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Jensen–Shannon Divergence
In probability theory and statistics, the Jensen– Shannon divergence is a method of measuring the similarity between two probability distributions. It is also known as information radius (IRad) or total divergence to the average. It is based on the Kullback–Leibler divergence, with some notable (and useful) differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence is a metric often referred to as Jensen–Shannon distance. Definition Consider the set M_+^1(A) of probability distributions where A is a set provided with some σ-algebra of measurable subsets. In particular we can take A to be a finite or countable set with all subsets being measurable. The Jensen–Shannon divergence (JSD) M_+^1(A) \times M_+^1(A) \rightarrow [0,\infty) is a symmetrized and smoothed version of the Kullback–Leibler divergence D(P \parallel Q). It is defined by :(P \parallel Q)= \fracD(P \parallel M)+\fracD(Q \parallel M), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix , the von Neumann entropy is : S = - \operatorname(\rho \ln \rho), where \operatorname denotes the trace and ln denotes the (natural) matrix logarithm. If is written in terms of its eigenvectors , 1\rangle, , 2\rangle, , 3\rangle, \dots as : \rho = \sum_j \eta_j \left, j \right\rang \left\lang j \ , then the von Neumann entropy is merely : S = -\sum_j \eta_j \ln \eta_j . In this form, ''S'' can be seen as the information theoretic Shannon entropy. The von Neumann entropy is also used in different forms ( conditional entropies, relative entropies, etc.) in the framework of quantum information theory to characterize the entropy of entanglement. Background John von Neumann established a rigorous mathematical framework for quantum me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genome Comparison
Comparative genomics is a field of biological research in which the genomic features of different organisms are compared. The genomic features may include the DNA sequence, genes, gene order, regulatory sequences, and other genomic structural landmarks. In this branch of genomics, whole or large parts of genomes resulting from genome projects are compared to study basic biological similarities and differences as well as evolutionary relationships between organisms. The major principle of comparative genomics is that common features of two organisms will often be encoded within the DNA that is evolutionarily conserved between them. Therefore, comparative genomic approaches start with making some form of alignment of genome sequences and looking for orthologous sequences (sequences that share a common ancestry) in the aligned genomes and checking to what extent those sequences are conserved. Based on these, genome and molecular evolution are inferred and this may in turn be put ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combines biology, chemistry, physics, computer science, information engineering, mathematics and statistics to analyze and interpret the biological data. Bioinformatics has been used for '' in silico'' analyses of biological queries using computational and statistical techniques. Bioinformatics includes biological studies that use computer programming as part of their methodology, as well as specific analysis "pipelines" that are repeatedly used, particularly in the field of genomics. Common uses of bioinformatics include the identification of candidates genes and single nucleotide polymorphisms (SNPs). Often, such identification is made with the aim to better understand the genetic basis of disease, unique adaptations, desirable properties (e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher Information Metric
In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, ''i.e.'', a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements. The metric is interesting in several respects. By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. It can also be understood to be the infinitesimal form of the relative entropy (''i.e.'', the Kullback–Leibler divergence); specifically, it is the Hessian of the divergence. Alternately, it can be understood as the metric induced by the flat space Euclidean metric, after appropriate changes of variable. When extended to complex projective Hilbert space, it becomes the Fubini–Study metric; when written in terms of mixed states, it is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bures Metric
In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is identical to the Fubini–Study metric when restricted to the pure states alone. Definition The Bures metric G may be defined as : (\rho, \rho+d\rho)2 = \frac\mbox( d \rho G ), where G is Hermitian 1-form operator implicitly given by : \rho G + G \rho = d \rho, which is a special case of a continuous Lyapunov equation. Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states and the use of the volume element as a candidate for the Jeffreys prior probability density for mixed quantum states. Bures distance The Bures distance is the finite ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Pure States
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the Spectrum#Energy spectrum, energy spectrum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holevo's Theorem
Holevo's theorem is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a quantum state (accessible information). It was published by Alexander Holevo in 1973. Accessible information As for several concepts in quantum information theory, accessible information is best understood in terms of a 2-party communication. So we introduce two parties, Alice and Bob. Alice has a ''classical'' random variable ''X'', which can take the values with corresponding probabilities . Alice then prepares a quantum state, represented by the density matrix ''ρX'' chosen from a set , and gives this state to Bob. Bob's goal is to find the value of ''X'', and in order to do that, he performs a measurement on the state ''ρ''''X'', obtaining a classical outcome, which we denote with ''Y''. In this context, the amount ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information
Quantum information is the information of the 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 observables cannot be precisely measured simultaneously, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrices
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 in quantum mechanics, 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 quantum entanglement, 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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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