Search

The quantum entanglement measure of concurrence is shown to be directly calculable from a Tomita- Takesaki modular operator framework constructed from the local von Neumann algebras of observables for two quantum systems. Specifically, the Tomita-Takesaki modular conjugation operator $J$ that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence $C$ of a pure bi-variate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to $J$ as both a symmetry operator and a quantitative measure of entanglement. This procedure is then demonstrated for a supersymmetric quantum mechanical system and a real scalar field interacting with two entangled spin-$\frac{1}{2}$ Unruh-DeWitt qubit detectors. For the latter system, the concurrence result is shown to be consistent with some known results on the Bell-CHSH inequality for such a system., Comment: 10 pages

We study the chaotic motion of an optomechanical system coupled to a non-Markovian environment. We show that the environmental memory time can significantly affect chaos in an enhancing way. In addition to classical chaotic motion, the quantum entanglement in the presence of chaos is investigated. It is found that both the environmental memory and chaos can lift up bipartite entanglement in a non-linear optomechanical system. These observations may help expand our understanding of the transition from classical to quantum dynamics.

In this paper, we demonstrate that it is possible to create an adiabatic quantum computing algorithm that solves the shortest path between any two vertices on an undirected graph with at most 3V qubits, where V is the number of vertices of the graph. We do so without relying on any classical algorithms, aside from creating a (V x V) adjacency matrix. The objective of this paper is to demonstrate the fact that it is possible to model large graphs on an adiabatic quantum computer using the maximum number of qubits available and random graph generators such as the Barabasi-Albert and the Erdos-Renyi methods which can scale based on a power law., Comment: 10 pages, 17 figures, and 4 tables

This paper deals with the second quantization of interacting relativistic Fermionic and Bosonic fields in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. The corresponding Feynman diagrams and a new $S^{\#}$-matrix theory is developed. In the special case of proton-proton Moller scattering via an exchange of a neutral meson, the explicit second order element $\langle f | S^{\#}_{(2)} |i \rangle$ is deduced. In the approximation of very low external three-momenta, a new Yukawa potential is explicitly derived from $\langle f | S^{\#}_{(2)} |i \rangle$. Moreover, it is rigorously proved that this new Yukawa potential is divergence-free. The mass parameter of the exchanged meson may be set to zero to obtain a type of scalar Boson exchange between hypothetical Fermions. This provides a limiting case of a new Coulomb type potential directly from the new singularity free Yukawa potential. A divergence-free Coulomb potential between two Fermions at two discrete points is shown to be proportional to the Euler beta function. Within this relativistic discrete phase space continuous time, a single quanta is shown to occupy the hyper-tori $S^{1}_{n^1} \times S^{1}_{n^3} \times S^{1}_{n^3}$ where $S^{1}_{n}$ is a circle of radius $\sqrt{2n+1}$., Comment: 24 pages, 4 figures, 1 table. This is a continuation of our previous work arXiv:1905.02524 about non-singular Coulomb potentials to the current paper on non-singular Yukawa potentials. As such, there is some overlap between introductory sections in order to keep this paper self-contained

Quantum entanglement is shown for causally separated regions along the radial direction by using a conformal quantum mechanical correspondence with conformal radial Killing fields of causal diamonds in Minkowski space. In particular, the theory of local von Neumann algebras and Tomita Takesaki modular operators is applied in the entanglement structure of causal diamonds in conformal quantum mechanics. The entanglement of local states in their respective causal regions is shown through the measures of concurrence and entanglement entropy using the Tomita Takesaki modular conjugation operator. A holographic entropy formula is derived for the conformal quantum mechanics causal diamond correspondence. A new connection is made between the thermal time flow defined by the modular group of automorphisms to the physical time flow in a causal diamond via the aforementioned correspondence. The thermal interpretation of these results via two-point thermal Green's functions and modular group flow supports the idea of a possible emergent theory of spacetime., Comment: 27 pages, 1 figure. Typos corrected. References added. Additional statements have been added for clarification

The modular operator approach of Tomita-Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a supersymmetric two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these supersymmetric systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a supersymmetric Jaynes Cummings Model., Comment: 19 Pages

The discrete phase space and continuous time representation of relativistic quantum mechanics is further investigated here as a continuation of paper I [1]. The main mathematical construct used here will be that of an area-filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as $\bar{S}^{1}_{n}$, a circle of radius $\sqrt{2n+1}$ where $n \in \{0,1,\cdots\}$. We interpret this two-dimensional Peano circle in our framework as a phase cell inside our two-dimensional discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell $\bar{S}^{1}_{n}$. The time evolution of this Peano circle sweeps out a two-dimensional vertical cylinder analogous to the world-sheet of string theory. Extending this to three dimensional space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fibre bundles. We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein-Gordon equation. The second quantized field quantas of this system can cohabit with the tiny Planck oscillators inside the $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ phase cells for eternity. Finally, a generalized free second quantized Klein-Gordon equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory., Comment: 22 pages, 11 figures; minor typos corrected and extra comments added about Peano circles and quantum gravity

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length $l$ is investigated. Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles $S^{1}_{n}$ of radii $2E_n =\sqrt{2n+1}$ within the discrete (1 + 1)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities $S^{1}_{n} \times \mathbb{R} \subset \mathbb{R}^2$ and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion $S^{\#}$ matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein-Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group $\mathcal{I} [O(3,1)]$ ., Comment: 16 pages, 4 figures

Credit ratings are one of the primary keys that reflect the level of riskiness and reliability of corporations to meet their financial obligations. Rating agencies tend to take extended periods of time to provide new ratings and update older ones. Therefore, credit scoring assessments using artificial intelligence has gained a lot of interest in recent years. Successful machine learning methods can provide rapid analysis of credit scores while updating older ones on a daily time scale. Related studies have shown that neural networks and support vector machines outperform other techniques by providing better prediction accuracy. The purpose of this paper is two fold. First, we provide a survey and a comparative analysis of results from literature applying machine learning techniques to predict credit rating. Second, we apply ourselves four machine learning techniques deemed useful from previous studies (Bagged Decision Trees, Random Forest, Support Vector Machine and Multilayer Perceptron) to the same datasets. We evaluate the results using a 10-fold cross validation technique. The results of the experiment for the datasets chosen show superior performance for decision tree based models. In addition to the conventional accuracy measure of classifiers, we introduce a measure of accuracy based on notches called "Notch Distance" to analyze the performance of the above classifiers in the specific context of credit rating. This measure tells us how far the predictions are from the true ratings. We further compare the performance of three major rating agencies, Standard $\&$ Poors, Moody's and Fitch where we show that the difference in their ratings is comparable with the decision tree prediction versus the actual rating on the test dataset.

Machine learning algorithms perform well on identifying patterns in many different datasets due to their versatility. However, as one increases the size of the dataset, the computation time for training and using these statistical models grows quickly. Quantum computing offers a new paradigm which may have the ability to overcome these computational difficulties. Here, we propose a quantum analogue to K-means clustering, implement it on simulated superconducting qubits, and compare it to a previously developed quantum support vector machine. We find the algorithm's accuracy comparable to the classical K-means algorithm for clustering and classification problems, and find that it has asymptotic complexity $O(N^{3/2}K^{1/2}\log{P})$, where $N$ is the number of data points, $K$ is the number of clusters, and $P$ is the dimension of the data points, giving a significant speedup over the classical analogue., Comment: Updated to the published version

Temporal modes of photonic quantum states provide a new framework to develop a robust free-space quantum key distribution (QKD) scheme in a maritime environment. We show that the high-dimensional temporal modes can be used to fulfill a persistent communication channel to achieve high photon-efficiency even in severe weather conditions. We identify the parameter regimes that allow for a high-fidelity quantum information transmission. We also examine how the turbulent environment affects fidelity and entanglement degree in various environmental settings.

Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing non-unitary, non-cyclic and non-adiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realisation of quantum gates due to its robustness against dynamical errors, however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a non-perturbative control scheme., Comment: 8 pages, 3 figures

This paper deals with the relativistic, quantized electromagnetic and Dirac field equations in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. In the consequent relativistic quantum electrodynamics, the corresponding Feynman diagrams and S#-matrix elements are derived. In the special case of electron-electron scattering (Moller scattering), the explicit second order element is deduced. Moreover, assuming the slow motions for two external electrons, the approximation of yields a divergence-free Coulomb potential., Comment: 19 Pages, 2 tables, 3 figures

Random numbers are a fundamental and useful resource in science and engineering with important applications in simulation, machine learning and cyber-security. Quantum systems can produce true random numbers because of the inherent randomness at the core of quantum mechanics. As a consequence, quantum random number generators are an efficient method to generate random numbers on a large scale. We study in this paper the applications of a viable source of unbiased quantum random numbers (QRNs) whose statistical properties can be arbitrarily programmed without the need for any post-processing and that pass all standard randomness tests of the NIST and Dieharder test suites without any randomness extraction. Our method is based on measuring the arrival time of single photons in shaped temporal modes that are tailored with an electro-optical modulator. The advantages of our QRNs are shown via two applications: simulation of a fractional Brownian motion, which is a non-Markovian process, and option pricing under the fractional SABR model where the stochastic volatility process is assumed to be driven by a fractional Brownian motion. The results indicate that using the same number of random units, our QRNs achieve greater accuracy than those produced by standard pseudo-random number generators. Moreover, we demonstrate the advantages of our method via an increase in computational speed, efficiency, and convergence.

We examine in this article the pricing of target volatility options in the lognormal fractional SABR model. A decomposition formula by Ito's calculus yields a theoretical replicating strategy for the target volatility option, assuming the accessibilities of all variance swaps and swaptions. The same formula also suggests an approximation formula for the price of target volatility option in small time by the technique of freezing the coefficient. Alternatively, we also derive closed formed expressions for a small volatility of volatility expansion of the price of target volatility option. Numerical experiments show accuracy of the approximations in a reasonably wide range of parameters.

The support vector machine (SVM) is a popular machine learning classification method which produces a nonlinear decision boundary in a feature space by constructing linear boundaries in a transformed Hilbert space. It is well known that these algorithms when executed on a classical computer do not scale well with the size of the feature space both in terms of data points and dimensionality. One of the most significant limitations of classical algorithms using non-linear kernels is that the kernel function has to be evaluated for all pairs of input feature vectors which themselves may be of substantially high dimension. This can lead to computationally excessive times during training and during the prediction process for a new data point. Here, we propose using both canonical and generalized coherent states to rapidly calculate specific nonlinear kernel functions. The key link will be the reproducing kernel Hilbert space (RKHS) property for SVMs that naturally arise from canonical and generalized coherent states. Specifically, we discuss the fast evaluation of radial kernels through a positive operator valued measure (POVM) on a quantum optical system based on canonical coherent states. A similar procedure may also lead to fast calculations of kernels not usually used in classical algorithms such as those arising from generalized coherent states., Comment: 9 pages, 3 figures

We present recent developments in the theory of Nambu mechanics, which include new examples of Nambu-Poisson manifolds with linear Nambu brackets and new representations of Nambu-Heisenberg commutation relations., Comment: 7 pages, LaTex

We demonstrate that the free motion of any two-dimensional rigid body colliding elastically with two parallel, flat walls is equivalent to a billiard system. Using this equivalence, we analyze the integrable and chaotic properties of this new class of billiards. This provides a demonstration that coin tossing, the prototypical example of an independent random process, is a completely chaotic (Bernoulli) problem. The related question of which billiard geometries can be represented as rigid body systems is examined., Comment: 16 pages, LaTex

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4)-Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique., Comment: 9 pages, Latex, Section four on the SO(4)-Kepler problem has been expanded