Information-theoretical analysis of topological entanglement entropy and multipartite correlations

Kato, Kohtaro; Furrer, Fabian; Murao, Mio

doi:10.1103/physreva.93.022317

Cited by 42 publications

(69 citation statements)

References 47 publications

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“…The amount of classical information that can be hidden with this scheme is then given by the JKL index. We compare this to a similar result in finite dimensions [12], where the TEE was shown to coincide with the optimal achievable rate of a (different) secret sharing scheme via the irreducible correlation. Based on this we argue that the JKL index is indeed closely related to the TEE.…”

Section: Introductionmentioning

confidence: 57%

“…Section 4 then is devoted to constructing a secret sharing scheme in the example of the toric code on the infinite plane and to explaining how it relates to the index. Furthermore we discuss how this is connected to recent work [12], where for finite two-dimensional lattice systems it was shown that there is a connection between the TEE and the irreducible correlation of certain secret sharing schemes. We also discuss the role of superselection sectors in our construction.…”

Section: Content Of This Workmentioning

confidence: 68%

“…The latter can be obtained as a Jones-Kosaki-Longo index associated to cone algebras [11]. In this paper we concentrate on the dashed arrows, arguing that by reinterpreting the topological entropy in terms of an information hiding task [12], it can be naturally connected to the cone index. The log of the index tells us something about how much information we can hide, and the index theory provides us with a channel  which describes the limited powers of an adversary Eve.…”

Section: Content Of This Workmentioning

confidence: 99%

“…There exists an interesting interplay between the TEE [5,6] and the irreducible correlation of the state, which provides the TEE with an operational interpretation as the achievable rate of a certain secret sharing scheme of classical information [12]. The irreducible correlation is a measure of correlations on multipartite quantum systems.…”

Section: Tee and The Irreducible Correlationmentioning

confidence: 99%

“…when optimising over all states that have the same l-RDM as ρ. Precise definitions can be found in [12,42].…”

Section: Tee and The Irreducible Correlationmentioning

confidence: 99%

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Jones index, secret sharing and total quantum dimension

2017

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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. 2 [9]. A total quantum dimension that is larger than the number of distinct particles signifies nonabelian anyons [10], since an anyon a is abelian if and only if d a =1. The quantum dimension d a of an anyon of OPEN ACCESS RECEIVED

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Section: Introductionmentioning

confidence: 57%

Section: Content Of This Workmentioning

confidence: 68%

Section: Content Of This Workmentioning

confidence: 99%

Section: Tee and The Irreducible Correlationmentioning

confidence: 99%

“…when optimising over all states that have the same l-RDM as ρ. Precise definitions can be found in [12,42].…”

Section: Tee and The Irreducible Correlationmentioning

confidence: 99%

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Jones index, secret sharing and total quantum dimension

2017

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Immersed figure-8 Annuli and Anyons

Shi

2024

Ann. Henri Poincaré

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Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit

Hayden

Penington

2020

Commun. Math. Phys.

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We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them; replacing classical bits by zero-bits makes teleportation asymptotically reversible. This decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applications to state merging, the quantum channel capacity, entanglement distillation, quantum identification and remote state preparation. The source of these results is the theory of approximate quantum error correction. The action of a quantum channel is reversible if and only if no information is leaked to the environment, a characterization that is useful even in approximate form. However, different notions of approximation lead to qualitatively different forms of quantum error correction in the limit of large dimension. We study the effect of a constraint on the dimension of the reference system when considering information leakage. While the resulting condition fails to ensure that the entire input can be corrected, it does ensure that all subspaces of dimension matching that of the reference are correctable. The size of the reference can be characterized by a parameter α; we call the associated resource an α-bit. Changing α interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of α-bits, including the applications above, and determine the α-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants.

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Information-theoretical analysis of topological entanglement entropy and multipartite correlations

Cited by 42 publications

References 47 publications

Jones index, secret sharing and total quantum dimension

Jones index, secret sharing and total quantum dimension

Immersed figure-8 Annuli and Anyons

Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit

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