In this work we present a study of improper mixtures from a quantum logical and geometrical point of view. We do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we consider the possibility of extending the lattice of projections in order to include improper mixtures as atoms of a new lattice. This lattice structure is constructed using the convex set of states instead of the traditional set of rays in the Hilbert space. We study the relationship between these two lattices and argue that this approach is suitable for the study of compound systems. Finally, we study the maps which can be defined between the extended lattices of a compound system and its subsystems. Key words: convex set of states, quantum logic, entanglement. Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)In this work we present a study of improper mixtures from a quantum logical and geometrical point of view. We do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we consider the possibility of extending the lattice of projections in order to include improper mixtures as atoms of a new lattice. This lattice structure is constructed using the convex set of states instead of the traditional set of rays in the Hilbert space. We study the relationship between these two lattices and argue that this approach is suitable for the study of compound systems. Finally, we study the maps which can be defined between the extended lattices of a compound system and its subsystems. Key words: convex set of states, quantum logic, entanglement. Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)

Jan 1, 200919 citations

On the homogeneity of the quantum transition probability

Gerd Niestegge

In the years 1952 and 1965, H.-C. Wang and U. Hirzebruch showed that the two-point homogeneous compact spaces with convex metrics are isometric to the spheres, the real, complex, octonion projective spaces and the Moufang plane and as well to the sets of the minimal idempotents or pure states in the simple Euclidean Jordan algebras. Here we reveal the physical meaning of these mathematical achievements for the quantum mechanical transition probability. We show that this transition probability features a maximum degree of homogeneity in all simple Euclidean Jordan algebras, which includes common finite-dimensional Hilbert space quantum theory. The atomic parts of these algebras or, equivalently, the extreme boundaries of their state spaces can be characterized by purely topological means. This is an important difference to many other recent approaches that aim to distinguish the entire state spaces among the convex compact sets. An interesting case with non-homogeneous transition probability arises, when the $E_6$ -symmetric bioctonionic projective plane is used as quantum logic.In the years 1952 and 1965, H.-C. Wang and U. Hirzebruch showed that the two-point homogeneous compact spaces with convex metrics are isometric to the spheres, the real, complex, octonion projective spaces and the Moufang plane and as well to the sets of the minimal idempotents or pure states in the simple Euclidean Jordan algebras. Here we reveal the physical meaning of these mathematical achievements for the quantum mechanical transition probability. We show that this transition probability features a maximum degree of homogeneity in all simple Euclidean Jordan algebras, which includes common finite-dimensional Hilbert space quantum theory. The atomic parts of these algebras or, equivalently, the extreme boundaries of their state spaces can be characterized by purely topological means. This is an important difference to many other recent approaches that aim to distinguish the entire state spaces among the convex compact sets. An interesting case with non-homogeneous transition probability arises, when the $E_6$-symmetric bioctonionic projective plane is used as quantum logic.

quant-pharXiv:2601.01936Jan 6, 20260 citations

Convex quantum logic

Federico Holik, César Massri

In this work we study the convex set of quantum states from a quantum logical point of view. We consider an algebraic structure based on the convex subsets of this set. The relationship of this algebraic structure with the lattice of propositions of quantum logic is shown. This new structure is suitable for the study of compound systems and shows new differences between quantum and classical mechanics. This differences are linked to the nontrivial correlations which appear when quantum systems interact. They are reflected in the new propositional structure, and do not have a classical analogue. This approach is also suitable for an algebraic characterization of entanglement.In this work we study the convex set of quantum states from a quantum logical point of view. We consider an algebraic structure based on the convex subsets of this set. The relationship of this algebraic structure with the lattice of propositions of quantum logic is shown. This new structure is suitable for the study of compound systems and shows new differences between quantum and classical mechanics. This differences are linked to the nontrivial correlations which appear when quantum systems interact. They are reflected in the new propositional structure, and do not have a classical analogue. This approach is also suitable for an algebraic characterization of entanglement.

Aug 14, 201617 citations

E_6

-symmetric bioctonionic projective plane is used as quantum logic.