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Category Of Finite Dimensional Hilbert Spaces
In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms. Whereas the theory described by the normal category of Hilbert spaces, Hilb, is ordinary quantum mechanics, the corresponding theory on finite dimensional Hilbert spaces is called fdQM. Properties This category * is monoidal, * possesses finite biproducts, and * is dagger compact. According to a theorem of Selinger, the category of finite-dimensional Hilbert spaces is complete in the dagger compact category In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from thei .... Many ideas from Hilbert spaces, such as the no-cloning theorem, hold in general for dagger compact categories. See that article for additional details. References Categorie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Spaces
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object (category Theory)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to function composition. Morphisms and objects are constituents of a category. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a partial operation, called ''composition'', on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition ( associativity of composition when it is defined, and existence of an identity morphism for every object). Morphisms and categories recur in much of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (category theory), object ''I'' that is both a left identity, left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagram (category theory), diagrams commutative diagram, commute. The ordinary tensor product makes vector spaces, abelian groups, module (mathematics), ''R''-modules, or algebra (ring theory), ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small category, small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules. Definition Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects ''A''1, ..., ''A''''n'' in C, their ''biproduct'' is an object A_1 \oplus \dots \oplus A_n in C together with morphisms *p_k \!: A_1 \oplus \dots \oplus A_n \to A_k in C (the ''projection morphisms'') *i_k \!: A_k \to A_1 \oplus \dots \oplus A_n (the ''embedding morphisms'') satisfying *p_k \circ i_k = 1_, the identity morphism of A_k, and *p_l \circ i_k = 0, the zero morphism A_k \to A_l, for k \neq l, and such that *\left( A_1 \oplus \dots \oplus A_n, p_k \right) is a product for the A_k, and *\left( A_1 \oplus \dots \oplus A_n, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dagger Compact Category
In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian category, Tannakian categories). They also appeared in the work of John Baez and James Dolan as an instance of semistrict ''k''-tuply monoidal category, monoidal n-category, ''n''-categories, which describe general topological quantum field theories, for ''n'' = 1 and ''k'' = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics. Overview Dagger compact categories can be used to express and verify some fundamental quantum computing, quantum information protocols, namely: quantum teleportation, teleportation, quantum gate teleportation, logic gate teleportation and quantum teleportation, entanglement swa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-cloning Theorem
In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computer, quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James L. Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by William Wootters and Wojciech H. Zurek as well as Dennis Dieks the same year). The aforementioned theorems do not preclude the state of one system becoming quantum entanglement, entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Hadamard transform#Quantum computing applications, Walsh–Hadamard gate to entangle two qubits without violating the no-cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categories In Category Theory
Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Category (Vaisheshika) * Stoic categories *Category mistake Science *Cognitive categorization, categories in cognitive science *Statistical classification, statistical methods used to effect classification/categorization Mathematics * Category (mathematics), a structure consisting of objects and arrows * Category (topology), in the context of Baire spaces * Lusternik–Schnirelmann category, sometimes called ''LS-category'' or simply ''category'' * Categorical data, in statistics Linguistics *Lexical category, a part of speech such as ''noun'', ''preposition'', etc. *Syntactic category, a similar concept which can also include phrasal categories *Grammatical category, a grammatical feature such as ''tense'', ''gender'', etc. Other * Category (chess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a " categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, for which monoidal categories can be considered an ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
