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Completely Positive Trace-preserving
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet. More formally, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to also include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.) Memoryless quantum channel We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional. The memoryless in the section title carries the same meaning as in classical information theory: ... [...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] |
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] |
Separable State
In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard. Separability of bipartite systems Consider first composite states with two degrees of freedom, referred to as ''bipartite states''. By a postulate of quantum mechanics these can be described as vectors in the tensor product space H_1\otimes H_2. In this discussion we will focus on the case of the Hilbert spaces H_1 and H_2 being finite-dimensional. Pure states Let \_^n\subset H_1 and \_^m \subset H_2 be orthonormal bases for H_1 and H_2, respectively. A basis for H_1 \otimes H_2 is then \, or in more compact notation \. From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Channel-state Duality
In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from ''A'' to C''n''×''n'', where ''A'' is a C*-algebra and C''n''×''n'' denotes the ''n''×''n'' complex entries, and positive linear functionals (states) on the tensor product :\mathbb^ \otimes A. Details Let ''H''1 and ''H''2 be (finite-dimensional) Hilbert spaces. The family of linear operators acting on ''Hi'' will be denoted by ''L''(''Hi''). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in ''L''(''Hi'') respectively. A quantum channel, in the Schrödinger picture, is a completely positive (CP for short), trace-preserving linear map :\Phi : L(H_1) \rightarrow L(H_2) that takes a state of system 1 to a state of system 2. Next, we describe the dual state correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No Teleportation Theorem
In quantum information theory, the no-teleportation theorem states that an arbitrary quantum state cannot be converted into a sequence of classical bits (or even an infinite number of such bits); nor can such bits be used to reconstruct the original state, thus "teleporting" it by merely moving classical bits around. Put another way, it states that the unit of quantum information, the qubit, cannot be exactly, precisely converted into classical information bits. This should not be confused with quantum teleportation, which does allow a quantum state to be destroyed in one location, and an exact replica to be created at a different location. In crude terms, the no-teleportation theorem stems from the Heisenberg uncertainty principle and the EPR paradox: although a qubit , \psi\rangle can be imagined to be a specific direction on the Bloch sphere, that direction cannot be measured precisely, for the general case , \psi\rangle; if it could, the results of that measurement would b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Holevo
Alexander Semenovich Holevo(russian: Алекса́ндр Семéнович Хóлево, also spelled as Kholevo and Cholewo) is a Soviet and Russian mathematician, one of the pioneers of quantum information science. Biography Steklov Mathematical Institute, Moscow, since 1969. He graduated from Moscow Institute of Physics and Technology in 1966, defended a PhD Thesis in 1969 and a Doctor of Science Thesis in 1975. Since 1986 A.S. Holevo is a Professor (Moscow State University and Moscow Institute of Physics and Technology). Research A.S. Holevo made substantial contributions in the mathematical foundations of quantum theory, quantum statistics and quantum information theory. In 1973 he obtained an upper bound for the amount of classical information that can be extracted from an ensemble of quantum states by quantum measurements (this result is known as '' Holevo's theorem''). A.S. Holevo developed the mathematical theory of quantum communication channels, the noncommutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naimark's Dilation Theorem
In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem. Some preliminary notions Let ''X'' be a compact Hausdorff space, ''H'' be a Hilbert space, and ''L(H)'' the Banach space of bounded operators on ''H''. A mapping ''E'' from the Borel σ-algebra on ''X'' to L(H) is called an operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets \, we have : \langle E (\cup _i B_i) x, y \rangle = \sum_i \langle E (B_i) x, y \rangle for all ''x'' and ''y''. Some terminology for describing such measures are: * ''E'' is called ''regular'' if the scalar valued measure : B \rightarrow \langle E (B) x, y \rangle is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets. * ''E'' is called ''bounded'' if , E, = \sup_B \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
POVM
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs (called projective measurements). In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory. They are extensively used in the field of quantum information. Definition In the simplest case, of a POVM with a finite number of elements acting on a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation. Details Suppose V, W are finite-dimensional vector spaces over a field, with dimensions m and n, respectively. For any space A, let L(A) denote the space of linear operators on A. The partial trace over W is then written as \operatorname_W: \operatorname(V \otimes W) \to \operatorname(V). It is defined as follows: For T\in \operatorname(V \otimes W), let e_1, \ldots, e_m , and f_1, \ldots, f_n , be bases for ''V'' and ''W'' respectively; then ''T'' has a matrix representation : \ \quad 1 \leq k, i \leq m, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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