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Bures Distance
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 may be defined as : _\text(\rho, \rho+d\rho)2 = \frac\mbox( d \rho G ), where G is the 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 ... [...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] |
Symmetric Logarithmic Derivative
The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information. Definition Let \rho and A be two operators, where \rho is Hermitian and positive semi-definite. In most applications, \rho and A fulfill further properties, that also A is Hermitian and \rho is a density matrix (which is also trace-normalized), but these are not required for the definition. The symmetric logarithmic derivative L_\varrho(A) is defined implicitly by the equation :i varrho,A\frac \ where ,YXY-YX is the commutator and \=XY+YX is the anticommutator. Explicitly, it is given by :L_\varrho(A)=2i\sum_ \frac \langle k \vert A \vert l\rangle \vert k\rangle \langle l \vert where \lambda_k and \vert k\rangle are the eigenvalues and eigenstates of \varrho, i.e. \varrho\vert k\rangle=\lambda_k\vert k\rangle and \varrho=\sum_k \lambda_k \vert k\rangle\langle k\vert. Formally, the map from operator A to operator L_\varrho(A) is a (linear) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gell-Mann Matrices
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the fundamental building blocks of the strongly interacting particles, and the renormalization group as a foundational element of quantum field theory and statistical mechanics. He played key roles in developing the concept of chirality in the theory of the weak interactions and spontaneous chiral symmetry breaking in the strong interactions, which controls the physics of the light mesons. In the 1970s he was a co-inventor of quantum chromodynamics (QCD) which explains the confinement of quarks in mesons and baryons and forms a large part of the Standard Model of elementary particles and forces. Murray Gell-Mann received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. Life and education Gell-Mann was born in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_x &= \begin 0&1\\ 1&0 \end, \\ \sigma_2 = \sigma_y &= \begin 0& -i \\ i&0 \end, \\ \sigma_3 = \sigma_z &= \begin 1&0\\ 0&-1 \end. \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore–Penrose Inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. The terms ''pseudoinverse'' and ''generalized inverse'' are sometimes used as synonyms for the Moore–Penrose inverse of a matrix, but sometimes applied to other elements of algebraic structures which share some but not all properties expected for an inverse element. A common use of the pseudoinverse is to compute a "best fit" ( least squares) approximate solution to a system of linear equations that lacks an exact solution (see below under § Applications). Another use is to find the minimum ( Euclidean) norm solution to a system of linear equations with multiple solutions. The pseu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). There are several notations, such as \mathbf^\mathrm or \mathbf^*, \mathbf', or (often in physics) \mathbf^. For real matrices, the conjugate transpose is just the transpose, \mathbf^\mathrm = \mathbf^\operatorname. Definition The conjugate transpose of an m \times n matrix \mathbf is formally defined by where the subscript ij denotes the (i,j)-th entry (matrix element), for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\mathbf^\mathrm = \left(\overline\right)^\operatorname = \overline where \mathbf^\operatorname denotes the transpose and \overline denotes the matrix with complex conjugated entries. Other na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, if r and \varphi are real numbers then the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a matrix ''A'', denoted vec(''A''), is the column vector obtained by stacking the columns of the matrix ''A'' on top of one another: \operatorname(A) = _, \ldots, a_, a_, \ldots, a_, \ldots, a_, \ldots, a_\mathrm Here, a_ represents the element in the ''i''-th row and ''j''-th column of ''A'', and the superscript ^\mathrm denotes the transpose. Vectorization expresses, through coordinates, the isomorphism \mathbf^ := \mathbf^m \otimes \mathbf^n \cong \mathbf^ between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix A = \begin a & b \\ c & d \end, the vectorization is \operatorname(A) = \begin a \\ c \\ b \\ d \end. The connection between the vectorization of ''A'' and the vectorization of its transpose is given by the commutation matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used term. The misattribution to Kronecker rather than Zehfuss wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The Pearson product-moment correlation coefficient, correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables. A distinction must be made between (1) the covariance of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramér–Rao Bound
In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, but has also been derived independently by Maurice Fréchet, Georges Darmois, and by Alexander Aitken and Harold Silverstone. It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance. An unbiased estimator that achieves this bound is said to be (fully) '' efficient''. Such a solution achieves the lowest possible mean squared error among all unbiased methods, and is, therefore, the minimum variance unbiased (MVU) estimator. However, in some cases, no unbiased technique exists which achieves the bound. This may occur either if for any unbiased ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi-squared Divergence
In probability theory, an f-divergence is a certain type of function D_f(P\, Q) that measures the difference between two probability distributions P and Q. Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of f-divergence. History These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes. ''f''-divergences were studied further independently by , and and are sometimes known as Csiszár f-divergences, Csiszár–Morimoto divergences, or Ali–Silvey distances. Definition Non-singular case Let P and Q be two probability distributions over a space \Omega, such that P\ll Q, that is, P is absolutely continuous with respect to Q (meaning Q>0 wherever P>0). Then, for a convex function f: , +\infty)\to(-\infty, +\infty/math> such that f(x) is finite for all x > 0, f(1)=0, and f(0)=\lim_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
