Jones index, secret sharing and total quantum dimension

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Fiedler, L.; Naaijkens, P.; Osborne, T.J.: Jones index, secret sharing and total quantum dimension. In: New Journal of Physics 19 (2017), Nr. 2, 23039. DOI:

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Sum total of downloads: 126

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.
License of this version: CC BY 3.0
Document Type: Text
Issue Date: 2017
Appears in Collections:Fakultät für Mathematik und Physik

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pos. country downloads
total perc.
1 image of flag of Germany Germany 100 79.37%
2 image of flag of United States United States 10 7.94%
3 image of flag of China China 5 3.97%
4 image of flag of Mexico Mexico 2 1.59%
5 image of flag of Japan Japan 2 1.59%
6 image of flag of Italy Italy 2 1.59%
7 image of flag of No geo information available No geo information available 1 0.79%
8 image of flag of Ukraine Ukraine 1 0.79%
9 image of flag of India India 1 0.79%
10 image of flag of United Kingdom United Kingdom 1 0.79%
    other countries 1 0.79%

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