Linear Algebra and Quantum Mechanics

2009 ◽  
pp. 19-32
Keyword(s):  
Quantum Mechanics ◽  
Linear Algebra

Linear Algebra and the Dirac Notation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Vector Spaces ◽  
Complex Vector ◽  
Dual Vector ◽  
Finite Dimensional ◽  
Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

scholarly journals From linear algebra to quantum information

2021 ◽  
pp. 032-047
Author(s):  
Yu LW ◽  
Wang NL ◽  
Kanemitsu S
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Linear Algebra ◽  
Smooth Transition ◽  
Quantum Computers ◽  
Hermite Operator ◽  
Hermite Matrix ◽  
Hermite Operators

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.

Introduction

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Quantum Mechanics ◽  
Computer Graphics ◽  
Computer Science ◽  
Control Systems ◽  
Linear Algebra ◽  
Mathematical Physics ◽  
Molecular Symmetry ◽  
Research Papers ◽  
Physics Journals

This chapter provides a brief overview of this volume, showing that this book can be used for a variety of ways and can be used to assist the reader toward a better understanding of quaternion linear algebra. Quaternions after all have become increasingly useful for practitioners in research, both in theory and applications. For example, a significant number of research papers on quaternions, perhaps even most of them, appear regularly in mathematical physics journals, and quantum mechanics based on quaternion analysis is mainstream physics. In engineering, quaternions are often used in control systems, and in computer science they play a role in computer graphics. Quaternion formalism is also used in studies of molecular symmetry. Hence, to give the reader a preliminary understanding of quaternions, this chapter also provides some notations and conventions to be used in the remainder of this volume, as well as some standard matrices.

scholarly journals How students apply linear algebra to quantum mechanics

2019 ◽  
Author(s):  
Charlotte Hillebrand-Viljoen ◽  
Spencer Wheaton
Keyword(s):  
Quantum Mechanics ◽  
Linear Algebra

The Mathematics of Quantum Mechanics

2019 ◽  
pp. 30-41
Author(s):  
Jeffrey A. Barrett
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Physical Properties ◽  
State Space ◽  
Linear Algebra ◽  
Linear Operators ◽  
Mathematical Representation ◽  
Quantum Mechanical ◽  
Standard Formulation ◽  
Schrödinger Picture

Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

An introduction to S Bases in S Linear Algebra

2019 ◽  
Vol 5 (2) ◽  
pp. 35
Author(s):  
David CARFI’
Keyword(s):  
Mathematical Model ◽  
Quantum Mechanics ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Lecture Note ◽  
Continuous Case ◽  
Euclidean Spaces ◽  
Hilbert Bases ◽  
Finite Dimensional

In this lecture note we define the S bases for the spaces of tempered distributions.These new bases are the analogous of Hilbert bases of separable Hilbert spaces for the continuous case (they are indexed by m-dimensional Euclidean spaces) and enjoy properties similar to those shown by algebraic bases in the finite dimensional case.The S bases are one possible rigorous and extremely manageable mathematical model for the "physical" bases used in Quantum Mechanics.

A simple proof of the strong subadditivity inequality

2005 ◽  
Vol 5 (6) ◽  
pp. 507-513
Author(s):  
M.A. Nielsen ◽  
D. Petz
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Linear Algebra ◽  
Simple Proof ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Math Phys

Arguably the deepest fact known about the von~Neumann entropy, the strong subadditivity inequality is a potent hammer in the quantum information theorist's toolkit. This short tutorial describes a simple proof of strong subadditivity due to Petz [Rep. on Math. Phys. \textbf{23} (1), 57--65 (1986)]. It assumes only knowledge of elementary linear algebra and quantum mechanics.

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