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The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The commutation theorem for tensor products states that
Von Neumann's traditional definition simply takes the "obvious" tensor product: to compute , first collect all simple tensors of the form such that ‖ ‖ <. The latter describes a pre-inner product through the polarization identity , so take the closed span of such simple tensors modulo that inner product's isotropy subspaces.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
Von Neumann's closest friend in the United States was the mathematician Stanisław Ulam. [76] Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up. [63] Ulam noted that von Neumann's way of thinking might not be visual, but more ...
The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most ...
It was introduced by John von Neumann in 1936. [2] Relation with the strong (operator) topology ... Topological tensor product – Tensor product constructions for ...
The state space of the entire quantum system is then the tensor product of the spaces for the two parts. :=. Let ρ AB be a density matrix acting on states in H AB. The von Neumann entropy of a density matrix S(ρ), is the quantum mechanical analogy of the Shannon entropy.
The Koopman–von Neumann (KvN) theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable wavefunctions. As its name suggests, the KvN theory is related to work [1] [2]: 220 by Bernard Koopman and John von Neumann.