On the Relation between Kullback Divergence and Fisher Information: From Classical Systems to Quantum Systems

2005 ◽  
pp. 399-419 ◽  
Author(s):  
Hiroshi Nagaoka
Keyword(s):  
Fisher Information ◽  
Quantum Systems ◽  
Classical Systems ◽  
Kullback Divergence

scholarly journals Quantum Shannon theory with superpositions of trajectories

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180903 ◽  
Author(s):  
Giulio Chiribella ◽  
Hlér Kristjánsson
Keyword(s):  
Transmission Lines ◽  
Internal State ◽  
Quantum Systems ◽  
Space Time ◽  
Shannon Theory ◽  
Classical Systems ◽  
Theory Of Information ◽  
Quantum Shannon Theory ◽  
Quantum Counterpart ◽  
Multiple Transmission

Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities arising when the information carriers are quantum systems. Traditionally, quantum Shannon theory has focused on scenarios where the internal state of the information carriers is quantum, while their trajectory is classical. Here we propose a second level of quantization where both the information and its propagation in space–time is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the counterintuitive phenomena taking place when information travels simultaneously through multiple transmission lines.

scholarly journals On A Quantum Proof Of The Equivalence of P And NP

2019 ◽  
Author(s):  
Adewale Oluwasanmi
Keyword(s):  
Polynomial Time ◽  
Solution Space ◽  
Quantum Systems ◽  
Satisfiability Problem ◽  
Quantum Mechanical ◽  
Correctness Proof ◽  
Classical Systems ◽  
Classical Architecture ◽  
General Quantum

We present an algorithm, along with a correctness proof, for solving the 3 Satisfiability problem that is inspired by quantum mechanical principles and that runs in polynomial time with respect to the size of the input problem. Even though we term both our algorithm and its associated proof as quantum (for reasons which we will demonstrate), it is intended to be run on standard classical architecture. In the article, we posit that the 3 Satisfiability problem has an intrinsic complex quantum form that can be programmed in order to build a model of the solution space for satisfiable instances or show that such a model cannot be constructed. This yields surprising results on the ability for classical systems to abstractly simulate general quantum systems.

scholarly journals The fundamental connections between classical Hamiltonian mechanics, quantum mechanics and information entropy

2020 ◽  
Vol 18 (01) ◽  
pp. 1941025
Author(s):  
Gabriele Carcassi ◽  
Christine A. Aidala
Keyword(s):  
Quantum Mechanics ◽  
Information Entropy ◽  
Quantum Systems ◽  
The State ◽  
Hamiltonian Mechanics ◽  
Internal Dynamics ◽  
Classical Systems ◽  
Arbitrary Precision

We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.

Loss and Dissipation: The RLC Circuit

2021 ◽  
pp. 230-246
Author(s):  
Duncan G. Steel
Keyword(s):  
Quantum Mechanics ◽  
Energy Loss ◽  
Quantum System ◽  
Equations Of Motion ◽  
Quantum Systems ◽  
Quantum Vacuum ◽  
Classical Systems ◽  
Rlc Circuit ◽  
Quantum Behavior ◽  
Continuum Of States

The effects of energy loss or dissipation is well-known and understood in classical systems. It is the source of heat in LCR circuits and in the application of brakes in a vehicle or why a struck bell does not ring indefinitely. Understanding quantum behavior begins with understanding the Hamiltonian for the problem. Classically, loss arises from a coupling of the Hamiltonian for an isolated quantum system to a continuum of states. We look at such a Hamiltonian and develop the equations of motion following the rules of quantum mechanics and find that even in a quantum system, this coupling leads to loss and non-conservation of probability in the otherwise isolated quantum system. This is the Weisskopf–Wigner formalism that is then used to understand the quantum LCR circuit. The same formalism is used in Chapter 15 for the decay of isolated quantum systems by coupling to the quantum vacuum and the resulting emission of a photon.

Characterizations of symmetric monotone metrics on the state space of quantum systems

2006 ◽  
Vol 6 (7) ◽  
pp. 597-605
Author(s):  
F. Hansen
Keyword(s):  
State Space ◽  
Quantum System ◽  
Fisher Information ◽  
Quantum Fisher Information ◽  
Classical Case ◽  
Riemannian Metric ◽  
Quantum Systems ◽  
The State ◽  
Stochastic Mappings

The quantum Fisher information is a Riemannian metric, defined on the state space of a quantum system, which is symmetric and decreasing under stochastic mappings. Contrary to the classical case such a metric is not unique. We complete the characterization, initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed and tractable formula for the set of Morozova-Chentsov functions. In addition, we provide a continuously increasing bridge between the smallest and largest symmetric monotone metrics.

scholarly journals On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-18 ◽  
Author(s):  
J. Hutchinson ◽  
J. P. Keating ◽  
F. Mezzadri
Keyword(s):  
Classical System ◽  
Companion Paper ◽  
Quantum Systems ◽  
Spin Systems ◽  
Two Dimensional ◽  
Critical Properties ◽  
One Dimensional ◽  
Classical Systems ◽  
Long Range Interactions ◽  
Quantum Hamiltonian

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.

scholarly journals QUANTUM CORNER — TRANSFER MATRIX DMRG

2008 ◽  
Vol 19 (08) ◽  
pp. 1145-1161 ◽  
Author(s):  
ERIK BARTEL ◽  
ANDREAS SCHADSCHNEIDER
Keyword(s):  
Thermal Properties ◽  
Transfer Matrix ◽  
Heisenberg Model ◽  
Quantum Systems ◽  
Test Cases ◽  
The Novel ◽  
Simple Test ◽  
Ising Chain ◽  
One Dimensional ◽  
Classical Systems

We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.

scholarly journals Confined Quantum Hard Spheres

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 775
Author(s):  
Sergio Contreras ◽  
Alejandro Gil-Villegas
Keyword(s):  
Computer Simulation ◽  
Monte Carlo ◽  
Computer Simulations ◽  
Hard Spheres ◽  
Quantum Systems ◽  
Confined Space ◽  
Free Energies ◽  
Classical Systems ◽  
Accessible Volume ◽  
Theoretical Results

We present computer simulation and theoretical results for a system of N Quantum Hard Spheres (QHS) particles of diameter σ and mass m at temperature T, confined between parallel hard walls separated by a distance Hσ, within the range 1≤H≤∞. Semiclassical Monte Carlo computer simulations were performed adapted to a confined space, considering effects in terms of the density of particles ρ*=N/V, where V is the accessible volume, the inverse length H−1 and the de Broglie’s thermal wavelength λB=h/2πmkT, where k and h are the Boltzmann’s and Planck’s constants, respectively. For the case of extreme and maximum confinement, 0.5<H−1<1 and H−1=1, respectively, analytical results can be given based on an extension for quantum systems of the Helmholtz free energies for the corresponding classical systems.

scholarly journals Fisher Information in Noisy Intermediate-Scale Quantum Applications

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 539
Author(s):  
Johannes Jakob Meyer
Keyword(s):  
Fisher Information ◽  
Quantum Algorithms ◽  
Quantum Fisher Information ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Quantum Devices ◽  
Intermediate Scale ◽  
Quantum Sensing ◽  
Near Term

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

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