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Negativity (quantum Mechanics)
In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability. It has shown to be an entanglement monotone and hence a proper measure of entanglement. Definition The negativity of a subsystem A can be defined in terms of a density matrix \rho as: :\mathcal(\rho) \equiv \frac where: * \rho^ is the partial transpose of \rho with respect to subsystem A * , , X, , _1 = \text, X, = \text \sqrt is the trace norm or the sum of the singular values of the operator X . An alternative and equivalent definition is the absolute sum of the negative eigenvalues of \rho^: : \mathcal(\rho) = \left, \sum_ \lambda_i \ = \sum_i \frac where \lambda_i are all of the eigenvalues. Properties * Is a convex function of \rho: :\mathcal(\sum_p_\rho_) \le \sum_p_\mathcal(\rho_) * Is an entanglement monotone: :\mathcal(P(\rho_)) \le \mathcal(\rho_) where P(\rho) is an arbitrary LOCC operation ... [...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] |
Quantum Entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurements of physical properties such as position, momentum, spin, and 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 gives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peres–Horodecki Criterion
The Peres–Horodecki criterion is a necessary condition, for the joint density matrix \rho of two quantum mechanical systems A and B, to be separable. It is also called the PPT criterion, for ''positive partial transpose''. In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres and the Horodecki family ( Michał, Paweł, and Ryszard) In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on entanglement witnesses. Definition If we have a general state \rho which acts on \mathcal_A \otimes \mathcal_B :\rho = \sum_ p^_ , i\rangle \langle j , \otimes , k\rangle \langle l, Its partial transpose (with respect to the B party) is defined as :\rho^ := (I \otimes T) (\rho) = \sum_ p^ _ , i\rangle \langle j , \otimes (, k\rangle \langle l, ) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Entanglement Monotone
In quantum information and quantum computation, an entanglement monotone is a function that quantifies the amount of entanglement present in a quantum state. Any entanglement monotone is a nonnegative function whose value does not increase under local operations and classical communication. Definition Let \mathcal(\mathcal_A\otimes\mathcal_B)be the space of all states, i.e., Hermitian positive semi-definite operators with trace one, over the bipartite Hilbert space \mathcal_A\otimes\mathcal_B. An entanglement measure is a function \mu:\to \mathbb_such that: # \mu(\rho)=0 if \rho is separable; # Monotonically decreasing under LOCC, viz., for the Kraus operator In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discusse ... E_i\otimes F_i corresponding to the LOCC \mathcal_, let p_i=\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Transpose
Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial dee" **Partial differential equation, a differential equation that contains unknown multivariable functions and their partial derivatives Other uses *Partial application, in computer science the process of fixing a number of arguments to a function, producing another function *Partial charge or net atomic charge, in chemistry a charge value that is not an integer or whole number *Partial fingerprint, impression of human fingers used in criminology or forensic science *Partial seizure or focal seizure, a seizure that initially affects only one hemisphere of the brain * Partial or Part score, in contract bridge a trick score less than 100, as well as other meanings * Partial or Partial wave, one sound wave of which a complex tone is composed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows and n columns and entries in the field K. A matrix norm is a norm on K^. This article will always write such norms with double vertical bars (like so: \, A\, ). Thus, the matrix norm is a function \, \cdot\, : K^ \to \R that must satisfy the following properties: For all scalars \alpha \in K and matrices A, B \in K^, *\, A\, \ge 0 (''positive-valued'') *\, A\, = 0 \iff A=0_ (''definite'') *\left\, \alpha A\right\, =\left, \alpha\ \left\, A\right\, (''absolutely homogeneous'') *\, A+B\, \le \, A\, +\, B\, (''sub-additive'' or satisfying the ''triangle inequality'') The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: *\left\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the quadratic function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function whose graph is shaped like a cup \cup, while a concave function's graph is shaped like a cap \cap. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a st ... [...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] |
