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The Einstein–Podolsky–Rosen paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided by quantum mechanics was incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.
Quantum entanglement is the physical 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 lacking in classical mechanics.
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.
In quantum mechanics, counterfactual definiteness (CFD) is the ability to speak "meaningfully" of the definiteness of the results of measurements that have not been performed. The term "counterfactual definiteness" is used in discussions of physics calculations, especially those related to the phenomenon called quantum entanglement and those related to the Bell inequalities. In such discussions "meaningfully" means the ability to treat these unmeasured results on an equal footing with measured results in statistical calculations. It is this aspect of counterfactual definiteness that is of direct relevance to physics and mathematical models of physical systems and not philosophical concerns regarding the meaning of unmeasured results.
Bell's theorem is a term encompassing a number of closely-related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories. The "local" in this case refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields can only occur at speeds no greater than the speed of light. "Hidden variables" are hypothetical properties possessed by quantum particles, properties that are undetectable but still affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."
A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. The experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables to explain the behavior of particles like photons and electrons. To date, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.
Elliott Hershel Lieb is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is
In quantum mechanics, superdeterminism is a loophole in Bell's theorem. By postulating that all systems being measured are correlated with the choices of which measurements to make on them, the assumptions of the theorem are no longer fulfilled. A hidden variables theory which is superdeterministic, hence, can fulfill Bell's notion of local causality and still violate the inequalities derived from Bell's theorem. This makes it possible to construct a local hidden-variable theory that reproduces the predictions of quantum mechanics, for which a few toy models have been proposed. The term superdeterminism is misleading. Superdeterministic models are deterministic in the usual sense. But in addition to being deterministic, they also postulate correlations between the state that is measured and the measurement setting.
In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics, due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model.
The Ahlswede–Daykin inequality, also known as the four functions theorem, is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics.
The Leggett–Garg inequality, named for Anthony James Leggett and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism is a classical worldview defined by the conjunction of two postulates:
- Macrorealism per se: "A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states."
- Noninvasive measurability: "It is possible in principle to determine which of these states the system is in without any effect on the state itself, or on the subsequent system dynamics."
The Leggett inequalities, named for Anthony James Leggett, who derived them, are a related pair of mathematical expressions concerning the correlations of properties of entangled particles. They are fulfilled by a large class of physical theories based on particular non-local and realistic assumptions, that may be considered to be plausible or intuitive according to common physical reasoning.
Jan Raphael Høegh-Krohn was a Norwegian mathematician.
In statistical mechanics, the Griffiths inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.
Jean Ginibre was a French mathematical physicist. He is known for his contributions to random matrix theory, statistical mechanics, and partial differential equations. With Martine Le Berre and Yves Pomeau, he provided a kinetic theory for the emission of photons by an atom maintained in an excited state by an intense field that creates Rabi oscillations. He received the Paul Langevin Prize in 1969.
Yves Pomeau, born in 1942, is a French mathematician and physicist, emeritus research director at the CNRS and corresponding member of the French Academy of sciences. He was one of the founders of the Laboratoire de Physique Statistique, École Normale Supérieure, Paris. He is the son of René Pomeau.
