IPQI 2014
Entanglement concentration
protocol
using linear optics
Anindita Banerjee
CAPSS, Bose Institute
Kolkata
Collaborators: Chitra
Shukla, Anirban Pathak
Outline
Introduction
Motivation
Linear optics
Example:Purification using PBS
ECP for CAT state
ECP for GHZ-like state
ECP for Four qubit state
ECP for n+1 qubit state of a particular form
Single qubit assisted ECP
Transformation Efficiency
Entanglement
Applications of entanglement
Teleportation
(require max entangled state between
two parties)
Dense coding
Quantum key distribution
Secure direct communication
IPQI 2014
!
Problems
Mixed state
Distributed Qubits interact with the environment
Gets noisy due to storage processing and transmissi
Less
entangled
state
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
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Motivation
ECP/EP
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]
IPQI 2014
Cross kerr
nonlinearities
Bell state
W state
Zhaop et al.
Sheng et al[W]
QED
Bell state
Romero et
al.
Schmidt Projective
method
Bell state
Bennet et al.
Bell state
Zhao et al.,[Experiment]
Yamamoto et al
[Experiment]
POVM
Bell state
Gu et al.
Linear optics
Bell state
Zhao et al.,
Yamamoto et al.,
Sheng et al.
S. Banyopadhyay
Entanglement
swapping
Bell state
Bose et al.
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Polarized Beam Splitter
Horizontally polarized photon is transmitted
and
Vertically polarized photon is reflected
I H1 >
I V1 >
I V3 >
I H4 >
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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
V. Vedral and M. B. Plenio
Local operation
Classical communication
Post selection
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Entanglement Swapping
Bose et al.
Alice
1
3
Bob
2
4
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
example
ψi
ψj
and
are orthogonal to each other
And belong to bell state
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General state
Non-maximally entangled (n + 1)-qubit state
Ψ0 are arbitrary n-qubit states that are mutually orthogonal.
Why is it important?
Applications
where Ψ0 and
Bell state
GHZ
GHZ-like
CAT states
Bidirectional quantum teleportation
Hierarchical quantum communication schemes (HQIS),
Hierarchical quantum secret sharing (HQSS)
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Four-qubits entangled states
There exist nine failies of states corresponding to nine different ways of
entangling four qubits.
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.
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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 is
possible in NMR based technologies.
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Two alternative ECPs for quantum states
ECP1
and
ECP2
need not be real
Requires Bell measurement
and
should be real
Single measurent
and particle swapping
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Entanglement transformation efficiency
s the amount of entanglement in the initial partially entangled state
E0
i
Ec
is the amount of entanglement of the state after concentration.
Ambiguity
E0
is the amount of entanglement of the state to be concentrated
OR
is the amount of entanglement of the entire initial state.
E 0 be the total initial entanglement
higher efficiency of single photon assisted ECPs over Bell-type state assisted ECPs
IPQI 2014
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.
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 and
Song defined a simple measure of tripartite entanglement as
where Mi-jk is a measure of entanglement between subsystem i and subsystem jk.
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Thus Sabın and Garca-Alcaine’s measure
of tripartite entanglement
For the Bell measurement protocol
For the single qubit assisted protocol
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Thank you