Haag duality and Jones-Kosaki-Longo index in Kitaev's quantum double models for finite abelian groups

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Haag duality for Kitaev's quantum double model for abelian groups

We prove Haag duality for conelike regions in the ground state representation corresponding to the translational invariant ground state of Kitaev's quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localised outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localised in disjoint regions commute.As an application we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher-Haag-Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double

Operator Algebras and Data Hiding in Topologically Ordered Systems

Presented at the QMath13 Conference: Mathematical Results in Quantum Theory, October 8-11, 2016 at the Clough Undergraduate Learning Commons, Georgia Tech.New Mathematical Topics Arising in Current Theoretical Physics - Sunday, October 9th, 2016, Skiles 202 - Chair: Robert Seiringe

Jones index, secret sharing and total quantum dimension

We study the total quantum dimension in the thermodynamic limit of topologically ordered systems. In particular, using the anyons (or superselection sectors) of such models, we define a secret sharing scheme, storing information invisible to a malicious party, and argue that the total quantum dimension quantifies how well we can perform this task. We then argue that this can be made mathematically rigorous using the index theory of subfactors, originally due to Jones and later extended by Kosaki and Longo. This theory provides us with a 'relative entropy' of two von Neumann algebras and a quantum channel, and we argue how these can be used to quantify how much classical information two parties can hide form an adversary. We also review the total quantum dimension in finite systems, in particular how it relates to topological entanglement entropy. It is known that the latter also has an interpretation in terms of secret sharing schemes, although this is shown by completely different methods from ours. Our work provides a different and independent take on this, which at the same time is completely mathematically rigorous. This complementary point of view might be beneficial, for example, when studying the stability of the total quantum dimension when the system is perturbed.ERC/QFTCMPSERC/SIQSDFG/EXC/201EU/Horizon 2020ERC/DQSI

Reply to Nielsen et al. social mindfulness is associated with countries’ environmental performance and individual environmental concern

info:eu-repo/semantics/publishedVersio

From local social mindfulness to global sustainability efforts?

info:eu-repo/semantics/publishedVersio

Social mindfulness and prosociality vary across the globe

Humans are social animals, but not everyone will be mindful of others to the same extent. Individual differences have been found, but would social mindfulness also be shaped by one’s location in the world? Expecting cross-national differences to exist, we examined if and how social mindfulness differs across countries. At little to no material cost, social mindfulness typically entails small acts of attention or kindness. Even though fairly common, such low-cost cooperation has received little empirical attention. Measuring social mindfulness across 31 samples from industrialized countries and regions (n = 8,354), we found considerable variation. Among selected country-level variables, greater social mindfulness was most strongly associated with countries’ better general performance on environmental protection. Together, our findings contribute to the literature on prosociality by targeting the kind of everyday cooperation that is more focused on communicating benevolence than on providing material benefits

Jones index, secret sharing and total quantum dimension

We study the total quantum dimension in the thermodynamic limit of topologically ordered systems. In particular, using the anyons (or superselection sectors) of such models, we define a secret sharing scheme, storing information invisible to a malicious party, and argue that the total quantum dimension quantifies how well we can perform this task. We then argue that this can be made mathematically rigorous using the index theory of subfactors, originally due to Jones and later extended by Kosaki and Longo. This theory provides us with a "relative entropy" of two von Neumann algebras and a quantum channel, and we argue how these can be used to quantify how much classical information two parties can hide form an adversary. We also review the total quantum dimension in finite systems, in particular how it relates to topological entanglement entropy. It is known that the latter also has an interpretation in terms of secret sharing schemes, although this is shown by completely different methods from ours. Our work provides a different and independent take on this, which at the same time is completely mathematically rigorous. This complementary point of view might be beneficial, for example, when studying the stability of the total quantum dimension when the system is perturbed

Firms as the source of innovation and growth The evolution of technological competence

Available from British Library Document Supply Centre-DSC:3597.9407(356) / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo

Haag duality for Kitaev’s quantum double model for abelian groups

We prove Haag duality for conelike regions in the ground state representation corresponding to the translational invariant ground state of Kitaev's quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localised outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localised in disjoint regions commute. As an application we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher-Haag-Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.Comment: 41 pages, 9 figures; final version, fixed statement about uniqueness of the ground state: only the translational invariant ground state is unique, corrected minor error