scholarly journals Hardy’s inequalities in finite dimensional Hilbert spaces

2021 ◽  
pp. 1
Author(s):  
Dimitar K. Dimitrov ◽  
Ivan Gadjev ◽  
Geno Nikolov ◽  
Rumen Uluchev
Keyword(s):  
Hilbert Spaces ◽  
Finite Dimensional ◽  
Hardy’S Inequalities

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 Some Characterizations of Finite-Dimensional Hilbert Spaces

1998 ◽  
Vol 223 (1) ◽  
pp. 364-365 ◽  
Author(s):  
M.A Navarro
Keyword(s):  
Hilbert Spaces ◽  
Finite Dimensional

scholarly journals Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2066
Author(s):  
Messaoud Bounkhel ◽  
Mostafa Bachar
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Space Setting ◽  
Nonlinear Anal ◽  
Finite Dimensional ◽  
In Trans ◽  
First Time ◽  
The Relationship ◽  
Primal Lower Nice Functions

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

scholarly journals Infinite dimensional generalizations of Choi’s Theorem

2019 ◽  
Vol 7 (1) ◽  
pp. 67-77
Author(s):  
Shmuel Friedland
Keyword(s):  
Hilbert Spaces ◽  
Quantum Channel ◽  
Trace Class ◽  
Natural Generalization ◽  
Linear Operators ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

scholarly journals A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

2006 ◽  
Vol 04 (02) ◽  
pp. 325-330 ◽  
Author(s):  
B. V. RAJARAMA BHAT
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Product Formula ◽  
Maximal Dimension ◽  
Maximum Dimension ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Minimum Dimension ◽  
Orthogonal Product ◽  
Product Vector

Consider a tensor product [Formula: see text] of finite-dimensional Hilbert spaces with dimension [Formula: see text], 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of [Formula: see text] with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

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.

Sein oder Nichtsein als Grundfrage der Quantenphysik / To he or not to be as basic question in quantum physics

1984 ◽  
Vol 39 (2) ◽  
pp. 113-131
Author(s):  
Fritz Bopp
Keyword(s):  
Hilbert Spaces ◽  
Quantum Physics ◽  
Basic Assumption ◽  
Finite Lattice ◽  
Basic Question ◽  
Classical Physics ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Basic Ideas ◽  
Finite Dimensional Hilbert Space

The question is often asked how to interprete quantum physics. That question does not arise in classical physics, since Newton's axioms are immediately connected with basic ideas and experiences. The same is possible in quantum physics, if we remember how elementary particle physicists describe their experiments. As Helmholtz has pointed out. the basic assumption of classical physics is that of geneidentity. That means: Bodies remain the same during their motion. Obviously, that is no longer true in quantum physics. Particles can be created and annihilated. Therefore creation and annihilation must be considered as basic processes. Motion only occurs, if a particle is annihilated in a certain point, if an equal one is created in an infinitesimally neighbouring point, and if this process is continuously going on during a certain time. Motions of that kind are compatible with the existence of some manifest creation and annihilation processes. If we accept this idea, quantum physics can be derived from first principles. As in classical physics, we know therefore what happens from the very beginning. Thus questions of interpretation become dispensable. A particular mathematical method is used to exhaust continua. The theory is formulated in a finite lattice, whose point density and extension equally go to infinity. All calculations are therefore performed in a finite dimensional Hilbert space. The results are however related to an infinite dimensional one. Earlier calculations may, therefore, be essentially correct, though they must be rejected in theories which are based on manifestly infinite dimensional Hilbert spaces. Here limiting processes do not occur in the state space. They are only admissible for numerical results.

scholarly journals von Neumann lattices in finite-dimensional Hilbert spaces

2008 ◽  
Vol 78 (1) ◽  
Author(s):  
M. Revzen ◽  
F. C. Khanna
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann ◽  
Finite Dimensional
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