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Entropy Of Entanglement
The entropy of entanglement (or entanglement entropy) is a measure of the degree of quantum entanglement between two subsystems constituting a two-part composite quantum system. Given a pure bipartite quantum state of the composite system, it is possible to obtain a reduced density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled. More mathematically; if a state describing two subsystems ''A'' and ''B'' , \Psi_\rangle=, \phi_A\rangle, \phi_B\rangle is a separable state, then the reduced density matrix \rho_A=\operatorname_B, \Psi_\rangle\langle\Psi_, =, \phi_A\rangle\langle\phi_A, is a pure state. Thus, the entropy of the state is zero. Similarly, the density matrix of ''B'' would also have 0 entropy. A reduced density matrix having a non-zero entropy is therefore a signal of the existence of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Monotone
In quantum information and quantum computation, an entanglement monotone or entanglement measure 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 E_i\otimes F_i corresponding to the LOCC \mathcal_, let p_i=\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math> and \rho_i=(E_i\otimes F_i)\rho (E_i\otimes F_i)^/\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math>for a given state \rho, then (i) \mu does not increase under the average over all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Of Formation
The entanglement of formation is a quantity that measures the entanglement of a bipartite quantum state. Definition For a pure bipartite quantum state , \psi\rangle_, using Schmidt decomposition, we see that the reduced density matrices of systems A and B, \rho_A and \rho_B, have the same spectrum. The von Neumann entropy S(\rho_A)=S(\rho_B) of the reduced density matrix can be used to measure the entanglement of the state , \psi\rangle_. We denote this kind of measure as E_(, \psi\rangle_)=S(\rho_A)=S(\rho_B) , and call it the entanglement entropy. This is also known as the entanglement of formation of a pure state. For a mixed bipartite state \rho_, a natural generalization is to consider all the ensemble realizations of the mixed state. We define the entanglement of formation for mixed states by minimizing over all these ensemble realizations, :E_f (\rho_)= \inf\left\ , where the infimum is taken over all the possible ways in which one can decompose \rho_ into pure states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix Renormalization Group
The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method, DMRG is an efficient algorithm that attempts to find the lowest-energy matrix product state wavefunction of a Hamiltonian. It was invented in 1992 by Steven R. White and it is nowadays the most efficient method for 1-dimensional systems. History The first application of the DMRG, by Steven R. White and Reinhard Noack, was a ''toy model'': to find the spectrum of a spin 0 particle in a 1D box. This model had been proposed by Kenneth G. Wilson as a test for any new renormalization group method, because they all happened to fail with this simple problem. The DMRG overcame the problems of previous renormalization group methods by connecting two blocks with the two sites in the middle rather than just adding a single site to a block at each step as well as by using the density matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the molar gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating Johnson–Nyquist noise, thermal noise in resistors. The Boltzmann constant has Dimensional analysis, dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann. As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "Physical constant, defining constants" that have been defined so as to have exact finite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly joules per kelvin, with the effect of defining the SI unit ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making up a substance. Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the Celsius scale with the unit symbol °C (formerly called ''centigrade''), the Fahrenheit scale (°F), and the Kelvin scale (K), with the third being used predominantly for scientific purposes. The kelvin is one of the seven base units in the International System of Units (SI). Absolute zero, i.e., zero kelvin or −273.15 °C, is the lowest point in the thermodynamic temperature scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the third law of thermodynamics. It would be impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermal Equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially uniform and temporally constant. Systems in thermodynamic equilibrium are always in thermal equilibrium, but the converse is not always true. If the connection between the systems allows transfer of energy as 'change in internal energy' but does not allow transfer of matter or transfer of energy as work, the two systems may reach thermal equilibrium without reaching thermodynamic equilibrium. Two varieties of thermal equilibrium Relation of thermal equilibrium between two thermally connected bodies The relation of thermal equilibrium is an instance of equilibrium between two bodies, which means that it refers to transfer through a selectively permeable par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Harmonic Oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known. One-dimensional harmonic oscillator Hamiltonian and energy eigenstates The Hamiltonian of the particle is: \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , where is the particle's mass, is the force constant, \omega = \sqrt is the angular frequency of the oscillator, \hat is the position operator (given by in the coordinate basis), and \hat is the momentum operator (given by \hat p = -i \hbar \, \partial / \partial x in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rényi Entropy
In information theory, the Rényi entropy is a quantity that generalizes various notions of Entropy (information theory), entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as diversity indices, index of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of Quantum entanglement, entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of can be calculated explicitly because it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Negativity
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 been 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 operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 been 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 LOCC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squashed Entanglement
Squashed entanglement, also called CMI entanglement (CMI can be pronounced "see me"), is an information theoretic measure of quantum entanglement for a bipartite quantum system. If \varrho_ is the density matrix of a system (A,B) composed of two subsystems A and B, then the CMI entanglement E_ of system (A,B) is defined by where K is the set of all density matrices \varrho_ for a tripartite system (A,B,\Lambda) such that \varrho_=tr_\Lambda (\varrho_). Thus, CMI entanglement is defined as an extremum of a functional S(A:B , \Lambda) of \varrho_. We define S(A:B , \Lambda), the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the “min” (minimum) in Eq.(1) by an “inf” (infimum). When \varrho_ is a pure state, E_(\varrho_)=S(\varrho_)=S(\varrho_), in agreement with the definition of entanglement of formation for pure states. Here S(\varrho) is the Von Neumann entropy of density matrix \varrho. Motivation for definition of C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Relative Entropy
In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy. Motivation For simplicity, it will be assumed that all objects in the article are finite-dimensional. We first discuss the classical case. Suppose the probabilities of a finite sequence of events is given by the probability distribution ''P'' = , but somehow we mistakenly assumed it to be ''Q'' = . For instance, we can mistake an unfair coin for a fair one. According to this erroneous assumption, our uncertainty about the ''j''-th event, or equivalently, the amount of information provided after observing the ''j''-th event, is :\; - \log q_j. The (assumed) average uncertainty of all possible events is then :\; - \sum_j p_j \log q_j. On the other hand, the Shannon entropy of the probability distribution ''p'', defined by :\; - \sum_j p_j \log p_j, is the real amount of uncertainty before observation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
