Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory

Abstract

In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.

PhD Thesis, 178 pages; University of Oxford (2015)

References in corpus (21)

• Quantum Mechanics as Quantum Information (and only a little more)
• Characterizing quantum theory in terms of information-theoretic constraints
• A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
• Logical Bell Inequalities
• A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
• A Topos Foundation for Theories of Physics: IV. Categories of Systems
• A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
• (Non-)Contextuality of Physical Theories as an Axiom
• Extreme violation of local realism in quantum hypergraph states
• Contextuality, Cohomology and Paradox
• A classification of multipartite states by degree of non-locality
• Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states
• Observables I: Stone Spectra
• The Born rule as structure of spectral bundles (extended abstract)
• A Comparison of Two Topos-Theoretic Approaches to Quantum Theory
• An infinite dimensional Schur-Horn theorem and majorization theory with applications to operator ideals
• Topos Analogues of the KMS State
• Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory
• Contextual Entropy and Reconstruction of Quantum States
• From Topology to Noncommutative Geometry: $K$-theory
• Subalgebras of orthomodular lattices

Cited by in corpus (2)

• A representation of the wave function on the three-dimensional space
• On Possibilistic Conditions to Contextuality and Nonlocality