51 research outputs found
Contextuality and the fundamental theorems of quantum mechanics
Contextuality is a key feature of quantum mechanics, as was first brought to
light by Bohr and later realised more technically by Kochen and Specker. Isham
and Butterfield put contextuality at the heart of their topos-based formalism
and gave a reformulation of the Kochen-Specker theorem in the language of
presheaves. Here, we broaden this perspective considerably (partly drawing on
existing, but scattered results) and show that apart from the Kochen-Specker
theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate
fundamentally to contextuality. We provide reformulations of the theorems using
the language of presheaves over contexts and give general versions valid for
von Neumann algebras. This shows that a very substantial part of the structure
of quantum theory is encoded by contextuality.Comment: v2: minor revisions, added definition of Bell presheaf, adjustment of
Bell's theorem in contextual for
A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
The Kochen-Specker theorem asserts the impossibility of assigning values to
quantum quantities in a way that preserves functional relations between them.
We construct a new type of valuation which is defined on all operators, and
which respects an appropriate version of the functional composition principle.
The truth-values assigned to propositions are (i) contextual; and (ii)
multi-valued, where the space of contexts and the multi-valued logic for each
context come naturally from the topos theory of presheaves.
The first step in our theory is to demonstrate that the Kochen-Specker
theorem is equivalent to the statement that a certain presheaf defined on the
category of self-adjoint operators has no global elements. We then show how the
use of ideas drawn from the theory of presheaves leads to the definition of a
generalized valuation in quantum theory whose values are sieves of operators.
In particular, we show how each quantum state leads to such a generalized
valuation.Comment: Clarification of situation for situation for operators with
continuous spectr
Category-Theoretic Interpretative Framework of the Complementarity Principle in Quantum Mechanics
This study aims to provide an analysis of the complementarity principle in quantum theory through the establishment of partial structural congruence relations between the quantum and Boolean kinds of event structure. Specifically, on the basis of the existence of a categorical adjunction between the category of quantum event algebras and the category of presheaves of variable Boolean event algebras, we establish a twofold complementarity scheme consisting of a generalized/global and a restricted/local conceptual dimension, where the latter conception is subordinate to and constrained by the former. In this respect, complementarity is not only understood as a relation between mutually exclusive experimental arrangements or contexts of comeasurable observables, as envisaged by the original conception, but it is primarily comprehended as a reciprocal relation concerning information transfer between two hierarchically different structural kinds of event structure that can be brought into partial congruence by means of the established adjunction. It is further argued that the proposed category-theoretic framework of complementarity naturally advances a contextual realist conceptual stance towards our deeper understanding of the microphysical nature of reality
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