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Wigner–Yanase Skew Information
The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder (or, equivalently, Ramsey) interferometer-based phase or parameter estimation. It is shown that the quantum Fisher information can also be a sensitive probe of a quantum phase transition (e.g. recognizing the superradiant quantum phase transition in the Dicke model). The quantum Fisher information F_ varrho,A of a state \varrho with respect to the observable A is defined as : F_ varrho,A2\sum_ \frac \vert \langle k \vert A \vert l\rangle \vert^2, where \lambda_k and \vert k \rangle are the eigenvalues and eigenvectors of the density matrix \varrho, respectively, and the summation goes over all k and l such that \lambda_k+\lambda_l>0. When the observable generates a unitary transformation of the system with a parameter \theta from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Metrology
Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum Squeezed coherent state, squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing. Mathematical foundations A basic task of quantum metrology is estimating the parameter \theta of the unitary dynamics : \varrho(\theta)=\exp(-iH\theta)\varrho_0\exp(+iH\theta), where \varrho_0 is the initial state of the system and H is the Hamiltonian of the system. \theta is estimated based on measurements on \varrho(\theta). Typically, the system is composed of many particles, and the Hamiltonian is a sum of single-particle terms : H=\sum_k H_k, where H_k ... [...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] |
Uncertainty Relation
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known. More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, ''x'', and momentum, ''p''. Such paired-variables are known as complementary variables or canonically conjugate variables. First introduced in 1927 by German physicist Werner Heisenberg, the formal inequality relating the standard deviation of position ''σx'' and the standard deviation of momentum ''σp'' was derived by Earle Hesse Kennard later that y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greenberger–Horne–Zeilinger State
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger (GHZ) state is an entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). Named for the three authors that first described this state, the GHZ state predicts outcomes from experiments that directly contradict predictions by every classical local hidden-variable theory. The state has applications in quantum computing. History The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. The following year Abner Shimony joined in and they published a three-particle version based on suggestions by N. David Mermin. Experimental measurements on such states contradict intuitive notions of locality and causality. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states. Definition The GHZ state is an entangled q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Metrological Gain
The quantum metrological gain is defined in the context of carrying out a metrological task using a quantum state of a multiparticle system. It is the sensitivity of parameter estimation using the state compared to what can be reached using separable states, i.e., states without quantum entanglement. Hence, the quantum metrological gain is given as the fraction of the sensitivity achieved by the state and the maximal sensitivity achieved by separable states. The best separable state is often the trivial fully polarized state, in which all spins point into the same direction. If the metrological gain is larger than one then the quantum state is more useful for making precise measurements than separable states. Clearly, in this case the quantum state is also entangled. Background The metrological gain is, in general, the gain in sensitivity of a quantum state compared to a product state. Metrological gains up to 100 are reported in experiments. Let us consider a unitary dynami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Depth
In quantum physics, entanglement depth characterizes the strength of multiparticle entanglement. An entanglement depth k means that the quantum state of a particle ensemble cannot be described under the assumption that particles interacted with each other only in groups having fewer than k particles. It has been used to characterize the quantum states created in experiments with cold gases. Definition Entanglement depth appeared in the context of spin squeezing. It turned out that to achieve larger and larger spin squeezing, and thus larger and larger precision in parameter estimation, a larger and larger entanglement depth is needed. Later it was formalized in terms of convex sets of quantum states, independent of spin squeezing as follows. Let us consider a pure state that is the tensor product of multi-particle quantum states , \Psi\rangle=, \phi_1\rangle\otimes, \phi_2\rangle\otimes ... \otimes, \phi_n\rangle. The pure state , \Psi\rangle is said to be k-producibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Entanglement
Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurement#Quantum mechanics, Measurements of physical properties such as position (vector), position, momentum, Spin (physics), spin, and polarization (waves), polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Science
Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum physics, including the limits of what can be achieved with quantum information. The term quantum information theory is sometimes used, but it does not include experimental research and can be confused with a subfield of quantum information science that deals with the processing of quantum information. Scientific and engineering studies Quantum teleportation, Quantum entanglement, entanglement and the manufacturing of quantum computers depend on a comprehensive understanding of quantum physics and engineering. Google and IBM have invested significantly in quantum computer hardware research, leading to significant progress in manufacturing quantum computers since the 2010s. Currently, it is possible to create a quantum computer with over 100 qub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable State
In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are multipartite quantum states that can be written as a tensor product of states in each space. The physical intuition behind these definitions is that product states have no correlation between the different degrees of freedom, while separable states might have correlations, but all such correlations can be explained as due to a classical random variable, as opposed to being due to entanglement. In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fidelity Of Quantum States
In quantum mechanics, notably in quantum information theory, fidelity quantifies the "closeness" between two density matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space. Definition The fidelity between two quantum states ''\rho'' and ''\sigma'', expressed as density matrices, is commonly defined as:R. Jozsa, ''Fidelity for Mixed Quantum States'', J. Mod. Opt. 41, 2315--2323 (1994). DOI: http://doi.org/10.1080/09500349414552171 :F(\rho, \sigma) = \left(\operatorname \sqrt\right)^2. The square roots in this expression are well-defined because both \rho and \sqrt\rho\sigma\sqrt\rho are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product. As ... [...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 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] |
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] |
