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Efficiency of Multi-Qubit W statesin Information Processing
Atul Kumar IPQI-2014IIT Jodhpur 25.02.2014
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A pure state of two-qubits is said to be entangled if
Apart from being central to the foundational aspects of quantum physics, entanglement has also been used as an efficient resource in communication protocols to perform tasks such as quantum teleportation, quantum cryptography, quantum secret sharing, quantum secure direct communication etc.
Our focus for this work is quantum teleportation and entanglement swapping
12 1 2
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Pictorial representation of original scheme
D. Bouwmeester et al, Nature 390 (1997), 575
2-animation-final2.exe
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For multiqubit systems some of the entangled resources are Cluster states Brown States
For two-qubit systems, the maximally entangled resources are Bell States;
12 12 12 12 12 12
1 101 10 , 00 112 2
For three-qubit systems, maximally entangled GHZ States and non-maximally entangled W states
123 123
1 000 1112GHZ
123 123 123
1 001 101 1003W
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Teleportation of a single qubit using three-qubit W state as a resource
The state to be teleported
Teleportation protocol is probabilistic using standard measurements and unitary transformations
Probability of teleportation depends on the unknown coefficients of state to be teleported
Alternately, one can also realize teleportation with success probability of 2/3 independent of and
0 1
2 2
1 1 12 1 1
Shi and Tomita, Phys. Lett. A 296, 161 (2002)
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In general, quantum teleportation using partially entangled states is always probabilistic
However, Agrawal and Pati proposed a new class of W type states for deterministic quantum teleportation of a single qubit
123
1 100 010 1 0012 2
i in ne n e
n
P. Agrawal and A. K. Pati, Phys. Rev. A 74,062320 (2006)
0 1 n
Projection basis used to realize teleportation protocol forn=1
1 123
1 123
1 010 001 2 10021 110 101 2 0002
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How is it possible?
We are performing a three qubit projective measurement to achieve the task
What if we perform standard two qubit and single qubit measurements?
Teleportation of single qubit is still probabilistic
Hence, to achieve teleportation of a single qubit using three-qubit W type of states one has to perform multiqubit measurements
Distinguishing these measurements is an issue
But nevertheless one can achieve perfect teleportation! S. Adhikari and S. Gangopadhyay, IJTP 48, 403
(2009)
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In this work, we address the following questions
Generalization of W type of states for perfect information transfer protocols
Given a four qubit W type state shared between two users, is it possible to let these users share a two qubit entangled state using entanglement swapping?
If so what is the degree of entanglement of the finally shared resource between the two users?
Comparison between the three and four-qubit W states in terms of concurrence of finally shared two-qubit states
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We generalize the three-qubit W type state for the case of four qubits
1234
1000 0100 1 001012 1 2 2 0010
i i
n i
e n e n
n e n
And then to the case of k qubits
12..
123...
1 10.. 010..( 2)(2 3) 2
1 0010.. ( 3) 00..01
( 2)( 3) ( 2) 1 00...1 ]2
in k
i i
ik
k ne kk n k
n e k n k e
k kk n e
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In order to teleport an unknown state Alice and Bob must share the four qubit W-like state such that Alice has qubits 1, 2 and 3 and Bob has qubit 4
1 123
1234
0100 0010 2 000112 2 2 1000a
1 123
1234
1100 1010 2 100112 2 2 0000a
Performing above four-qubit measurements on Alice’s qubits, perfect teleportation can be achieved
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Projection basis required if the shared state is a k qubit shared entangled resource
However, for practical cases we have analyzed the efficiency of W-type states for bi-partite entanglement sharing between the sender and the receiver
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For this we use the four-qubit state as
such that qubits 1, 2 and 3 are with Alice and qubit 4 is with Bob
We further consider that Alice has a two-qubit entangled state
The idea is to establish an entanglement between Alice’s qubit a and Bob’s qubit 4
1 1000 0100 1 0010 2 2 00012 1n n n nn
00 11ab
Concurrence 22 1
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For this Alice measures her qubits in Bell basis
We consider two cases where Alice projects her qubits onto Bell states and or and
After these measurements, Alice shares one of the following states with Bob
1b
23
2b
13
2 2 01 10n 2 2 01 10n n or
The concurrence of the shared bi-partite states between Alice and Bob can be given as
2
2
2 1 2 2(2 1) 1
nn
2
2
2 1 2 2( 2)
n nn n
or
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For n=1, both set of measurements yield identical results
For or the degree of entanglement
is unity i.e. the shared state is maximally entangled!
Hence, for a given value of users can in fact share maximum entanglement
Above two cases are compared to ascertain the measurements to be performed for a given value of state parameter
We found different ranges of the state parameter for a given n to obtain concurrence of the shared state
2 12 3n
2
3 2nn
2
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C
alpha
n=10
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Why do we need to perform the two-qubit or single qubit measurements if Alice can share the initially entangled two qubit state with Bob?
Initially entangled pair with Alice
If Alice sends the qubit b to Bob through amplitude damping channel where the channel is represented by Kraus operators
The shared state in this case would be
00 11ab
Concurrence 22 1
0 11 0 0,0 1 0 0p pK K
' 2
2 2
(1 ) 00 00 (1 ) 00 11 11 00
10 10 11 11p p p
p
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For a specific case of 1/ 2p
' 21/ 2
2 2
(1/ 2) 00 00 (1/ 2) 00 11 11 00
/ 2 10 10 11 11
Concurrence 22 1
Case ICase IITwo qubit pure stateTwo qubit state after ADC
n=1
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n=1 n=2
n=5 n=10
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We have also compared the efficacy of our states with three-qubit W-type states
Hence, one can share a bi-partite maximally entangled state for certain value of state parameters
For certain ranges of state parameter four-qubit W-type states are more efficient resources in comparison to the three-qubit W-type states
1 100 010 1 0012 2n n nn
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C
alpha
4 qubit state3 qubit state4 qubit state3 qubit state
n=10
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Acknowledgement
Mr. Parvinder SinghDr. Satyabrata Adhikari
IIT Jodhpur
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Thank You