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Gleason's theorem is often cited as establishing the Born rule from the structure of Hilbert space, yet its original proof is mathematically sophisticated and rarely accessible to physicists. In this article we present a simple route to the…

We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for $d=2$…

We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason's theorem only relies upon the consistent assignment of probabilities to the outcomes of…

Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully…

As it is known, Gleason's theorem is not applicable for a two-dimensional Hilbert space since in this situation Gleason's axioms are not strong enough to imply Born's rule thus leaving room for a dispersion-free probability measure i.e.,…

In this paper, a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, is presented and the guidelines to how to extend…

This Chapter develops a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, what is fundamental in the transition from classical to quantum physics is the recognition that…

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it.…

Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at…

This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…

The machinery of quantum mechanics is fully capable of describing a single realistic world. Here we discuss the converse: in spite of appearances, and indeed numerous claims to the contrary, any quantum mechanical model can be mimicked, up…

The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory on "nonlocality without entanglement." For multipartite quantum systems, each of dimension three or greater, the…

We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism,…

Quantum processes cannot be reduced, in a nontrivial way, to classical processes without specifying the context in the description of a measurement procedure. This requirement is implied by the Kochen-Specker theorem in the…

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually…

In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own…

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and…

This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued…

The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with…

We study how meaningful physical predictions can arise in nonperturbative quantum gravity in a closed Lorentzian universe. In such settings, recent developments suggest that the quantum gravitational Hilbert space is one-dimensional and…