Sheaf-theoretic methods in quantum mechanics and quantum information theory

• Main Author: Constantin, Carmen Maria
• Other Authors: Abramsky, Samson ; Doering, Andreas ; Coecke, Bob
• Published: University of Oxford 2015
• Subjects: 530.12
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.712082

In this thesis we use the language of sheaf theory in an attempt to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - which separate quantum from classical physics. We first present, based on the work of Abramsky and Brandenburger [2], 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. One of the original contributions of this thesis is to use these insights in order to classify a particular class of multipartite entangled states, which we have named balanced states with functional dependencies. Almost all of these states turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We then further extend this result by showing that in fact all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Moreover, 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 [62], in which many quan- tum mechanical concepts, such as states, observables, and propositions about these, can be expressed. This presentation is followed by another original contribution in which we 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.