Mathematical picture language program

Jaffe, Arthur; Liu, Zhengwei

doi:10.1073/pnas.1710707114

Cited by 8 publications

(10 citation statements)

References 31 publications

(55 reference statements)

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“…Some Context: Manin and Feynman introduced the concept of quantum simulation for quantum systems [60,21,61]. One can use an abstract language L with words and grammar, along with a simulation S : L → R to map onto some interesting mathematical area R, as discussed in [36].…”

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confidence: 99%

“…The simulation S of elementary pictures in L leads us to concepts that may have no physical meaning in R. We call them virtual concepts in R, as discussed in [36]. In this paper we introduce one additional notion inspired by quantum information, that one does not find in PAPPA: a dotted line dividing pictures that correspond to different physical regions, see §4.5.…”

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confidence: 99%

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Holographic software for quantum networks

2018

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We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.

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confidence: 99%

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confidence: 99%

Holographic software for quantum networks

2018

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“…The quantum symmetries could be finite or infinite, discrete or continuous, commutative or noncommutative. In certain contexts F can be defined pictorially-as in the picture language program (19). QFA is the study of structures involving F.…”

Section: Qfamentioning

confidence: 99%

“…In section 4 in ref. 19, one finds expressions for the various |Max n basis states, as well as their relation to, and expressions for, the |GHZ n basis states. Also see ref.…”

Section: Asmentioning

confidence: 99%

Quantum Fourier analysis

Jaffe

Jiang

Liu

et al. 2020

Proc. Natl. Acad. Sci. U.S.A.

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Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

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“…In an earlier paper [JL17], we gave a new proof of the reflectionpositivity (RP) property for Hamiltonians, see Definition 2.1. We presented that proof within the framework of a picture language [JL18]. Our language includes a geometric transformation F s , that we call the string Fourier transform (SFT).…”

Section: Introductionmentioning

confidence: 99%

Reflection Positivity and Levin-Wen Models

Jaffe¹,

Liu²

2019

Preprint

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We give an algebraic formulation of our pictorial proof of the reflection positivity property for Hamiltonians. We apply our methods to the widely-studied Levin-Wen models and prove the reflection positivity property for a natural class of those Hamiltonians, both with respect to vacuum and to bulk excitations.The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin-Wen models.with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

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Mathematical picture language program

Cited by 8 publications

References 31 publications

Holographic software for quantum networks

Holographic software for quantum networks

Quantum Fourier analysis

Reflection Positivity and Levin-Wen Models

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