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Multiphoton Entanglement

Eli Megidish

Quantum Optics Seminar ,2010

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Outline

Multipartite Entanglement Multipartite detection QST & Entanglement

Witness GHZ W states Cluster states – one way quantum computer

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N=2:

Multi partite entanglement

1 2 3| | |H V V

1 2 1 2

1| | | | |

2H H V V

1 2| |H V Separable

N=3:

Separable Biseparable 1 23| |H

Pure state is called genuine tripartite entangled if it is not fully separable nor biseparable :

1 2 3 1 2 3

1| | | | | | |

2GHZ H H H V V V

1 2 3 1 2 3 1 2 3

1| | | | | | | | | |

3W H V V V H V V V H

Pure state is called genuine biipartite entangled if it is not fully separable :

In this talk ,N>2, and will focus on GHZ,W states and cluster states.

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Quantum state tomography

n particles density matrix is given by:

1 2 1 2

1 2

3

, ,{ , } 0

1

2 n n

n

i i i i i ini i i

r

Any two level system density matrix:

i

3

0

1

2 i ii

r

/HH HV

VH VV

reconstruction of the density matrix by measurements. 4n real parameters, 4n

one photon density matrix:

are the pauli matrices.

1. BS. 0

1 1| | | |

2 2 HH VVH H V V

For two photon one has to measure: 1 2i j , 0 3.i j

2. PBS in H/V basis.

3. PBS in P/M basis.

4. PBS in R/L basis.

0

1| |

2HH zH H

0

1 1 1| | | |

2 2 2HH HV VH VV yR R H iV H iV i i

0

1 1 1| | | |

2 2 2HH HV VH VV xP P H V H V

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Measures of entanglement using density matrix:

Fidelity- a measure of the state overlap:

21 2 1 2 1,F Tr

,

, ,

N M

ij kli j k l

i j k l

Peres- Horodecki criterion:

,

, ,

AN MT

ji kli j k l

i j k l

Define the partial transpose:For a 2x2 or 2x3 system the density matrix

If the the partial transpose has no negative eigenvalues than the state is separable!

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Entanglement witnessA witness operator detects genuine n-partite entanglement of a pure state

| |W I 2

maxB

|

0BTr W

B denotes the set of all biseparable states.

| | 0Tr W

exp 0W Tr W signifies a multipartite entanglement.

Bell state witness: 1 2 1 2

1| | |

2H V VH

2 2

1 2 1 2

1

2H V V H

1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2

| | | | | |1 1| |

2 2 | | | | | |

H H H H VV VV PP PPW I

M M M M R L R L L R L R

We don’t construct the quantum state but we can detect genuine multipartite entanglement with ~N measurements.

Therefore, by definition:

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Greenberger Horne Zeilinger state

1| | |

2

n n

nGHZ H V

3 1 2 3 1 2 3

1| | |

2GHZ H H H VV V

Properties:

3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 1 2 1 2 1 2| | | | | | | |Tr H H H VV V H H H VV V H H H H VV VV

When losing one particle we get a completely mixed state.

Maximally entangled

When we measure one particle in P/M basis we reduce the entanglement

1 2 3 1 2 3 1 2 3 2 3 1 2 3 2 3| | | | | | | |H H H VV V P H H V V M H H V V

, , ' , ' , ' , , ' , ' , ' 2Ma b c a b c a b c a b cS E E E E

4MGHZS

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GHZ sources

Two random photons superimposed on a PBS are projected into a Bell state

2 2 3 2 3

1| | | |

2GHZ H H V V

1 64

But what if one photon is a part of a bell state

12 3 3 1 2 1 2 3 3

1| | | | | | |

2PBSH V H H VV H V

3 1 2 3 1 2 3

1| | |

2GHZ H H H VV V But what if the other photon is also a part of a bell state

4 1 2 3 4 1 2 3 4

1| | |

2GHZ H H H H VV VV

In this way 5 6| ,|GHZ GHZ was created.

2 3 5

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GHZ sources - Experiment

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1 2 1 2 3 4 3 4

1 2

4 1 2 3 4 1 2 3 4

1 1| | | | |

2 2

1| | |

2

st nd

PBS

pass pass

H V V H H V V H

GHZ H V V H V H H V

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1| |

2M H V V H H V V H V H H V V H H V

Confirmation of the GHZ state is done using a measurement in the P/M basis.

PDC in a double pass configuration:

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Experiment six photons GHZ state

3 pairs of entangled photons are created using PDC:

Fidelity= 0.593 0.025

exp 0.095 0.036Tr W

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GHZ Bell Theorem without inequalities

1 2 3 1 2 3

1| | | | | | |

2GHZ H H H V V V

1 2 3 1 2 3

1 2 3 1 2 3

1| (| | | | | |

2| | | | | | )

GHZ R L P L R P

R R M L L M

1. Individual and two photons measurement are random. 2. Given any two results of measurement on any two photons, we

can predict with certainty the result of the corresponding measurement performed on the third photon.

Symmetry: In every one of the yyx,yxy and xyy experiment, third photon measurement (circular and linear polarization) is predicted with certainty.

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GHZ Bell Theorem without inequalities, Local realism

1, 1iY

1, 1iX

1 2 3 1YY X

Assume that each photon carries elements of reality for both x and y measurement that determine the specific individual measurement result:

for P / M polarization

for R / L polarization

In order to explain the quantum predictions:

1 2 3 1 2 3

1 2 3 1 2 3

1| (| | | | | |

2| | | | | | )

GHZ R L P L R P

R R M L L M

1 2 3 1Y X Y 1 2 3 1X Y Y

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iX

1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1

X X X X Y Y Y X Y YY X

X X X

1i iYY

But what if we decided to measure XXX?

Local realism:

Independent on the measurement bases on the other photons.

Possible results:

1 2 3 1 2 3

1| (| | | | | | )

2GHZ H H H V V V

1 2 3| | |P P M 1 2 3| | |M M M

1 2 3| | |M P P 1 2 3| | |P M P

Quantum Mech.:

1 2 3 1 2 3

1 2 3 1 2 3

1| (| | | | | |

2| | | | | | )

GHZ P P P P M M

M P M M M P

1 1| | | ,| | |

2 2P H V M H V

Possible results:

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Results

yyx

yxy

xyy Exp:

LHV:

QuantumMechanics:

Quantum mach. is right 85% of the times!

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W state

3 1 2 3 1 2 3 1 2 3

1| | | |

3W H H V H V H V H H

, , ' , ' , ' , , ' , ' , ' 2Ma b c a b c a b c a b cS E E E E

3.046 2 2MWMax S

“Less” entangled compared to GHZ state

“Less” fragile to photon loss than GHZ state

1 2 1 2 1 2

1| | | |n n n nW H H V H V H V H H

n

3 1 2 3 2 3 1 2 3

tan

1 1| | | | | |

3 3MixtureEn gled

W H H V V H V H H

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Experiment

Two indistinguishable pairs are created:

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0

1 1| | | | |

2 21| 2 | |

2

a b a b a b a b

a a b b a a b b a a b b

H V V H H V V H

H H V V H V H V V V H H

2H VT TWe choose only

0 0 0| | |t a a bV H V H

of the times we get136

3

1| | | |

3a b c a b c a b cW H H V H V H V H H

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Results- state characterization

, , { , }

HHHHHH

ijki j k H V

CP

C

These probabilities are also obtained from

Incoherent mixture

1| | | |

3M a b c a b a b c a b c a b c a b cH H V H H V H V H H V H V H H V H H

Equally weighted mixture of bisparable states 1 1 1

3 3 3B a bc b ac c ab

a a aH H

bc Bell state between modes b and c.

To confirm the desired state we measure the correlation in the R/L bases.

31 18 4 8B wM

ijk RRR RRRP P P

exp

0.321 0.021RRRP

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Results- Entanglement properties

1| , | |

2ji

j j c jk R k e L

| , ,j

j j j j j jk

k k k

1jk Measurement basis

, ,j a b c

3

1| | | |

3a b c a b c a b cW H H V H V H V H H

Correlation function: , ,

, ,

, , , ,

, ,a b c

a b c

a b c a a b b c c

a b c k k k a b ck k k

E

k k k P

, , , ,a b ck k k a b cP is the probability for a threefold coincidence with the specific results and a

specific phase settings.

For w state:

2 1, , cos cos cos cos

3 3a b c a b c a b cE

2

, , 3, , , , , , ,a b ck k k a b c a a b b c cP k k k W

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Results- Entanglement properties

, 0, 0 cos 1a b c aE V

2 2, , 0 cos

2 3 2 3a b c aE V

exp 0.864 0.019V

exp 0.481 0.029V

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cH

cVCorrelations between photons in mode a and b, depending on the measurement result of photon in mode c.

Hab

Vab

Two photon state tomography:

Peres- Horodecki criterion:exp

0.5

0.348 0.019

H

H

exp

0

0.113 0.062

V

V

Bipartite entanglement

Results- Robustness of Entanglement

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Graph state

A graph state is a multipartite entangled state that can be repressed by a graph.Qubit- vertex and there’s an edged between interacting (entangled) qubits.

,G V E

,

V

aba b E

G U

Given a graph the state vector for the corresponding graph is prepared as follows:

1. Prepare the qubits at each vertex in the pure state with the state vector 2. Apply the phase gate to all the vertices a,b in G. abU

.

1 jkUj k j k In the computational basis:

Cluster states are a subset of the graph states that can be fitted into a cubic lattice.

Two graphs are equivalent if under Stochastic Local Operation and Classical Communication (SLOCC) one transforms to the other.

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Cluster state

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1 2

1 2 1 2

1 1| | | 0 |1 | 0 |1 | 00 | 01 |10 |11

2 21 1| 0 |1 | 0 | 0 |1 |1 | | 0 | |1

2 2

U

Bell

231231 2 3 1 2 1 2 3 3 1 2 3 1 2 3

| | | | | 0 | |1 | 0 |1 | | 0 | | |1 | |

| 000 | 001 | 010 | 011 |100 |101 |110 |111

UU

U

GHZ

1 2 3 4| | | | | 0000 | 0100 |1000 |1100 | 0010 | 0110 |1010 |1110

| 0001 | 0101 |1001 |1101 | 0011 | 0111 |1001 |1111

U

1 2

1 2 3

1 2 3

4

1 2 3 4

1 2 3 4

4

| | | | | 0000 | 0100 |1000 |1100 | 0010 | 0110 |1010 |1110

| 0001 | 0101 |1001 |1101 | 0011 | 0111 |1001 |1111 |

U

GHZ

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Cluster state

Single particle measurements on a cluster state:

Measurement in the computational basis have the effect of disentangling the qubit from the cluster. Remove the vertex and its edges.

| ,|B

3 1 2 3 1 2 3| | |GHZ H H H VV V

1 2 3 1 2 3 2 1 3 1 3 2 1 3 1 3| | | | | | | |H H H VV V P H H VV M H H VV

Measurement in the basis:

| 0 ,|1

1| | 0 |1

2ie

0 | | | ,|P M

Pauli error in the case

1 3

1 2 3

1 2 3

1 3

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Cluster state , How much entanglement is in there?

- A state is maximally connected if any pair of qubits can be projected, with certainty into pure Bell state by local measurements on a subset of the other qubits.

-The persistency of entanglement is the minimum number of local measurements such that, for all measurement outcomes, the state is completely disentangled.

1 2 3

4

1 2 3 4

1eP 2eP

Cluster states are maximally connected and has persistency2e

NP

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One way Quantum computer

inp

uts

out

pu

ts

A new model (architecture) for quantum computer based on highly entangled state, cluster states.Computation is done using single qubit measurements. Classical feed forward make a Quantum One Way computer deterministic. Protocol:• Prepare the cluster state needed.• Encode the logic.• Single qubit measurements along the cluster (feed forward)• Read the processed qubits.

Universal set includes: single qubit rotation and CNot/CPhase operation

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One way Quantum computer – Building blocks

1 2 1 2 1 2

1| | | 0 |1 | 0 |1 | 0 | |1 |

2Ua b a b

1 2

1| | 0 |1

2

1| 0 | | |1 | |

2 2

i

i

B e

e

1 2 1 2 2 2 2 21 1| 0 | |1 | | | | |i ia b a be a be

1 If we measured 2 2 2

2 2 2 21 1| | | |

i i iia be e ae be

This is equivalent to the operation on the encoded qubit:

2 2 2 2 2| 0 |1 | 0 |1 | 0 |1 | |zi i i i i

z

Rotation

R a b e a b ae be ae be

Special case: 2 210 | |a b The encoded qubit is teleported along the chain.

1 If we measured classical feed forward in needed to correct the pauli errors.

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1 2 3 4

Mesurement Readout

zR zR zR

, ,z z z z x zR R R R R R U

3D -qubit rotation on the bloch sphere.

Using single photon measurement in the appropriate basis:

Classical feed forward make a Quantum One Way computer deterministic.

One way Quantum computer – Building blocks

zR

zR

zR

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12

3 4

1 2 3 4| | | | | 0 0 | 0 1 |1 0 |1 1U

Measure particles 2,3 in the base

If the outcomes are

2,3 1,4 1,4 1,4 1,4 2,3 1,4 1,4| | | 00 | 01 |10 |11 | | 0 |1

This is equivalent to the operation on the encoded qubits:

2 3 2 3 2 3 2 3 2 3

2 3 2 3 2 3 2 3 2 3 2 3

| | | 0 | 0 | 0 |1 |1 | 0 |1 |1

| 0 | 0 | 0 |1 |1 | 0 |1 |1 | 0 | |1 |

CPhase CPhase

Consider the cluster:

These kind of operations generates entanglement.

One way Quantum computer – Building blocks

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1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

| | | | | | | | |

| | | | | | | |

cluster H H H H H H V V

V V H H V V V V

Accounting for all possible 2 pair generated in PDC which are super imposed on a PBS:

This state is equivalent to the four qubit linear cluster under the local unitary operation:

1 2 3 4I I

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3 4

1 2 3 4

One way Quantum computer – Experiment

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The rotated photon is left on photon 4. Photon 4 was characterized using QST.

, ,02 4

2

Input state:

in

Theory Exp.

0.86 0.03 , 0.85 0.04 , 0.83 0.03fidelity

out x z inR R

1 2 3 4

Single qubit rotation was presented using three qubit linear cluster

One way Quantum computer – Experiment

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1 42 3

2 3 1 4 1 4

10 1

2

out in

in out

CPhase

CPhase gate was presented using:

The twp photon density matrix was reconstructed using QST.

2.47 0.08 2S

0.84 0.03fidelity

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3 4Measure photons 2,3 in

Theory Exp.

One way Quantum computer – Experiment

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We can simulate any network computation using the appropriate cluster and measurements !

So far the largest cluster state generated 6 (photons) 8 (ions).

One way Quantum computer – Summary

inp

uts

out

pu

ts

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“Measurement of qubits ”, James, PRA,64, 052313 (2001).“Experimental detection of multipartite entanglement using witness operator”, Bourennane, PRL, 92, 087902, (2004).“Observation of three- photon Greenberger-Horne-Zeilinnger entanglement”, Bouwmeester, PRL , 82,1345, (1999) “Experimental test of quantum nonlocality in three photon Greenberger-Horne-Zeilinger entanglement”, – Pan Nature 403, 169-176 (2000).“Experimental Demonstration of Four-Photon Entanglement and High-Fidelity Teleportation” ,Pan, PRL, 86, 4435 (2001).“ Experimental entanglement of six photons in graph states” , Pan, nature, 3,91,(2007).“ Experimental realization of three qubit entangled W state”, Weinfurter, RPL,7,077901, (2004).“Entanglement in graph states and its applications” Briegel, arxive: quan“Persistent entanglement in arrays of interacting particles ”, Briegel, PRL, 86, 910(2001)“A one way quantum computer” , Briegel, PRL, 86, 5188, (2001).“Experimental one way quantum computer” Zeilinger, nature, 434, 169, (2005).

References:

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Summary

• Multipartite entanglement characteriation• Multipartite entanglement properties.• Bell theory without inequalities.• One way quantum computer.

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1 2 3 1 2 3

1 2 3 1 2 3

| | | ' | | ' | ' | | '

| ' | | | ' | | ' |

T

T

H H H V V V H

H H H V V V H

1| (| | | | ) (| ' | ' | ' | ' )

2 a b a b a b a bH V V H H V V H

3

3

1| | | | |

21 1

| | | | | |2 2

a T b b

a a a b b

H H V V V

V V H H H H

Experiment:

Restricting for 4 fold coincidence:

1 2 3 1 2 3| | | | | | | |TGHZ H H H V V V H

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| H

|V

| P

|M | L

| R

2 21

| cos | sin |iH e V

0 1

1 0xx

| | |H V

0

0y

iy

i

1 0

0 1zz

Polarization base

| ,|H V

1 1| | | ,| | |

2 2P H V M H V

1 1| | | ,| | |

2 2L H i V R H i V

Measurement Operator

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Two random polarized photon can be entangled using a Bell state projection:

1 2 1 2

1| | | | |

2H H V V

For the case that two photons emerge from

1 2

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1 2 1 2

1| | | | |

2H V V H

1 2

40

)a(

Delayline

Delayline

41

Delayline

)b(

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EPR EPR EPR

1 2 3 4 5 6

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44

45

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