scholarly journals The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

2021 ◽  
pp. 531-550
Author(s):  
Renaud Vilmart
Keyword(s):  
Quantum Measurement ◽  
Graphical Calculus ◽  
Linear Maps ◽  
Quantum Operators ◽  
Rewrite System

AbstractWe show that the formalism of “Sum-Over-Path” (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has the structure of a dagger-compact PROP. Several consequences arise from this observation:– Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two– A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement.We also enrich the rewrite system so as to get the completeness of the Clifford fragments of both the initial formalism and its enriched version.

Monoidal Categories

2019 ◽  
pp. 29-60
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Hilbert Spaces ◽  
Monoidal Category ◽  
Monoidal Categories ◽  
Graphical Calculus ◽  
The Core ◽  
Basic Language ◽  
Linear Maps ◽  
Visual Notation

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory of monoidal categories, including braidings, symmetries and coherence. They form the core of this book, as they provide the basic language with which the rest of the material will be developed. We introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work with them. We also introduce the monoidal categories Hilb of Hilbert spaces and linear maps, Set of sets and functions and Rel of sets and relations, which will be used as running examples throughout the book.

Nano-Mechanical Oscillator and Basic Dynamics: Part I

2021 ◽  
pp. 5-31
Author(s):  
Duncan G. Steel
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Harmonic Oscillator ◽  
Quantum Measurement ◽  
Dynamical Behavior ◽  
Quantum Radiation ◽  
Energy Plus ◽  
Quantum Hamiltonian ◽  
Quantum Operators ◽  
Quantum Technology

This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian represents the total energy of the system, the sum of kinetic energy plus potential energy, written in canonical coordinates and conjugate momenta, and where these variables become time independent quantum operators. The nature of the potential energy for the nano-vibrator, following Hooke’s law, serves to localize the particle. The relevance of the nano-vibrator Hamiltonian—sometimes called the harmonic oscillator Hamiltonian—is perhaps one of the most important Hamiltonians in quantum systems. Not only can it be extended to cover things like phonons in solids, vibrations in molecules, and the behavior of bosons, but it is also the basis for leading to the concept of a photon, the quantum radiation field, and the quantum vacuum. This chapter provides the basic introduction for vibration of a particle or a nano-rod and looks at the wave-like behavior that emerges from the solution to the time independent Schrödinger equation. When we include the time evolution, we can observe dynamical behavior and begin to examine the meaning of quantum measurement.

Quantum Measurement

1992 ◽  
Author(s):  
Vladimir B. Braginsky ◽  
Farid Ya Khalili ◽  
Kip S. Thorne
Keyword(s):  
Quantum Measurement

Quantum Measurement

1996 ◽  
Vol 193 (Part_1_2) ◽  
pp. 226-227
Author(s):  
H. Schmiedel
Keyword(s):  
Quantum Measurement

scholarly journals Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bao-ning Du ◽  
Min-xin Huang
Keyword(s):  
Large Class ◽  
Bethe Ansatz ◽  
Difference Equations ◽  
The Other ◽  
Elementary Functions ◽  
Quantum Dilogarithm ◽  
Thermodynamic Bethe Ansatz ◽  
Dilogarithm Function ◽  
Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

Sign in / Sign up

Export Citation Format

Share Document