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Typical Subspace
In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory. Unconditional quantum typicality Consider a density operator \rho with the following spectral decomposition: : \rho=\sum_p_( x) \vert x\rangle \langle x\vert . The weakly typical subspace is defined as the span of all vectors such that the sample entropy \overline( x^) of their classical label is close to the true entropy H( X) of the distribution p_( x) : : T_^\equiv\text\left\ , where : \overline( x^) \equiv-\frac\log( p_( x^) ) , :H( X) \equiv-\sum_p_( x) \log p_( x) . The projector \Pi_^ onto the typical subspace of \rho is defined as : \Pi_^\equiv\sum_\vert x^\rangle \langle x^\vert , where we have "overloaded" the symbol T_^ to refer also to the set of \delta-typical sequences: : T ... [...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] |
Schumacher Compression
Schumacher or Schuhmacher is an occupational surname (German, "shoemaker", pronounced , both variants can be used as surnames, with Schumacher being the more popular one, however, only the variant with three "h"s can also be used as a job description in modern German spelling). The variant Schumaker is also commonly seen in the USA. Still another variant is Shumacher. Notable people with the surname include: Science * Benjamin Schumacher, American theoretical physicist, married to Carol * Carol Schumacher, Bolivian-American mathematician, married to Benjamin * E. F. Schumacher (1911–1977), British economist * Eugen Schuhmacher (1906–1973), German zoologist and pioneer of animal documentaries * Heinrich Christian Schumacher (1780–1850), German astronomer * Heinrich Christian Friedrich Schumacher, (1757–1830), German-Danish surgeon, botanist, malacologist and anatomist * William S. Massey (1920–2017), American mathematician Sports * Anton Schumacher (born 1938), Ger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Typical Set
In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself. This has great use in compression theory as it provides a theoretical means for compressing data, allowing us to represent any sequence ''X''''n'' using ''nH''(''X'') bits on average, and, hence, justifying the use of entropy as a measure of information from a source. The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases. (Weakly) typical sequences (weak typicality, entropy typicality) If a sequence ''x''1, ..., ''x''''n'' is drawn from an i.i.d. distribution ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering (field), information engineering, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a dice, die (with six equally likely outcomes). Some other important measures in information theory are mutual informat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent ''mixed states''. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state. Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information. Definition and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition Of Spectrum (functional Analysis)
The spectrum of a linear operator T that operates on a Banach space X (a fundamental concept of functional analysis) consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectrum has a standard decomposition into three parts: * a point spectrum, consisting of the eigenvalues of T; * a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of T-\lambda a proper dense subset of the space; * a residual spectrum, consisting of all other scalars in the spectrum. This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen. Decomposition into point spectrum, continuous spectrum, and residual spectrum For bounded Banach space operators Let ''X'' be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication. The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names ''thermodynamic function'' and ''heat-potential''. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of hea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P : V \to V such that P^2 = P. When V has an inner product and is complete (i.e. when V is a Hilbert space) the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Entropy
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y given that the value of another random variable X is known. Here, information is measured in shannons, nats, or hartley Hartley may refer to: Places Australia *Hartley, New South Wales *Hartley, South Australia **Electoral district of Hartley, a state electoral district Canada *Hartley Bay, British Columbia United Kingdom *Hartley, Cumbria *Hartley, Plymou ...s. The ''entropy of Y conditioned on X'' is written as \Eta(Y, X). Definition The conditional entropy of Y given X is defined as where \mathcal X and \mathcal Y denote the support sets of X and Y. ''Note:'' Here, the convention is that the expression 0 \log 0 should be treated as being equal to zero. This is because \lim_ \theta\, \log \theta = 0. Intuitively, notice that by definition of Expected value, expected value and of Conditional Probability, conditional proba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Capacity
In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel. Holevo, Schumacher, and Westmoreland proved the following least upper bound on the classical capacity of any quantum channel \mathcal: : \chi(\mathcal) = \max_ I(X;B)_ where \rho^ is a classical-quantum state of the following form: : \rho^ = \sum_x p_X(x) \vert x \rangle \langle x \vert^X \otimes \rho_x^A , p_X(x) is a probability distribution, and each \rho_x^A is a density operator that can be input to the channel \mathcal. Achievability using sequential decoding We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate I(X;B) for communicating classical data over a quantum channel). We first review the minimal amount of quantum mechanics needed for the theorem. We then cover quantum typicality, and finally we prove the theorem using a ... [...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] |
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