46,572 research outputs found
The Many Classical Faces of Quantum Structures
Interpretational problems with quantum mechanics can be phrased precisely by
only talking about empirically accessible information. This prompts a
mathematical reformulation of quantum mechanics in terms of classical
mechanics. We survey this programme in terms of algebraic quantum theory.Comment: 24 page
Statistical equilibrium of tetrahedra from maximum entropy principle
Discrete formulations of (quantum) gravity in four spacetime dimensions build
space out of tetrahedra. We investigate a statistical mechanical system of
tetrahedra from a many-body point of view based on non-local, combinatorial
gluing constraints that are modelled as multi-particle interactions. We focus
on Gibbs equilibrium states, constructed using Jaynes' principle of constrained
maximisation of entropy, which has been shown recently to play an important
role in characterising equilibrium in background independent systems. We apply
this principle first to classical systems of many tetrahedra using different
examples of geometrically motivated constraints. Then for a system of quantum
tetrahedra, we show that the quantum statistical partition function of a Gibbs
state with respect to some constraint operator can be reinterpreted as a
partition function for a quantum field theory of tetrahedra, taking the form of
a group field theory.Comment: v3 published version; v2 18 pages, 4 figures, improved text in
sections IIIC & IVB, minor changes elsewher
Generalized probabilities in statistical theories
In this review article we present different formal frameworks for the
description of generalized probabilities in statistical theories. We discuss
the particular cases of probabilities appearing in classical and quantum
mechanics, possible generalizations of the approaches of A. N. Kolmogorov and
R. T. Cox to non-commutative models, and the approach to generalized
probabilities based on convex sets
Quantum Correlations in the Minimal Scenario
In the minimal scenario of quantum correlations, two parties can choose from
two observables with two possible outcomes each. Probabilities are specified by
four marginals and four correlations. The resulting four-dimensional convex
body of correlations, denoted , is fundamental for quantum
information theory. It is here studied through the lens of convex algebraic
geometry. We review and systematize what is known and add many details,
visualizations, and complete proofs. A new result is that is
isomorphic to its polar dual. The boundary of consists of
three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds
of exposed extreme points. These share all basic properties with the usual
maximally CHSH-violating correlations. These patches are separated by cubic
surfaces of non-exposed extreme points. We provide a trigonometric
parametrization of all extreme points, along with their exposing Tsirelson
inequalities and quantum models. All non-classical extreme points (exposed or
not) are self-testing, i.e., realized by an essentially unique quantum model.
Two principles, which are specific to the minimal scenario, allow a quick and
complete overview: The first is the pushout transformation, the application of
the sine function to each coordinate. This transforms the classical polytope
exactly into the correlation body , also identifying the boundary
structures. The second principle, self-duality, reveals the polar dual, i.e.,
the set of all Tsirelson inequalities satisfied by all quantum correlations.
The convex body includes the classical correlations, a cross
polytope, and is contained in the no-signaling body, a 4-cube. These polytopes
are dual to each other, and the linear transformation realizing this duality
also identifies with its dual.Comment: We also discuss the sets of correlations achieved with fixed Hilbert
space dimension, fixed state or fixed observable
On the lattice structure of probability spaces in quantum mechanics
Let C be the set of all possible quantum states. We study the convex subsets
of C with attention focused on the lattice theoretical structure of these
convex subsets and, as a result, find a framework capable of unifying several
aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent
principle. We also encounter links with entanglement witnesses, which leads to
a new separability criteria expressed in lattice language. We also provide an
extension of a separability criteria based on convex polytopes to the infinite
dimensional case and show that it reveals interesting facets concerning the
geometrical structure of the convex subsets. It is seen that the above
mentioned framework is also capable of generalization to any statistical theory
via the so-called convex operational models' approach. In particular, we show
how to extend the geometrical structure underlying entanglement to any
statistical model, an extension which may be useful for studying correlations
in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416
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