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In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system.The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable".
As Garrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". [168] Von Neumann combined traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work ...
The formalism of density operators and matrices was introduced in 1927 by John von Neumann [29] and independently, but less systematically, by Lev Landau [30] and later in 1946 by Felix Bloch. [31] Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
In physics, the von Neumann entropy, named after John von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics , and it is the quantum counterpart of the Shannon entropy from classical ...
Every commutative von Neumann algebra is isomorphic to L ∞ (X) for some measure space (X, μ) and conversely, for every σ-finite measure space X, the *-algebra L ∞ (X) is a von Neumann algebra. Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C*-algebras is sometimes ...
In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. [1] The theory was further developed by Dorothy Maharam (1958) [ 2 ] and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). [ 3 ]
In quantum information theory, quantum mutual information, or von Neumann mutual information, after John von Neumann, is a measure of correlation between subsystems of quantum state. It is the quantum mechanical analog of Shannon mutual information .
In descriptive set theory the Jankov–von Neumann uniformization theorem is a result saying that every measurable relation on a pair of standard Borel spaces (with respect to the sigma algebra of analytic sets) admits a measurable section. It is named after V. A. Jankov and John von Neumann.
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