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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

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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

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2b

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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

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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