Entanglement concentration protocol

using linear optics

Anindita BanerjeeCAPSS, Bose Institute

Kolkata

IPQI 2014

Collaboratorsollaborators: Chitra Shukla, Anirban Pathak

IntroductionMotivationLinear opticsExample:Purification using PBS ECP for CAT stateECP for GHZ-like stateECP for Four qubit state ECP for n+1 qubit state of a particular formSingle qubit assisted ECP Transformation Efficiency

Outline

Applications of entanglement

TeleportationDense codingQuantum key distributionSecure direct communication

Entanglement

(require max entangled state between two parties)

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The IDEA is that the two distant parties Alice and Bob are supplied with finite ensemble of pure states

from which they wish to extract the maximally entangled states (MESs).

Entanglement concentration transforms a pure non maximally entangled state into MES

Entanglement distillation transforms a mixed non maximally entangled state into MES

Distributed Qubits interact with the environment

Gets noisy due to storage processing and transmission

Mixed state Less entangled state

Problems!

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Bell state: Bose et al., Zhao et al., Yamamoto et al., Sheng et al. , Sheng and Zhou, Gu et al., Deng

GHZ state: Chaudhury and Dhara [GHZ] and Zhou et al.[GHZ]

W state: Sheng et al.[W state], Ling-yan He

Cluster state: Chaudhury and Dhara [Cluster], Ting-Ting Xu et al. , Zhau et al.[Cluster]

ECP/EP

Motivation

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Schmidt Projective method Bell state Bennet et al.

Entanglement swappingBell state

Bose et al.

POVM Bell state Gu et al.

QEDBell state

Romero et al.

Cross kerr nonlinearities

Bell stateW state

Zhaop et al. Sheng et al[W]

Linear optics

Bell state

Zhao et al.,

Yamamoto et al.,

Sheng et al. S.

Banyopadhyay

Bell stateZhao et al.,

[Experiment] Yamamoto et al

[Experiment]

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Polarized Beam Splitter

Horizontally polarized photon is transmitted and Vertically polarized photon is reflected

I H1 >

I H4 >

I V1 > I V3 >

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Purification of Bell state using PBS

a1 b1

a2 b2

Source pair

Target pair

NATURE |VOL 423 | 22 MAY 2003 417--421

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Three ingredients involved

Local operation

Classical communication

Post selection

V. Vedral and M. B. Plenio

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Entanglement Swapping

Bose et al.

1 2

3 4

Alice Bob

Bell measurement

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ECP for partially entangled cat state

Bell and GHZ are special cases

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ECP for partially entangled GHZ-like state

GHZ-like state

ψi ψ jand are orthogonal to each other And belong to bell state

example

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General state

Applications

Non-maximally entangled (n + 1)-qubit state where and are arbitrary n-qubit states that are mutually orthogonal.

Ψ 0Ψ 0Ψ 0

Ψ 0Ψ 0Ψ 0

Why is it important?Bell stateGHZGHZ-likeCAT states

Bidirectional quantum teleportationHierarchical quantum communication schemes (HQIS),Hierarchical quantum secret sharing (HQSS)

IPQI 2014

F. Verstraete, J. Dehaene, B. De Moor and H. Verschelde, “Four qubits can be entangled in nine different ways”, Phys. Rev. A 65 (2002) 052112.L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani and W. Rubens, “Four-qubit entanglement classification from string theory”, Phys. Rev. Lett. 105 (2010) 100507.

Four-qubits entangled statesThere exist nine failies of states corresponding to nine different ways of entangling four qubits.

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Four-qubit entangled states

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Sigle qubit assisted ECP for general state

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Optical circuit using linear optics Bell measurement

Bell states

Further, the CNOT can be implemented using optical circuits implemented by J. L. O’Brien et al. Thus in general ECPs proposed here can be realized optically. These ECPs may be practically realized using NMR as Bell measurement ispossible in NMR based technologies.

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Two alternative ECPs for quantum states

ECP1 ECP2

and need not be real and should be real

Requires Bell measurement

and particle swapping

Single measurent

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Entanglement transformation efficiency

is the amount of entanglement in the initial partially entangled state is the amount of entanglement of the state after concentration.

E0

Ec

Ambiguity is the amount of entanglement of the state to be concentrated ORis the amount of entanglement of the entire initial state.

E0

be the total initial entanglementE0

higher efficiency of single photon assisted ECPs over Bell-type state assisted ECPs

IPQI 2014

Yu and Song established that any good measure MA-B of bi-partite entanglement can be generalized to multipartite systems, by considering bipartite partitions of the multipartite system. Yu andSong defined a simple measure of tripartite entanglement as

where Mi-jk is a measure of entanglement between subsystem i and subsystem jk.

Sheng et al. (von Neumann entropy ) as a measure of entanglement,But von Neumann entropy is a good measure of entanglement for bipartite systems only.

How to find for an ECP that is designed for multipartite case? Interestingly, the problem is equivalent to provide a quantitative measure of multiparite entanglement.

Let us choose tangle as a measure of entanglement.

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Thus Sabın and Garca-Alcaine’s measure of tripartite entanglement

For the single qubit assisted protocolFor the Bell measurement protocol

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