly, we give a weak uncertainty principle for it and we show that the portion of the q-windowed Bessel Fourier transform lying outside some set of finite measure cannot be arbitrarily too small. Then, we verify a version of Heisenberg–Pauli–Weyl type uncertainty inequalities for the q-windowed Bessel Fourier transform and its generalization. Finally, using the kernel reproducing theory, given by Saitoh (Theory of reproducing kernels and its applications. Longman Scientific and Technical, Harlow, 1988), we will be able to realize the natural and powerful approximation problems that lead to the q-windowed Bessel Fourier transform inverses.

On noise-resolution uncertainty in quantum field theory

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.

Uncertainty relations on nilpotent Lie groups.

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.