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Choi's Theorem On Completely Positive Maps
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's " Radon–Nikodym" theorem for completely positive maps. Statement Choi's theorem. Let \Phi : \mathbb^ \to \mathbb^ be a linear map. The following are equivalent: :(i) is -positive (i.e. \left (\operatorname_n\otimes\Phi \right )(A)\in\mathbb^\otimes\mathbb^ is positive whenever A\in\mathbb^\otimes\mathbb^ is positive). :(ii) The matrix with operator entries ::C_\Phi= \left (\operatorname_n\otimes\Phi \right ) \left (\sum_E_\otimes E_ \right ) = \sum_E_\otimes\Phi(E_) \in \mathbb ^ :is positive, where E_ \in \mathbb^ is the matrix with 1 in the -th entry and 0s elsewhere. (The matrix ''C''Φ is sometimes called the ''Choi matrix'' of .) :(iii) is completely positive. Proof (i) implies (ii) We observe that if :E=\sum_E_\ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Positive Map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear map \phi: A\to B is called positive map if \phi maps positive elements to positive elements: a\geq 0 \implies \phi(a)\geq 0. Any linear map \phi:A\to B induces another map :\textrm \otimes \phi : \mathbb^ \otimes A \to \mathbb^ \otimes B in a natural way. If \mathbb^\otimes A is identified with the C*-algebra A^ of k\times k-matrices with entries in A, then \textrm\otimes\phi acts as : \begin a_ & \cdots & a_ \\ \vdots & \ddots & \vdots \\ a_ & \cdots & a_ \end \mapsto \begin \phi(a_) & \cdots & \phi(a_) \\ \vdots & \ddots & \vdots \\ \phi(a_) & \cdots & \phi(a_) \end. We say that \phi is k-positive if \textrm_ \otimes \phi is a positive map, and \phi is called completely positive if \phi is k-positive for all k. Properties * Positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viacheslav Belavkin
Viacheslav Pavlovich Belavkin (russian: Вячеслав Павлович Белавкин; 20 May 1946 – 27 November 2012) was a Russian-British professor in applied mathematics at the University of Nottingham. An active researcher, he was one of the pioneers of quantum probability. His research spanned areas such as quantum filtering, quantum information and quantum chaos. Biography He was born in Lviv, and graduated from Moscow State University in 1970 where his teachers include Evgeny Lifshitz, Victor Pavlovich Maslov, Andrey Kolmogorov and Ruslan L. Stratonovich. In the 1980s Belavkin held visiting professorship in the Dublin Institute for Advanced Studies, and the Volterra Centre in Rome before taking up an appointment at the University of Nottingham in 1992. He was promoted to a Chair in Mathematical Physics in 1996. He and Ruslan L. Stratonovich were awarded the State Prize of the Russian Federation (formerly the Lenin Prize) for outstanding achievements in science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as :\nu(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the ''n''-dimensional Euclidean space (corresponding to our stan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term. It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting Observable, observables cannot be precisely mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kraus Operator
In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel. Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving. Quantum operations are formulated in terms of the density operator description of a quantum mechani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Kraus (physicist)
Karl Kraus (21 March 1938 – 9 June 1988) was a German theoretical physicist who made major contributions to the foundations of quantum physics. Life and work Kraus was born in 1938 in Hohenelbe/Giant Mountains, today Vrchlabí. After the war, he grew up in Elsterwerda and attended local schools. He studied physics from 1955 to 1960 at the Humboldt University of Berlin (East) and the Free University of Berlin (West). He graduated in 1962 with a thesis about Lorentz's theory of gravity, carried out under the supervision of Kurt Just. Kraus then joined as an assistant to Günther Ludwig at the University of Marburg, where he qualified in 1966. In 1971, he accepted a professorship at the Institute of Physics of the University of Würzburg, where he established a mathematical physics working group on the topic of the foundations of quantum theory. In 1980 Kraus spent a sabbatical year at UT Austin with John Archibald Wheeler, Arno Böhm, George Sudarshan, William Wootters, and Wojc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Schmidt Operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_H, where \ is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm \, \cdot\, _\text is identical to the Frobenius norm. , , ·, , is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if \_ and \_ are such bases, then \sum_i \, Ae_i\, ^2 = \sum_ \left, \langle Ae_i, f_j\rangle \^2 = \sum_ \left, \langle e_i, A^*f_j\rangle \^2 = \sum_j\, A^* f_j\, ^2. If e_i = f_i, then \sum_i \, Ae_i\, ^2 = \sum_i\, A^* e_i\, ^2. As for any bounded operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stinespring Factorization Theorem
In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra ''A'' as a composition of two completely positive maps each of which has a special form: #A *-representation of ''A'' on some auxiliary Hilbert space ''K'' followed by #An operator map of the form ''T'' ↦ ''V*TV''. Moreover, Stinespring's theorem is a structure theorem from a C*-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms. Formulation In the case of a unital C*-algebra, the result is as follows: :Theorem. Let ''A'' be a unital C*-algebra, ''H'' be a Hilbert space, and ''B''(''H'') be the bounded operators on ''H''. For every completely positive ::\Phi : A \to B(H), :there exists a Hilbert space ''K'' and a unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Operation
In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel. Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving. Quantum operations are formulated in terms of the density operator description of a quantum mechanica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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