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.

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 BASICS OF QUANTUM MECHANICS, GEOMETRIZATION AND SOME APPLICATIONS TO QUANTUM INFORMATION

2008 ◽  
Vol 05 (06) ◽  
pp. 989-1032 ◽  
Author(s):  
JESÚS CLEMENTE-GALLARDO ◽  
GIUSEPPE MARMO
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Hilbert Spaces ◽  
Functional Independence ◽  
Practical Application ◽  
Finite Dimensional ◽  
Entanglement Witnesses ◽  
Geometric Formulation

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

scholarly journals Separation of quantum, spatial quantum, and approximate quantum correlations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 389
Author(s):  
Salman Beigi
Keyword(s):  
Mathematical Model ◽  
Hilbert Spaces ◽  
Quantum Correlations ◽  
Quantum Measurements ◽  
Finite Dimensional ◽  
Nonlocal Correlations ◽  
Local Quantum

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e., Cq(4,4,2,2)≠Cqs(4,4,2,2). We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e., Cqs(4,4,3,3)≠Cqa(4,4,3,3).

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

2021 ◽  
Vol 3 (1) ◽  
pp. 12
Author(s):  
Ariel Caticha
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Information Geometry ◽  
Complex Numbers ◽  
Born Rule ◽  
Statistical Manifold ◽  
Finite Dimensional ◽  
Starting Point

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

ALTERNATIVE HAMILTONIAN DESCRIPTIONS FOR QUANTUM SYSTEMS AND NON-HERMITIAN OPERATORS WITH REAL SPECTRUM

2002 ◽  
Vol 17 (24) ◽  
pp. 1589-1599 ◽  
Author(s):  
FRANCO VENTRIGLIA
Keyword(s):  
Inverse Problem ◽  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Systems ◽  
Theoretical Physics ◽  
Inner Product ◽  
Real Spectrum ◽  
Standard Methods ◽  
Finite Dimensional

Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

Completeness of Graphical Languages for Mixed State Quantum Mechanics

2021 ◽  
Vol 2 (4) ◽  
pp. 1-28
Author(s):  
Titouan Carette ◽  
Emmanuel Jeandel ◽  
Simon Perdrix ◽  
Renaud Vilmart
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Monoidal Category ◽  
Quantum Information Processing ◽  
Quantum Circuits ◽  
Quantum Operations ◽  
Symmetric Monoidal Category ◽  
Finite Dimensional ◽  
General Quantum ◽  
New Construction

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.

Quotients of Essentially Euclidean Spaces

2019 ◽  
Vol 62 (1) ◽  
pp. 71-74
Author(s):  
Tadeusz Figiel ◽  
William Johnson
Keyword(s):  
Normed Space ◽  
Quotient Space ◽  
Euclidean Spaces ◽  
Quantitative Version ◽  
Finite Dimensional ◽  
Dimensional Normed Space

AbstractA precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

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