Complete effect-preparation structures: Attempt of an unification of two different approaches to axiomatic quantum mechanics

1985 ◽  
Vol 90 (2) ◽  
pp. 161-184 ◽  
Author(s):  
G. Cattaneo ◽  
G. Nisticò
Keyword(s):  
Quantum Mechanics ◽  
Axiomatic Quantum Mechanics

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

Representations of Groups as Automorphisms on Orthomodular Lattices and Posets

1971 ◽  
Vol 23 (4) ◽  
pp. 659-673 ◽  
Author(s):  
Stanley P. Gudder
Keyword(s):  
Quantum Mechanics ◽  
General Setting ◽  
Symmetry Groups ◽  
Orthomodular Lattices ◽  
Induced Representations ◽  
Complete Study ◽  
Logic Approach ◽  
Physical Symmetry ◽  
Groups Of Automorphisms ◽  
Axiomatic Quantum Mechanics

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.

scholarly journals QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY

2020 ◽  
Author(s):  
Siddharth Sharma
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Matrix ◽  
Pythagorean Theorem ◽  
Quantum Entropy ◽  
Density Matrices ◽  
Relative Quantum ◽  
Axiomatic Quantum Mechanics ◽  
Measure Entropy

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

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