Quantum Logic in Algebraic Approach

Non-unitary Evolution of Quantum Logics

“…On the basis of this observation, in this work, we study the impact of a non unitary evolution on the logical structure of the system by continuing previous works [31]. That is, we study how the logical structure of quantum properties corresponding to relevant observables, becomes essentially Boolean by using an algebraic approach [17].…”

“…Pure states can be represented by one dimensional projection operators or equivalently, as density matrices ρ such that ρ 2 = ρ [17]. In order to establish the algebraic structure underlying the logical propositions associated to a quantum system, let us take a deeper look into the formal structure of quantum mechanics.…”

“…The rigorous formulation of quantum mechanics was achieved after a series of papers by von Neumann, Jordan, Hilbert and Nordheim [16]. Projection operators play a key role in the axiomatization, and this is related to the spectral decomposition theorem [17,18], which associates a projection valued measure to any quantum observable represented by a self adjoint operator [19]. The set of projection operators can be endowed with an orthomodular (non-Boolean) lattice structure [20,22] and was named quantum logic [21], in contrast with the distributive structure of classical propositional systems [23].…”

“…The complete description of a quantum system involves non-commutative operators: in the standard approach to quantum mechanics, these are generated by the set of bounded operators B(H ) on a Hilbert space H [17,23]. As a consequence, the lattice of quantum properties will be non-distributive [43,23].…”

On the Interpretation of Probabilities in Generalized Probabilistic Models

“…Later, it was discovered that this was strongly related to the nonexistence of joint distributions for noncommuting observables. These peculiarities and formal aspects of the probabilities involved in quantum theory have been vastly studied in the literature [5][6][7][8][9][10][11]. One of the most important axiomatizations in probability theory is due to Kolmogorov [12].…”