Kohtaro Kato Fabian Furrer, Mio Murao University of Tokyo

1 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) 2 Outline

Anyon models

Entanglement entropy in anyonic systems • Previous research • My research

Entanglement distillation in anyonic systems

Summary

3 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)

4 Fusion space

We restrict

: Fusion space

Commutatively of fusion rules Associativity of fusion rules

Braiding F-matrix

5 Quantum dimension Def. Quantum dimension A positive number corresponding to a label

Th. Dim. of fusion space of anyons

n

6 Example of anyon models • Toric anyon {1,m,e,ε}

• Ising anyon {1,σ,ψ}

• Fibonacci anyon {1,τ} Non-Abelian

7 States of anyonic Hilbert space

• Fibonacci anyon

1 qubit !

2 qubit + NC

8 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

9 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

10 The method of TQFT (topological quantum field theory)

Fusion space TQFT Vector of fusion space Wilson line operator (diagram)

11 Quantum trace

12 Previous research*

“Entanglement entropy’’ from quantum trace n

Replica trick

Result

Topological entanglement entropy :quantum dimension of the total charge of system A

*K. Hikami. Ann. Phys. 323, 1729 (2008) 13 Examples Ex: Fibonacci anyon model Alice Bob

(Line: τ anyon)

14 Unnaturality of T.E.E.

Even the state is “separable’’,

and (mixed) Where is randomness? Is it useful for Q.I.P. as usual entanglement? 15 My Research

Let’s start from fusion space ! (Linear Algebra)

+ Restriction (From fusion rules) Tensor product structure!

Define partial trace and E.E. by using the embedding map i

There is the relationship* between our partial trace (linear algebra) and quantum trace (category theory)

Ex.

*Bonderson, Ph.D thesis (2007) Entanglement entropy (from trace)

We treat as a state in and calculate entanglement entropy.

Then obviously,

Where is T.E.E ?17

17 State copy

When we consider N copies of a state …

n+n (τ)anyons

N-copy total charge=1 (1 x 1=1)

Physical space (Nn+Nn anyons)

To calculate E.E. of N copies, we have to treat a state as a state of physical space.

Embed in physical space.

18 Embedded state

n

Extra degrees of N freedom

19 Change of basis

20 Result①

T.E.E. is reappeared !! 21 Distillation protocol

Q. Does have some operational meanings ? i.e. Can we use to quantify a resource of quantum information protocols ?

A. Yes ! Local Operations and Classical Communications Entanglement distillation protocol LOCC

Alice Bob Alice Bob

n copies m copies 22 Distillation protocol of anyons

PVM

PVM Local braidings

M PVM

23 Result②

LOCC

24 Summary

We studied entanglement entropy in non-abelian anyonic systems from view of quantum information. Result①  Topological entanglement entropy is the asymptotic correction of the entanglement entropy of N-copied state. Result②  We can use as a function which quantify a resource for entanglement distillation protocol. (Operationally meaningful)

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