Geometry of quantum states

Convex Quantum Logic

“…We think that this approach is suitable in order to consider decoherence or entangled systems from a quantum logical and algebraic point of view. Furthermore, taking the convex set of states as an starting point could be of interest if we take into account that there exists a formulation of QM in terms of convex sets (see [15], [16] and [17]). This is an independent formulation of QM and has the advantage that it can include models of theories which cannot be represented by Hilbert spaces, as is the case of non linear generalizations of quantum mechanics.…”

Nambu-type generalization of the Dirac equation

“…Unfortunately, in practical applications the idea turns out to possess ambiguities as well (some of them are mentioned in [5] in the context of two-level atoms). The third direction was initiated by Mielnik [8] who proposed to discuss the probability interpretation of generalized theories at the abstract level of convex figures of states. A practical difficulty is that it is very difficult to apply such ideas in concrete situations since we have to know the "shape" of the figure of mixed states.…”

Mielnik’s probability spaces and characterization of inner product spaces

“…In [3] B. Mielnik pointed out the insufficiency of the classical approach and proposed a geometric approach to the foundation of general quantum mechanics.…”

“…Let B, and B2 be two bases in S; then, as shown in [3], B, and B2 have the same cardinal number. The cardinal number is called the dimension of (5, p).…”