arXiv subjects

Rupak Chatterjee

Publications and source records attributed to Rupak Chatterjee.

At least 19 recordsLinked to original sources

arXiv2026

Modular Self-Duality, Symmetrized Relative Entropy, and Bogoliubov--Kubo--Mori Susceptibility in Quantum Field Theory

We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral $U(1)$ current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.

hep-th↗

arXiv2025

Entanglement harvesting and curvature of entanglement: A modular operator approach

An operator-algebraic framework based on Tomita-Takesaki modular theory is used to study aspects of quantum entanglement via the application of the modular conjugation operator $J$. The entanglement structure of quantum fields is studied through the protocol of entanglement harvesting whereby quantum correlations evolve through the time evolution of qubit detectors coupled to a Bosonic field. Modular conjugation operators are constructed for Unruh-Dewitt type qubits interacting with a scalar field such that initially unentangled qubits become entangled. The entanglement harvested in this process is directly quantified by an expectation value involving $J$ offering a physical application of this operator. The modular operator formalism is then extended to the Markovian open system dynamics of coupled qubits by expressing entanglement monotones as functionals of a state $\rho$ and its modular reflection $J\rho J$. The second derivative of such functionals with respect to an external coupling parameter, termed the curvature of entanglement, provides a natural measure of entanglement sensitivity. At points of modular self-duality, the curvature of entanglement coincides with the quantum Fisher information measure. These results demonstrate that the modular conjugation operator $J$ captures both the harvesting of entanglement from quantum fields and the curvature of entanglement in coupled qubit dynamics providing parallel modular structures that connect these systems.

hep-th↗

arXiv2025

Solving the Traveling Salesman Problem via Different Quantum Computing Architectures

We study the application of emerging photonic and quantum computing architectures to solving the Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem. We investigate several approaches: Simulated Annealing (SA), Quadratic Unconstrained Binary Optimization (QUBO-Ising) methods implemented on quantum annealers and Optical Coherent Ising Machines, as well as the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Phase Estimation (QPE) algorithm on gate-based quantum computers. QAOA and QPE were tested on the IBM Quantum platform. The QUBO-Ising method was explored using the D-Wave quantum annealer, which operates on superconducting Josephson junctions, and the Quantum Computing Inc (QCi) Dirac-1 entropy quantum optimization machine. Gate-based quantum computers demonstrated accurate results for small TSP instances in simulation. However, real quantum devices are hindered by noise and limited scalability. Circuit complexity grows with problem size, restricting performance to TSP instances with a maximum of 6 nodes. In contrast, Ising-based architectures show improved scalability for larger problem sizes. SQUID-based Ising machines can handle TSP instances with up to 12 nodes, while entropy computing implemented in hybrid optoelectronic components extend this capability to 18 nodes. Nevertheless, the solutions tend to be suboptimal due to hardware limitations and challenges in achieving ground state convergence as the problem size increases. Despite these limitations, Ising machines demonstrate significant time advantages over classical methods, making them a promising candidate for solving larger-scale TSPs efficiently.

quant-ph↗

arXiv2024

Tomita-Takesaki theory and quantum concurrence

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.

quant-ph↗

arXiv2022

Chaos in Optomechanical Systems coupled to a Non-Markovian environment

We study the chaotic motion of a semi-classical optomechanical system coupled to a non-Markovian environment with a finite correlation time. We show that the non-Markovian environment can significantly enhance chaos, by studying the emergence of chaos using Lyapunov exponent with the changing non-Markovian parameter. It is observed that non-Markovian environment characterized by the Ornstein-Uhlenbeck type noise can modify the generation of chaos with different environmental memory times. As a comparison, the crossover properties from Markov to non-Markovian regimes are also discussed. Our findings indicate that the quantum memory effects on the onset of chaos may become a useful property to be investigated in quantum manipulations and control.

quant-ph↗

arXiv2022

Optimization on Large Interconnected Graphs and Networks Using Adiabatic Quantum Computation

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.

quant-ph↗

arXiv2022

A Modular Operator Approach to Entanglement of Causally Closed Regions

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.

hep-th↗

arXiv2022

Discrete Phase Space-Continuous Time Relativistic Klein-Gordon and Dirac Equations, and a New Non-Singular Yukawa Potential

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 $\sqrt{2n+1}$.

quant-ph↗

arXiv2021

Modular Operators and Entanglement in Supersymmetric Quantum Mechanics

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.

quant-ph↗

arXiv2021

Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fibre bundles

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.

quant-ph↗

arXiv2020

Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

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 $\mathcal{I} [O(3,1)]$ .

quant-ph↗

arXiv2020

A comparative study of forecasting Corporate Credit Ratings using Neural Networks, Support Vector Machines, and Decision Trees

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.

q-fin.RM↗

arXiv2019

Geometric decoherence in diffusive open quantum systems

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.

quant-ph↗

arXiv2019

Quantum Unsupervised and Supervised Learning on Superconducting Processors

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.

quant-ph↗

arXiv2019

High-dimensional temporal mode propagation in a turbulent environment

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.

quant-ph↗

arXiv2019

Discrete Phase Space, Relativistic Quantum Electrodynamics, and a Non-Singular Coulomb Potential

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.

physics.gen-ph↗

arXiv2018

Quantum Systems for Monte Carlo Methods and Applications to Fractional Stochastic Processes

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.

quant-ph↗

arXiv2018

Target volatility option pricing in lognormal fractional SABR model

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.

q-fin.CP↗