arXiv:1911.01470 · Physical Review Research

Verlinde formula from entanglement

Bowen Shi

Published May 6, 2020• 21 citations

Abstract

This work derives mutual braiding statistics of anyons from a set of axioms about entanglement entropy, which have been shown to imply the fusion rules of anyons. More specifically, the author defines the topological $S$ matrix from a quantum state and show that it satisfies the Verlinde formula.

DOI: 10.1103/physrevresearch.2.023132

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arXiv:1911.01470 · Physical Review Research

Verlinde formula from entanglement

Bowen Shi

Published May 6, 2020• 21 citations

arXiv HTML PDF

Abstract

S

matrix from a quantum state and show that it satisfies the Verlinde formula.

DOI: 10.1103/physrevresearch.2.023132

Related papers

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Juven Wang, Xiao-Gang Wen, Shing–Tung Yau

It is known that quantum statistics of quasiparticles in 2+1 spacetime dimensions beget anyons, beyond the familiar statistics of bosons and fermions, while Verlinde formula dictates the consistent anyon statistics. In 3+1 dimensions, although quasiparticles cannot be anyonic, extended quasi-excitations like strings can have anyonic string statistics. In this work, we first introduce the fusion and braiding data of particle and string excitations creatable from the operators of (world-) lines/sheets, and all are well-defined in gapped states of matter with intrinsic topological orders. We then apply the geometric-topology surgery theory to derive a set of quantum surgery formulas analogous to Verlinde's constraining the fusion and braiding quantum statistics of anyonic particle and anyonic string excitations in 2+1 and 3+1 dimensions, essential for the theory of bulk topological orders and potentially correlated to bootstrap boundary physics.It is known that quantum statistics of quasiparticles in 2+1 spacetime dimensions beget anyons, beyond the familiar statistics of bosons and fermions, while Verlinde formula dictates the consistent anyon statistics. In 3+1 dimensions, although quasiparticles cannot be anyonic, extended quasi-excitations like strings can have anyonic string statistics. In this work, we first introduce the fusion and braiding data of particle and string excitations creatable from the operators of (world-) lines/sheets, and all are well-defined in gapped states of matter with intrinsic topological orders. We then apply the geometric-topology surgery theory to derive a set of quantum surgery formulas analogous to Verlinde's constraining the fusion and braiding quantum statistics of anyonic particle and anyonic string excitations in 2+1 and 3+1 dimensions, essential for the theory of bulk topological orders and potentially correlated to bootstrap boundary physics.

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FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ and $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $|i-j|>1$ . - Hecke algebra relations: $(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0$ , with $q = e^{2\pi i/k}$ for SU(2)_k anyons. - Fusion rules: $\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . CONNECTION: - Fibonacci anyon quantum dimension $d_\tau = \phi = 1.618$ (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: $e^{\pm 3\pi i/5}$ , related to golden ratio via $ 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group \( B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $ and $ \sigma_i \sigma_j = \sigma_j \sigma_i $ for $ |i-j|>1 $. - Hecke algebra relations: $ (\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0 $, with $ q = e^{2\pi i/k} $ for SU(2)_k anyons. - Fusion rules: $ \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. CONNECTION: - Fibonacci anyon quantum dimension $ d_\tau = \phi = 1.618 $ (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: $ e^{\pm 3\pi i/5} $, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in nonchiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in the search for suitable platforms realizing quantum computing devices.We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in nonchiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in the search for suitable platforms realizing quantum computing devices.

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FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity under braid group representations, enabling topological quantum computation via braiding operations that encode quantum gates. MATH: Fusion algebra for anyon types a, b, c: $a \times b = \sum_c N_{ab}^c c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ (Yang-Baxter equation). For Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . Braid matrices yield $R$ -matrix eigenvalues: $e^{\pm i\pi/5}$ , $e^{\pm i3\pi/5}$ . CONNECTION: Fibonacci anyon quantum dimension $d_\tau = \phi = 1.618$ (golden ratio). Fusion matrix eigenvalues involve $\phi^{-1} = 0.618$ . Braid group representations correspond to $SU(2)_3$ Chern-Simons theory, linked to $E_8$ root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebras exhibit self-similarity under braid group representations, enabling topological quantum computation via braiding operations that encode quantum gates. MATH: Fusion algebra for anyon types a, b, c: $ a \times b = \sum_c N_{ab}^c c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). Braid group $ B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $ (Yang-Baxter equation). For Fibonacci anyons: fusion rule $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Braid matrices yield $ R $-matrix eigenvalues: $ e^{\pm i\pi/5} $, $ e^{\pm i3\pi/5} $. CONNECTION: Fibonacci anyon quantum dimension $ d_\tau = \phi = 1.618 $ (golden ratio). Fusion matrix eigenvalues involve $ \phi^{-1} = 0.618 $. Braid group representations correspond to $ SU(2)_3 $ Chern-Simons theory, linked to $ E_8 $ root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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S

matrix from a quantum state and show that it satisfies the Verlinde formula.

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

\phi = (1+\sqrt{5})/2 \approx 1.618

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. - Hecke algebra relations:

(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0

, with

q = e^{2\pi i/k}

for SU(2)_k anyons. - Fusion rules:

\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). - Fibonacci anyon fusion:

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. CONNECTION: - Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). - Braid matrix eigenvalues for Fibonacci anyons:

e^{\pm 3\pi i/5}

, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

a \times b = \sum_c N_{ab}^c c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

(Yang-Baxter equation). For Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. Braid matrices yield

R

-matrix eigenvalues:

e^{\pm i\pi/5}

e^{\pm i3\pi/5}

. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). Fusion matrix eigenvalues involve

\phi^{-1} = 0.618

. Braid group representations correspond to

SU(2)_3

Chern-Simons theory, linked to

E_8

root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com