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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)
IPQI 2014
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!
IPQI 2014
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
IPQI 2014
Three ingredients involved
Local operation
Classical communication
Post selection
V. Vedral and M. B. Plenio
IPQI 2014
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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IPQI 2014
IPQI 2014
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
IPQI 2014
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