Convex prestructures, a mathematical framework that extends the usual concept of convex subset of a real linear space, are employed to generalize methods used in the study of axiomatic quantum mechanics. A brief summary of the mathematical framework of convex prestructures is given. Convex prestructures are classified, and those which are isomorphic to a convex subset of a real linear space are characterized. The operational quantum mechanics of Davies and Lewis is generalized within the framework of convex prestructures and the existence of a physically motivated orthomodular poset is given. Mielnik's beams and filters are also discussed within the framework of convex prestructures. An error in Mielnik's formulation of beam mixtures is pointed out and it is shown that his beam mixtures and one classification of convex prestructures are equivalent. Also his concept of a filter is generalized in the framework of convex prestructures and geometric requirements needed on the set of normalized states so that they may correspond to a physical system are investigated. Finally, Mackey's axioms are discussed and reformulated in the language of P-convex structures.
Convex prestructures, a mathematical framework that extends the usual concept of convex subset of a real linear space, are employed to generalize methods used in the study of axiomatic quantum mechanics. A brief summary of the mathematical framework of convex prestructures is given. Convex prestructures are classified, and those which are isomorphic to a convex subset of a real linear space are characterized. The operational quantum mechanics of Davies and Lewis is generalized within the framework of convex prestructures and the existence of a physically motivated orthomodular poset is given. Mielnik's beams and filters are also discussed within the framework of convex prestructures. An error in Mielnik's formulation of beam mixtures is pointed out and it is shown that his beam mixtures and one classification of convex prestructures are equivalent. Also his concept of a filter is generalized in the framework of convex prestructures and geometric requirements needed on the set of normalized states so that they may correspond to a physical system are investigated. Finally, Mackey's axioms are discussed and reformulated in the language of P-convex structures.
What picture does one see, looking at a physical theory from a distance, so that the details disappear? Since quantum mechanics is a statistical theory, the most universal picture which remains after the details are forgotten is that of a convex set. Bogdan Mielnik Convex sets Our object is to understand the geometry of the set of all possible states of a quantum system that can occur in nature. This is a very general question, especially since we are not trying to define ‘state’ or ‘system’ very precisely. Indeed we will not even discuss whether the state is a property of a thing, or of the preparation of a thing, or of a belief about a thing. Nevertheless we can ask what kind of restrictions are needed on a set if it is going to serve as a space of states in the first place. There is a restriction that arises naturally both in quantum mechanics and in classical statistics: the set must be a convex set . The idea is that a convex set is a set such that one can form ‘mixtures’ of any pair of points in the set. This is, as we will see, how probability enters (although we are not trying to define ‘probability’ either). From a geometrical point of view a mixture of two states can be defined as a point on the segment of the straight line between the two points that represent the states that we want to mix. We insist that given two points belonging to the set of states, the straight line segment between them must belong to the set too. This is certainly not true of any set.
It is well - known that the set of states of a given quantum mechanical system is to be closed from the point of view of the operational approach if we want to make mixed states or convex combinations. That is, s1 and s2 are states, then the same is to be true for λs1+(1−λ)s2, where 0<λ<1. We can define a convex combination of elements in a linear space, but unfortunately, in the general case the linear space is artificial for the set of states and has no physical meaning, but the procedure of forming the mixtures of states has a natural meaning. This is why we provide an abstract definition of the mixtures, which is independent of the linearity notion. We call this space a convex structure. In the work we consider state spaces, generalized state spaces, in which we select pure states, define operations and effects associated with the operations. We also consider ultraproducts of the sequences of these structures, operations and effects.
We present a mathematical framework for quantum mechanics in which the basic entities and operations have physical significance. In this framework the primitive concepts are states and effects and the resulting mathematical structure is a convex effect algebra. We characterize the convex effect algebras that are classical and those that are quantum mechanical. The quantum mechanical ones are those that can be represented on a complex Hilbert space. We next introduce the sequential product of effects to form a convex sequential effect algebra. This product makes it possible to study conditional probabilities and expectations.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.
We sketch the so called convex, or operational, framework for statistical physical theories, based on the convex set formed by the states of the physical system. A notion of observable is introduced which encompasses the ones usually adopted in classical and in quantum mechanics. In this framework a structure of effect algebra naturally arises. We also discuss a notion of extension of a descriptive physical model, pointing at the possibility of constructing classical extension of quantum mechanics.