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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] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LOCC
LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received. Mathematical properties The formal definition of the set of LOCC operations is complicated due to the fact that later local operations depend in general on all the previous classical communication and due to the unbounded number of communication rounds. For any finite number r\geq1 one can define \operatorname_r, the set of LOCC operations that can be achieved with r rounds of classical communication. The set becomes strictly larger whenever r is increased and care has to be taken to define the limit of infinitely many rounds. In particular, the set LOCC is not topologically closed, that is there are quantum operations that can be approximate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Witness
In quantum information theory, an entanglement witness is a functional which distinguishes a specific entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the entangled state is strictly outside the range of possible expectation values of any separable state. Details Let a composite quantum system have state space H_A \otimes H_B. A mixed state ''ρ'' is then a trace-class positive operator on the state space which has trace 1. We can view the family of states as a subset of the real Banach space generated by the Hermitian trace-class operators, with the trace norm. A mixed state ''ρ'' is separable if it can be approximated, in the trace norm, by states of the form :\xi = \sum_ ^k p_i \, \rho_i^A \otimes \rho_i^B, where \rho_i^A and \rho_i^B are pure states on the subsystems ''A'' and ''B'' respectively. So t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Myrheim
Jan Myrheim (born 14 February 1948) is a Norwegian physicist. He was born in Røyrvik. He took the cand.real. at the University of Oslo in 1972 and took the dr.philos. degree at the University of Trondheim in 1994. He was then appointed as a professor of theoretical physics at the Norwegian University of Science and Technology. He had then worked at the Norwegian Institute of Technology since 1985, except the years 1987 to 1990. Together with Jon Magne Leinaas he discovered that in one and two spatial dimensions, there is a possibility of having fractional quantum statistics. This is of particular importance in two dimensions where fractional statistics particles, usually referred to as anyons, play an important role in the theory of the fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Magne Leinaas
Jon Magne Leinaas (born 11 October 1946) is a Norway, Norwegian theoretical physicist. He was born in Oslo. He took the cand.real. at the University of Oslo in 1970 and the dr.philos. degree at the same institution in 1980. He was a fellow at Nordita, and at CERN, and held a faculty position at the University of Stavanger, before he was appointed as a professor of theoretical physics at the University of Oslo in 1989. He is a fellow of the Norwegian Academy of Science and Letters, and thRoyal Swedish Academy of Sciences Together with Jan Myrheim he discovered that in one and two spatial dimensions, there is a possibility of having fractional quantum statistics. This is of particular importance in two dimensions where fractional statistics particles, usually referred to as anyons, play an important role in the theory of the fractional quantum Hall effect. The duo shared the Fridtjof Nansen Excellent Research Award in Science in 1993. He resides at Gjettum, a suburb of Oslo. Refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Product
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the following diagram commutes: *: Whether a product exists may depend on C or on X_1 and X_2. If it does exist, it is unique up to canonical is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Hilbert Space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by :w \sim v if and only if v = \lambda w for some non-zero complex number \lambda. The equivalence classes of v for the relation \sim are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. Overview The physical significance of the projective Hilbert space is that in quantum theory, the wave functions \psi and \lambda \psi represent the same ''physical state'', for any \lambda \ne 0. It is conventional to choose a \psi from the ray so that it has unit norm, \langle\psi, \psi\rangle = 1, in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine \psi within the ray, since \psi could be multiplied by any \lambda with absolute value 1 (the U(1) action ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Criterion
In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a ''separability criterion''. It was first proved and independently formulated in 1999. Violation of the reduction criterion is closely related to the distillability of the state in question. Details Let ''H''1 and ''H''2 be Hilbert spaces of finite dimensions ''n'' and ''m'' respectively. ''L''(''Hi'') will denote the space of linear operators acting on ''Hi''. Consider a bipartite quantum system whose state space is the tensor product : H = H_1 \otimes H_2. An (un-normalized) mixed state ''ρ'' is a positive linear operator (density matrix) acting on ''H''. A linear map Φ: ''L''(''H''2) → ''L''(''H''1) is said to be positive if it preserves the cone of positive elements, i.e. ''A'' is positive implied ''Φ''(''A'') is also. From the one-to-one correspondence between positive maps and entang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Criterion
In quantum mechanics, in particular 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 th ..., the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a ''separability criterion''. The result Consider a quantum mechanical system composed of ''n'' subsystems. The state space ''H'' of such a system is the tensor product of those of the subsystems, i.e. H = H_1 \otimes \cdots \otimes H_n. For simplicity we will assume throughout that all relevant state spaces are finite-dimensional. The criterion reads as follows: If ρ is a separable mixed state acting on ''H'', then the range of ρ is spanned by a set of product vectors. Proof In general, if a matrix ''M'' is of the form M = \sum_i v_i v_i^*, the rang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
