Cauchy-Schwarz Research Articles

Evidence of the Quantum Optical Nature of High-Harmonic Generation

High-harmonic generation is a light up-conversion process occurring in a strong laser field, leading to coherent bursts of extreme ultrashort broadband radiation [Lewenstein , Phys. Rev. A , 2117 (1994)]. As a new perspective, we propose that ultrafast strong-field electronic or photonic processes such as high-harmonic generation can potentially generate nonclassical states of light well before the decoherence of the system occurs [Gorlach , Nat. Commun. , 4598 (2020); Stammer ., Phys. Rev. Lett. , 123603 (2022)]. This could address fundamental challenges in quantum technology such as scalability, decoherence, or the generation of massively entangled states [Lewenstein , Luca Argenti Michael Chini, 27 (2024)]. Here, we report experimental evidence of the nonclassical nature of the harmonic emission in several semiconductors excited by a femtosecond infrared laser. By investigating single- and double-beam intensity cross-correlation [Loudon, Rep. Prog. Phys. , 913 (1980)], we measure characteristic nonclassical features in the single-photon statistics. We observe two-mode squeezing in the generated harmonic radiation, which depends on the laser intensity that governs the transition from super-Poissonian to Poissonian photon statistics. The measured violation of the Cauchy-Schwarz inequality realizes a direct test of multipartite entanglement in high-harmonic generation [Wasak, Phys. Rev. A , 033616 (2014)]. This result is supported by the theory of multimodal detection and the Hamiltonian from which the effective squeezing modes of the harmonics can be derived [Gonoskov , Phys. Rev. B , 125110 (2024); Christ New J. Phys. , 033027 (2011)]. With this work, we show experimentally that high-harmonic generation is a new quantum bosonic platform that intrinsically produces nonclassical states of light with unique features such as multipartite broadband entanglement or multimode squeezing. The source operates at room temperature, using standard semiconductors and a standard commercial fiber laser, opening up new routes for the quantum industry, such as optical quantum computing, communication, and imaging. Published by the American Physical Society 2024

Uncertainty of quantum channels based on symmetrized ρ-absolute variance and modified Wigner-Yanase skew information

We present the uncertainty relations in terms of the symmetrized ρ-absolute variance, which generalizes the uncertainty relations for arbitrary operators (not necessarily Hermitian) to quantum channels. By employing the quantity ∣U ρ ∣(Φ) proposed by Zhang et al (2023) Quantum Inf. Process. 22, 456, we also establish tighter uncertainty relations for quantum channels by using the Cauchy-Schwarz inequality. Detailed examples are provided to illustrate the tightness of our results.

NEW GENERALIZATION AND REFINEMENT OF THE DIAMOND-α INTEGRAL CAUCHY–SCHWARTZ INEQUALITY ON TIME SCALES

This paper is concerned with new generalizations and refinements of the diamond-[Formula: see text] integral Cauchy–Schwartz inequality on time scales. The new generalization and refinement of the Cauchy–Schwartz inequality related to diamond integral on time scales are also considered. These obtained inequalities unify continuous inequalities and their corresponding discrete forms associated with Cauchy–Schwartz inequality.

Harmonic analysis approach to the proof of Heisenberg inequality

the Heisenberg uncertainty Principle is a fundamental principle in quantum mechanics, which was developed by the German physicist Werner Heisenberg and was proposed by him in 1927. This principle states that for a pair of physical quantities that share phase space, such as position and momentum, it is impossible to accurately measure their values at the same time. There are several variants of it in harmonic analysis studies, and the article will introduce some of them in R^1 space and L^2 space. In the process of providing the Heisenberg inequality, the article proved the Plancherel identity and Schwartz inequality by using Fourier transform and inverse Fourier transform. Finally, author solved the equation of the wave function (x) . The famous physicists Heisenberg proposed one of the more novel ideas in quantum mechanics the existence of unobservable orbits cannot be assumed, which did bring great influence in quantum mechanics. The article will introduce the conception of Heisenberg inequality and try to finish the proof.

Enhanced Cauchy–Schwarz inequality and some of its statistical applications

Enhanced Cauchy–Schwarz inequality and some of its statistical applications

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries

Abstract A class of unital qubit maps displaying diagonal unitary and orthogonal symmetries is analyzed. Such maps have already found a lot applications in quantum information theory. We provide a complete characterization of this class of maps showing intricate relation between positivity, operator Schwarz inequality, and complete positivity. Finally, it is shown how to generalize the entire picture beyond unital case (so called generalized Schwarz maps). Interestingly, the first example of Schwarz but not completely positive map found by Choi belongs to our class. As a case study we provide a full characterization of Pauli maps. Our analysis leads to generalization of seminal Fujiwara–Algoet conditions for Pauli quantum channels.

Efficiency bounds for bipartite information-thermodynamic systems.

In this paper, we introduce an approach to derive a lower bound for the entropy production rate of a subsystem by utilizing the Cauchy-Schwarz inequality. It extends to establishing comprehensive upper and lower bounds for the efficiency of two subsystems. These bounds are applicable to a wide range of Markovian stochastic processes, which enhances the accuracy in depicting the range of energy conversion efficiency between subsystems. Empirical validation is conducted using a two-quantum-dot system model, which serves to confirm the effectiveness of our inequality in refining the boundaries of efficiency.

Generalized Cauchy–Bunyakovsky–Schwarz Inequalities and Their Applications

Generalized Cauchy–Bunyakovsky–Schwarz Inequalities and Their Applications

On an inequality related to the volume of a parallelepiped

The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

Incipient fault detection and isolation with Cauchy–Schwarz divergence: A probabilistic approach

To monitor the dynamics and non-stationarity inherent in industrial processes, we propose a novel incipient fault detection and isolation scheme grounded in a probabilistic perspective, using the Cauchy–Schwarz (CS) divergence. Our innovation lies in the utilization of marginal CS divergence for incipient fault detection and the conditional CS divergence for fault isolation. This approach neither require prior parametric assumptions about the underlying data distribution nor depend on historical fault data, while simultaneously providing explanatory diagnostics. Beyond this, we develop a change point detection-base diagnosis technique for practical engineering applications. This online process monitoring technique guarantees timely intervention to uphold process stability and safety. We demonstrate the compelling performance, higher detection rate and lower alarm rate, of the CS divergence over prevalent divergence-based approaches, such as Kullback–Leibler divergence, Wasserstein distance and Mahalanobis distance. We also illustrate the explanatory insights offered by conditional CS divergence in fault isolation on synthetic data, benchmarks of continuous stirred-tank reactor process and continuous stirred-tank heater process and even a real-world continuous catalytic reforming process. Code of this CS divergence based fault detection and isolation is available at https://github.com/Feiya-Lv/Incipient-Fault-Detection-and-Isolation-with-Cauchy--Schwarz-Divergence.

Cauchy-Schwarz inequality for shifted quantum integral operator

We show that Cauchy-Schwarz inequality for shifted quantum integral operator does not hold in general and we prove the conditions under which it is valid. We apply it to Ostrowski type inequalities for shifted quantum integral operator.

On the validity of the Cauchy–Schwarz inequality for the bracket map

Due to its properties, the bracket map associated with a dual integrable unitary representation of a locally compact group can be viewed as a certain operator-valued inner product; however, in the non-commutative setting, the Cauchy–Schwarz property for bracket is no longer present in its full strength. In this paper, we show that fulfillment of the property, even in weaker forms, has strong consequences on the underlying group \(G\) and the corresponding von Neumann algebra \(\mbox{VN}(G)\). In particular, we show that for unimodular group \(G\), positive elements of the \(L^1\) space over \(\mbox{VN}(G)\) which are affiliated with the commutant of \(\mbox{VN}(G)\) are precisely those for which the weaker variant of the inequality is fulfilled, and that the validity of the Cauchy–Schwarz property for the appropriate set of elements indicates the existence of a closed abelian subgroup or abelian von Neumann subalgebra of \(\mbox{VN}(G)\).

Monotonicity of optimized quantum f-divergence

Optimized quantum f-divergence was first introduced by Wilde and further explored by Li and Wilde later. Wilde raised the question of whether the monotonicity of optimized quantum f-divergence can be generalized to maps that are not quantum channels. In this paper, we answer this question by generalizing the monotonicity of optimized quantum f-divergences to positive trace preserving maps satisfying a Schwarz inequality. Any 2-positive maps satisfy such a Schwarz inequality. The main tool in this paper is the Petz recovery map.

A Note on the Laplacian Energy of the Power Graph of a Finite Cyclic Group

In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime.

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

The uncertainty principle and quantum wave functions that are their own Fourier transforms

We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.

NONCLASSICAL PROPERTIES OF TWO-MODE DISPLACED FOCK STATE

In this paper, we investigate a variety of nonclassical properties of a two-mode displaced Fock state (TMDFS). Nonclassical properties such as sum squeezing, difference squeezing, antibunching effect, and the Cauchy-Schwarz inequality will be discussed analytically for the TMDFS. The results show that by changing the Fock state parameters between modes and the displacement parameter, the TMDFS exhibits enhanced nonclassical behaviors and makes a valuable contribution to quantum optics.

Refinement of the Cauchy-Schwartz inequality with refinements and generalizations of the numerical radius type inequalities for operators

Motivated by the results previously reported, the current work aims at developing new numerical radius upper bounds of Hilbert space opera- tors by offering new improvements to the well-known Cauchy-Schwarz inequal- ity. In particular, a novel Lemma (3.1) is given, which is utilized to further generalize several vector and numerical radius type inequalities, as well as pre- viously given extensions of the Cauchy-Schwartz inequality. Specifically, (2.5) (2.8) (1.6) have been generalized by (4.3) (4.1) (4.2)

An improved energy‐efficient driving strategy for routes with various gradients and speed limits

AbstractWith the increasing concerns about railway energy efficiency, two typical driving strategies have been used in actual train operation. One includes a sequence of full power traction, cruising, coasting, and full braking (CC). The other uses coasting–remotoring (CR) to replace cruising in CC. However, energy‐saving performance by CC and CR, which can be affected by route parameters of gradients and speed limits, has not been fully compared and studied. This paper analyses the energy distribution of CC and CR considering various route parameters and proposes an improved strategy for different gradients and speed limits. The detailed energy flow of CC and CR is analysed by Cauchy–Bunyakovsky–Schwarz inequality and the generalised Hölder's inequality, and then, a novel driving strategy CC_CR is designed. To verify the theoretical results and the effectiveness of the proposed strategy, three simulators with CC, CR, and CC_CR driving modes have been developed and implemented into case studies of four scenarios as well as a real‐world metro line. Simulation results demonstrate that CR can only outperform CC on routes with steep downhill and CC_CR is always the best strategy. The energy savings of CC_CR can be as much as 15% more than CR and 42% greater than CC.