Effect algebras ([1], [2]) and D-posets ([3]) are equivalent systems important in quantum structures. In the paper an independent sequence of observables is defined by such a way that a very general version of the central limit theorem may be proved.
Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H H correspond to direct product decompositions H ≃ A × A ⊥ H≃A×A⊥ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.
The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Central Limit Theorem and to prove as well its convergence in a novel way.
Abstract The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C ∗ -algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism (Choi 1975 Lin. Alg. Appl. 10 285) further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical structure in C ∗ -algebras. We explain in detail how this dependence gives rise to the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiołkowski (1972 Rep. Math. Phys. 3 275). We then overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi’s theorem for Jamiołkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C ∗ -algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.