In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.
In quantum mechanics, each physical system is associated with a Hilbert space. For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional. In the approach codified by John von Neumann, a measurement upon a physical system is represented by a self-adjoint operator on that Hilbert space sometimes termed an "observable". The eigenvectors of such an operator form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1. In the language of von Weizsäcker, a density operator is a "catalogue of probabilities": for each measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator. The procedure for doing so is the Born rule, which states that
where is the density operator, and is the projection operator onto the basis vector corresponding to the measurement outcome .
The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis.
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This lecture describes advanced concepts and applications of quantum optics. It emphasizes the connection with ongoing research, and with the fast growing field of quantum technologies. The topics cov
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
The Born rule (also called Born's rule) is a postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated by German physicist Max Born in 1926.
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SPRINGER HEIDELBERG2018
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