Entropic Uncertainty Principle Research Articles

Information-theoretic measures and Compton profile of H atom under finite oscillator potential

Information-theoretic measures for nl (2L) states of a H atom (with n=1−10 and l=0−2 , where n and l denote principal and angular momentum quantum numbers) have been investigated within a quantum dot by utilizing the Ritz variational principle, with the help of a Slater-type basis set. A well-established two-parameter (depth and width) model of finite oscillator potential is used to simulate the dot environment. The variationally optimized position (r)-space wave function is utilized to determine the momentum (p)-space wave function, leading to the generation of p-space radial density distribution. We explore the impact of cavity parameters on quantum information theoretic measures, such as Shannon (S) and Fisher information (I) entropy, in the ground as well as the excited state. The results of S were also used to test the Bialynicki–Birula–Mycielski inequality, related to the entropic uncertainty principle for the confined H atom. Some simple new fitting laws pertaining to S and I have been proposed. Furthermore, the p-space radial density is employed to derive the Compton profile of the confined H atom. Possible tunability of S,I and Compton profiles with respect to the parameters is noted.

Entropic uncertainty principle for mixed states

The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this paper, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurement bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying quantum state. These results can be directly used to bound the extractable randomness of a source-independent quantum random number generator in the presence of fully quantum attacks, to certify entanglement between trusted parties, or to bound the entanglement of a system with an untrusted environment. Published by the American Physical Society 2024

Continuous Krishna–Parthasarathy entropic uncertainty principle

Continuous Krishna–Parthasarathy entropic uncertainty principle

Entropic uncertainty in quantum-state cryptography : A mathematical framework for quantum-resilient encryption

In the escalating race between cryptographic security and quantum computing capabilities, the need for robust encryption methodologies that can withstand the prowess of quantum algorithms is more pressing than ever. This paper introduces a novel cryptographic framework, grounded in the principles of quantum mechanics and the entropic uncertainty principle, to forge a path towards quantum-resilient encryption. At the heart of this approach lies the integration of the entropic uncertainty inherent in quantum states, a fundamental aspect often overlooked in traditional cryptographic strategies. By harnessing this intrinsic uncertainty of quantum mechanics, we propose a mathematical framework that not only challenges the conventional paradigms of encryption but also sets a new benchmark for security in the quantum computing era. The paper delves into the theoretical underpinnings of quantum mechanics relevant to cryptography, with a particular focus on the entropic uncertainty principle. This principle, which posits a natural limit on the precision with which certain pairs of physical properties can be known, serves as the cornerstone of our proposed encryption method. We meticulously develop and outline a mathematical model that leverages this principle, ensuring that the encrypted information remains secure against the formidable computational capabilities of quantum algorithms. We contrast our approach with existing cryptographic methods, highlighting the enhanced security features offered by the entropic uncertainty-based model. The findings underscore the potential of this framework to serve as a resilient encryption mechanism in a landscape increasingly dominated by quantum computing technologies. This research paves the way for a new era of encryption, one that embraces the uncertainty of quantum mechanics as its shield against the threats posed by quantum computing.

Sharp Stability for LSI

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier–Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability inequality, the remainder is in terms of the entropy, not a metric. Recently, a stability result for H1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an Lp norm. Afterward, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H1 convergence is identified in this paper. Moreover, an explicit H1 bound via a moment assumption is shown. Additionally, the Lp stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

Modified entropies as the origin of generalized uncertainty principles

The Heisenberg uncertainty principle is known to be connected to the entropic uncertainty principle. This correspondence is obtained employing a Gaussian probability distribution for wave functions associated to the Shannon entropy. Independently, due to quantum gravity effects the Heisenberg uncertainty principle has been extended to a Generalized Uncertainty Principle (GUP). In this work, we show that GUP has been derived from considering non-extensive entropies, proposed by one of us. We found that the deformation parameters associated with S+ and S− entropies are negative and positive respectively. This allows us to explore various possibilities in the search of physical implications. We conclude that non-extensive statistics constitutes a signature of quantum gravity.

Rényi Entropy, Signed Probabilities, and the Qubit.

The states of the qubit, the basic unit of quantum information, are 2 × 2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

On Quantum Entropy.

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.

The Quantum Condition Space

AbstractThe fundamental properties of quantum physics are exploited to evaluate event probabilities with projection measurements. Next, to study what events can be specified by quantum methods, the concept of the condition space is introduced, which is found to be the dual space of the classical outcome space of bit strings. Like the classical outcome space generates the quantum state space, the condition space generates the quantum condition space being the central idea of this work. The quantum condition space permits the existence of entangled conditions having no classical equivalent. In addition, the quantum condition space is related to the quantum state space by a Fourier transform guaranteed by the Pontryagin duality, and therefore an entropic uncertainty principle can be defined. The quantum condition space offers a novel perspective of understanding quantum states with the duality picture. Furthermore, the quantum conditions have physical meanings and realizations of their own and thus may be studied for purposes beyond the original motivation of characterizing events for probability evaluation. Finally, the relation between the condition space and quantum circuits provides insights into how quantum states are collectively modified by quantum gates, which may lead to deeper understanding of the complexity of quantum circuits.

Mutually testing source-device-independent quantum random number generator

Quantum random numbers have an incomparable advantage over pseudo-random numbers since randomness originates from intrinsic property of quantum mechanics. The generation rate and the security of quantum random numbers are two significant indicators of a quantum random number generator (QRNG) for practical applications. Here we propose a mutually testing source-device-independent QRNG by simultaneously measuring a pair of conjugate quadratures from two separate parts of an untrusted continuous-variable quantum state. The amounts of randomness of the quadratures can be mutually estimated by each other via entropic uncertainty principle. Instead of randomly toggling between the conjugate quadratures of one state for collecting different types of data, two quadratures can generate check data and raw bits simultaneously and continuously in this mutually testing manner, which enhances the equivalent generation rate of private random bits to around 6 Gbit/s with a 7.5 mW laser beam. Moreover, the overall security is also improved by adjusting the conditional min-entropy in real time according to the continually monitored fluctuations of the local oscillator and the randomly measured electronic noise of homodyne detectors.

Uncertainty Principle for the Two-Sided Quaternion Windowed Linear Canonical Transform

In this paper, we investigate the (two-sided) quaternion windowed linear canonical transform (QWLCT) and study the uncertainty principles associated with the QWLCT. First, several important properties of the QWLCT such as bounded, shift, modulation and orthogonality relations are presented based on the spectral representation of the quaternionic linear canonical transform (QLCT). Second, Pitt’s inequality and the Lieb inequality for the QWLCT are explored. Moreover, we study different kinds of uncertainty principles for the QWLCT, such as the logarithmic uncertainty principle, the entropic uncertainty principle, the Lieb uncertainty principle and Donoho–Stark’s uncertainty principle. Finally, we provide a numerical example and a potential application to signal recovery by using Donoho–Stark’s uncertainty principle associated with the QWLCT.

Uncertainty principles for the two‐sided offset quaternion linear canonical transform

The offset quaternion linear canonical transform (OQLCT) provides a more general framework for a number of linear integral transforms in signal processing and optics, such as quaternion Fourier transform (QFT), fractional quaternion Fourier transform (FrQFT), and linear canonical transform (QLCT). We devote this paper to various different of uncertainty principles (UPs) for the two‐sided OQLCT, which including logarithmic UP, Heisenberg‐type UP, Hardy's UP, Beurling's UP, Entropic UP, Donoho‐Stark's UP, and Local UP. Moreover, we also prove Lieb's UP for the two‐sided short‐time offset quaternion linear canonical transform (SOQLCT).

Deficit estimates for the Logarithmic Sobolev inequality

Carlen showed that the Beckner-Hirschman entropic uncertainty principle, which is an improvement of Heisenberg's uncertainty principle, is equivalent to a quantitative logarithmic Sobolev inequality involving the entropy of the Fourier-Wiener transform. In this work, we identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. Inter-alia, we prove a new inequality relating the entropy of the Fourier-Wiener transform with the Fisher information and Wasserstein metric via the Monge-Ampère equation. Moreover, we investigate the dependence on the dimension of the stability constant and show that in certain configurations it can be taken independently and identify spaces in which the dimension-dependence cannot be removed. The techniques are based on optimal transport theory and Fourier analysis. Lastly, we discuss a probabilistic approach to stability estimates based on Cramér's convolution theorem.

Entropic uncertainty principle, partition function and holographic principle derived from Liouville’s Theorem

An entropic version of Liouville’s Theorem is defined in terms of the conjugate variables (“hyperbolic position” and “entropic momentum”) of an entropic Hamiltonian. It is used to derive the Holographic Principle as applied to holomorphic structures that represent maximum entropy configurations. The Bekenstein-Hawking expression for black hole entropy is a consequence. Based on the entropic commutator derived from Liouville’s Theorem and the same entropic conjugate variables, an entropic Uncertainty Principle (in units of Boltzmann’s constant) isomorphic to the kinematic Uncertainty Principle (in units of Planck’s constant) is also derived. These formal developments underpin the previous treatment of Quantitative Geometrical Thermodynamics (QGT) which has established (entirely on geometric entropy grounds) the stability of the double-helix, the double logarithmic spiral, and the sphere. Since in the QGT formalism the Boltzmann and Planck constants are quanta of quantities orthogonal to each other in Minkowski spacetime, a solution of the Schrödinger Equation is demonstrated isomorphic to a probability term of an entropic Partition Function, where both are defined by path integrals obeying the stationary principle: this isomorphism represents an important symmetry of the formalism. The geometry of a holomorphic structure must also exhibit at least C2 symmetry.

On uncertainty principle for the two-sided quaternion linear canonical transform

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose five different forms of uncertainty principles for the two-sided QLCT, including logarithmic uncertainty principle, Heisenberg-type uncertainty principle, local uncertainty principle, Benedicks–Amrein–Berthier uncertainty principle and entropic uncertainty principle. These consequences actually describe the quantitative relationships of a quaternion-valued signal in arbitrary two different QLCT domains, and they have great applications in signal recovery and physical quantum.

Sequences of lower bounds for entropic uncertainty relations from bistochastic maps

Given two orthonormal bases and , the basic form of the entropic uncertainty principle is stated in terms of the sum of the Shannon entropies of the probabilities of measuring and onto a given quantum state. State independent lower bounds for this sum encapsulate the degree of incompatibility of the observables diagonal in the and bases, and are usually derived by extracting as much information as possible from the unitary operator U connecting the two bases. Here we show a strategy to derive sequences of lower bounds based on alternating sequences of measurements onto and . The problem can be mapped into the multiple application of bistochastic processes that can be described by the powers of the unistochastic matrices directly derivable from U. By means of several examples we study the applicability of the method. The results obtained show that the strategy can allow for an advantage both in the pure state and in the mixed state scenario. The sequence of lower bounds is obtained with resources which are polynomial in the dimension of the underlying Hilbert space, and it is thus suitable for studying high dimensional cases.

The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

Generalized entropic uncertainty principles on the complex signals in terms of fractional Hilbert transform

Uncertainty principle is of much signification to physics and so on. Hilbert transform (HT) plays the role of changing the real signal into the complex signal, which is related to the Fourier transform (FT) in traditional time-frequency domain. The fractional Hilbert transform (FRHT) plays the same role as the traditional counterpart in the fractional Fourier transform (FRFT) domain as well as in optics. Different from the general complex signals, the complex signals derived from HT and FRHT have the new properties such as the absence of negative frequency and so on. In this paper, the generalized entropic uncertainty relations in terms of Shannon entropy, Rényi entropy and Tsallis entropy with respect to the complex signals derived from FRHT are demonstrated for the first time based on the mathematical principles, including the special cases for the traditional HT. Interestingly, these novel derived uncertainty bounds proved to be different from and much lower (e.g., always less by ln2) in some cases than that of the traditional complex (and real) signals, which reveals the new properties of these complex signals derived from FRHT for the further potential valuable application. Moreover, the relationships between the uncertainty principles from the two types of complex signals are established. Finally, the numerical simulation is performed in this paper to show the efficiency of the proposed principles.

Minimizers of the entropic uncertainty principle for locally compact quantum groups

In this paper, we give an alternative proof of the Hirschman-Beckner uncertainty principle for locally compact quantum groups and characterize the minimizers of the uncertainty principles.

Uncertainty Principles for the Offset Linear Canonical Transform

The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we extend some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.