Topological Quantum Computation with Non-Abelian Anyons

25 Pages·2013·2.18 MB·English

Kohtaro Kato
Fabian Furrer, Mio Murao
University of Tokyo

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Motivation
 Entanglement is an important resource for quantum
information protocols such as quantum teleportation.
 In a topological phase, entanglement entropy has an
characteristic term called topological entanglement
entropy* which depends on the quantum dimension
of anyons.
Q.
Can we use topological entanglement entropy as a resource for
quantum information protocols?
*A.Kitaev et al., Phys. Rev. Lett. 96, 110404 (2006)
M.Levin et al., Phys. Rev. Lett. 96, 110405 (2006)
K. Hikami. Ann. Phys. 323, 1729 (2008)
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Outline
 Anyon models

 Entanglement entropy in anyonic systems

• Previous research
• My research

 Entanglement distillation in anyonic systems

 Summary
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Anyon models
• Charges
• Fusion rules
Anyon model
• Braiding
• R- & F- matrices
vacuum
Charges A finite label set
Fusion rules
Fusion of representation theory
SU(2)
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Fusion space
We restrict
: Fusion space
Commutatively of fusion rules Associativity of fusion rules
Braiding
F-matrix
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Quantum dimension
Def. Quantum dimension
A positive number corresponding to a label
Th. Dim. of fusion space of anyons
n
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Example of anyon models
• Toric anyon {1,m,e,ε}
• Ising anyon {1,σ,ψ}
• Fibonacci anyon {1,τ} Non-Abelian
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States of anyonic Hilbert space
• Fibonacci anyon
1 qubit !
2 qubit + NC
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Entanglement entropy
Alice Bob
Entanglement entropy is the unique entanglement
measure of bipartite pure states.
is “entangled’’
:Density matrix
:Reduced density matrix
Def. Entanglement entropy of system A
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E.E. of anyonic systems
Anyonic system : The fusion space does not have tensor product structure!!!
Alice Bob
How to define “entanglement entropy’’ ?
A1. using TQFT (previous research)

A2. embedding (my research)
Non-abelian anyons
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