Download Multi-photon entanglement (GHZ and W

Multiphoton Entanglement
Eli Megidish
Quantum Optics Seminar ,2010
1
Outline





Multipartite Entanglement
Multipartite detection QST & Entanglement
Witness
GHZ
W states
Cluster states – one way quantum computer
2
Multi partite entanglement
N=2:
Separable
| H 1 | V  2
Pure state is called genuine biipartite entangled if
it is not fully separable :
| 
1
| H1  | H 2  | V1  | V2  
2
N=3:
Separable
| H 1 | V  2 | V  3
Biseparable
| H 1 |    23
Pure state is called genuine tripartite entangled if it is not fully separable nor biseparable :
1
 | H 1 | H  2 | H  3  | V  1 | V  2 | V  3 
2
1
|W 
 | H 1 | V  2 | V  3  | V  1 | H  2 | V  3  | V  1 | V  2 | H  3 
3
| GHZ  
In this talk ,N>2, and will focus on GHZ,W states and cluster states.
3
Quantum state tomography
Any two level system density matrix:
3
1
  n  ri1 ,i2 , in  i1   i2
2 {i1 ,i2 in }0
3
1

2
4n
n particles density matrix is given by:
 r
i 0
real parameters,  i
i
i
are the pauli matrices.
4n
reconstruction of the density matrix by
1
2
 H |  |H
 V |  |V
  12  
HH
1
 z  0
2
measurements.
 /
   HH
 VH
one photon density matrix:
1. BS.
  in
 HV 
VV 
 VV    0

2. PBS in H/V basis.
H |  | H   HH 
3. PBS in P/M basis.
P |  |P 
1
1
1
H  V |  | H  V    HH   HV  VH  VV     x   0
2
2
2
4. PBS in R/L basis.
R |  |R 
1
1
1
H  iV |  | H  iV    HH  i  HV  i VH  VV  
 y  0
2
2
2
For two photon one has to measure:

 i1 2j
i, j  0
3.


4
Measures of entanglement using density matrix:
Fidelity- a measure of the state overlap:
 
F  1 , 2   Tr
1 2 1

2
Peres- Horodecki criterion:
For a 2x2 or 2x3 system the density matrix
N
M
   ij ,kl i
i, j
k ,l
j  k l
Define the partial transpose:

TA
N
M
i, j
k ,l
   ji ,kl i
j  k l
If the the partial transpose has no negative eigenvalues than the state is separable!
5
Entanglement witness
A witness operator detects genuine n-partite entanglement of a pure state
W   I  |  |
  max  
2
B
B denotes the set of all biseparable states.

|

Tr W  B  0

Therefore, by definition:

W  Tr W  exp  0
Tr W |   |  0
signifies a multipartite
entanglement.
We don’t construct the quantum state but we can detect genuine multipartite entanglement with
~N measurements.
|   
Bell state witness:
    H1V2
2
   V1 H 2
1
| H1V2  | V1H 2  
2
2

1
2
VV
PP

1
1  | H1H 2 H1H 2 |  | VV
1 2
1 2 |  | PP
1 2
1 2 |


W  I  |   | 

2
2   | M1M 2 M1M 2 |  | R1L2 R1L2 |  | L1R2 L1R2 | 
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Greenberger Horne Zeilinger state


1
n
n
|H
 |V
2
1
| GHZ3 
| H1H 2 H 3  | VV
1 2V3
2
| GHZ n 

Properties:
Maximally entangled
S M  E a , b ,  'c   E a ,  'b , c   E  'a , b , c   E  'a ,  'b ,  'c   2
SGHZ M  4
When losing one particle we get a completely mixed state.

Tr3 | H1H 2 H 3  | VV
1 2V3


H1H 2 H 3 |  VV
H1H 2 |  | VV
1 2V3 |  | H1 H 2
1 2 VV
1 2 |
When we measure one particle in P/M basis we reduce the entanglement
| H1 H 2 H 3  | VV
1 2V3 | P1  | H 2 H 3  | V2V3   | M 1 | H 2 H 3  | V2V3

7
GHZ sources
Two random photons superimposed on a PBS are projected into a Bell state
1
2
| GHZ 2  |    
3
1
| H 2 H 3  | V2V3  
2
|   12 | H  3  | V  3  
4
5
6
But what if one photon is a part of a bell state
1
PBS

| H1H 2  | VV
1 2  | H  3  | V  3  
2
| GHZ3  
1
| H1H 2 H 3  | VV
1 2V3  
2
| GHZ 4  
1
| H1H 2 H 3 H 4  | VV
1 2V3V4  
2
In this way | GHZ5 ,| GHZ6 
But what if the other photon is also a part of a bell state
was created.
8
GHZ sources - Experiment
PDC in a double pass configuration:
|  
1
1
PBS 23

| H1V2  | V1H 2    | H 3V4  | V3 H 4   
2
2
1st pass
| GHZ 4  
M

2nd pass
1
| H1V2V3 H 4  | V1H 2 H 3V4  
2
1
| H1V2V3 H 4  H1V2V3 H 4  | V1H 2 H 3V4  V1H 2 H 3V4
2

Confirmation of the GHZ state is done using a
measurement in the P/M basis.
9
Experiment six photons GHZ state
3 pairs of entangled photons
are created using PDC:
Fidelity= 0.593  0.025


Tr W  exp  0.095  0.036
10
GHZ Bell Theorem without inequalities
| GHZ  
1
 | H 1 | H  2 | H  3  | V  1 | V  2 | V  3 
2
| GHZ  
1
(| R1 | L 2 | P 3  | L1 | R 2 | P 3 
2
| R1 | R 2 | M  3  | L1 | L 2 | M  3 )
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
Assume that each photon carries elements of reality for both x and y measurement that
determine the specific individual measurement result:
X i 1, 1
Yi  1, 1
for P / M polarization
for R / L polarization
In order to explain the quantum predictions:
Y1Y2 X 3  1
Y1 X 2Y3  1
1
| GHZ   (| R1 | L 2 | P 3  | L1 | R 2 | P 3 
2
| R1 | R 2 | M  3  | L1 | L 2 | M  3 )
X 1Y2Y3  1
12
But what if we decided to measure XXX?
Local realism:
Xi
Quantum Mech.:
Independent on the measurement
bases on the other photons.
| GHZ  
YY
i i 1
X1 X 2 X 3   X1Y2Y3 Y1 X 2Y3 YY
1 2 X3 
| P 
1
(| H 1 | H  2 | H  3  | V 1 | V  2 | V  3 )
2
1
1
| H  | V   ,| M   | H  | V  
2
2
 X1 X 2 X 3  1
Possible results:
| M 1 | M  2 | M  3
| P1 | P 2 | M  3
| P1 | M  2 | P 3
| M 1 | P 2 | P 3
Possible results:
1
| GHZ   (| P1 | P 2 | P 3  | P1 | M  2 | M  3 
2
| M 1 | P 2 | M  3  | M 1 | M  2 | P 3 )
13
Results
yyx 
yxy 
xyy 
Quantum
Mechanics:
LHV:
Exp:
Quantum mach. is right 85% of the 14
times!
W state
1
| H1H 2 Vn   | H1V2 H n  | V1H 2
n
1
| W3 
| H1H 2V3  | H1V2 H 3  | V1H 2 H 3 
3
| Wn 
Hn

“Less” entangled compared to GHZ state
S M  E a , b ,  'c   E a ,  'b , c   E  'a , b , c   E  'a ,  'b ,  'c   2
Max  SW M   3.046  2 2
“Less” fragile to photon loss than GHZ state
| W3 
1
1
| H1 | H 2V3  | V2 H 3  
| V1 | H 2 H 3
3
3
En tan gled
Mixture
15
Experiment
Two indistinguishable pairs are created:

1
| H a0Vb0  | Va0 H b 0
2
| 




1
| H a0Vb0  | Va0 H b0
2

1
| H a0 H a0Vb0Vb0  2 | H a0Va0 H b0Vb0  | Va0Va0 H b0 H b0
2
We choose only
1
36
| W3 
|  | Vt | H a0Va0 H b0


TH  2TV
of the times we get
1
| H a HbVc  | H aVb H c  | Va H b H c
3

16
Results- state characterization
PHHH 
CHHH

i , j , k { H ,V }
Cijk
These probabilities are also obtained from
Incoherent mixture
1
 M  | H a H bVc
3

H a H bV |  | H aVb H c H aVb H c  | Va H b H c Va H b H c

Equally weighted mixture of bisparable states
1
1
1
 B   a  bc  b   ac  c   ab
3
3
3
a  H a H a
bc Bell state between modes b and c.
To confirm the desired state we measure
the correlation in the R/L bases.
P 
ijk
M
 1   PRRR    1   PRRR    3
8
4
8
B
w
 PRRR 
exp
 0.321  0.021
17
Results- Entanglement properties
| W3 
1
| H a HbVc  | H aVb H c  | Va H b H c
3
Measurement basis
| k j , j 


1
i
| Rc  k j e j | L
2

k j  1
 j   k j | k j , j k j , j
j  a , b, c
kj
Correlation function:
E a , b , c    a a  ,  b b  ,  c c  


ka kb kc  Pka ,kb ,kc a , b , c 
ka , kb , kc
Pka ,kb ,kc a ,b ,c 
is the probability for a threefold coincidence with the specific results and a
specific phase settings.
Pka ,kb ,kc a , b , c  
 ka , a  ,  kb , b  ,  kc , c  W3
2
For w state:
2
1
E a , b , c    cos a  b  c   cos a  cos b  cos c 
3
3
18
Results- Entanglement properties
E a , b  0, c  0   cos a   V  1
Vexp  0.864  0.019

2

2



E  a , b  , c  0    cos  a    V 
2
3
2
3



Vexp  0.481  0.029
19
Results- Robustness of Entanglement
Hc
Correlations between photons in mode a and
b, depending on the measurement result of
photon in mode c.
Two photon state tomography:
Peres- Horodecki criterion:
Vc
 abH
V
 ab
H  0.5
V  0
Hexp  0.348  0.019
Vexp  0.113  0.062
Bipartite entanglement
20
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 
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 U ab to all the vertices a,b in G.
In the computational basis:
U
j k 
  1
G 

jk
a ,bE
j k
U ab 
V
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
21
(SLOCC) one transforms to the other.
Cluster state
|  1| 
1
| 00  | 01  |10  |11  
2
2
1
1
 | 0  |1  | 0  | 0  |1 |1 
|  1 | 0 2  |  1 |1 2   Bell
2
2

1
| 0  |1
2

| 0
U12
 |1  


1
2
 
1
2
3
U 23
U12
|  1 |  2 |  3 
 |  1 | 0 2  |  1 |1 2  | 0 3  |1 3  
|  1 | 0 2 |  3  |  1 |1 2 |  3  | GHZ3
U

 | 000  | 001  | 010  | 011  |100  |101  |110  |111
1
|  1|  2 |  3| 
4
2
3
4
U

 | 0000  | 0100  |1000  |1100  | 0010  | 0110  |1010  | 1110
 | 0001  | 0101  |1001  |1101  | 0011  | 0111  |1001  |1111
1
|  1|  2 |  3| 
4
U

 | 0000  | 0100  |1000  |1100  | 0010  | 0110  |1010  | 1110
 | 0001  | 0101  |1001  |1101  | 0011  | 0111  |1001  |1111 
 | GHZ 4
2
3
4
22
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.
| 0
1
,|1 
2
1
3
3
| GHZ3 | H1H2 H3  | VV
1 2V3
Measurement in the basis:
B    |  ,| 

  0  |   |   | P ,| M
|  
1
1
| 0  ei |1

2
2
3


1
| H1 H 2 H 3  | VV
1 2V3 | P2  | H1 H 3  | VV
1 3   | M 2 | H1 H 3  | VV
1 3
3

Pauli error in the case
23
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
1
2
3
4
4
Pe  1
Pe  2
Cluster states are maximally connected and has persistency
Pe 
N
2
24
One way Quantum computer
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)
Universal set includes: single
qubit rotation and
CNot/CPhase operation
outputs
Read the processed qubits.
inputs
•
25
One way Quantum computer – Building blocks
| 1 | 
2

1
 a | 0  b |1
2
| 0
U
 |1  
  a | 0 1 |  2  b |1 1 | 

2

 b |1 1 | 
If we measured
2

2


a | 0 1| 

1
1
| 0  ei |1 
2
1
ei


|0 
| |
|1 
| |
2
2
B    |  
2
 

1
a| 
2
 bei | 
  

1
1

   a| 

2
1
a| 
2
 bei | 
2
 bei | 
e
2
i

2
2


i

  1  ae 2 | 

 be
2
i

2
|

2 

This is equivalent to the operation on the encoded qubit:
 Rz    a | 0  b |1   e
i
 z
Rotation
If we measured
Special case:

1
2

i
 i 2
 a | 0  b |1     ae | 0  be 2 |1



i
i

2
2
  ae |   be | 

classical feed forward in needed to correct the pauli errors.
  0  
1
a| 
2
 b| 
2

The encoded qubit is teleported along the chain.
26
One way Quantum computer – Building blocks
Using single photon measurement in the appropriate basis:
Mesurement
Readout
Rz 
1
2
3
Rz  
Rz 
4
Rz Rz  Rz   Rz  Rx   Rz   U  ,  ,  
3D -qubit rotation on the bloch sphere.
Classical feed forward make a Quantum One Way computer deterministic.
27
One way Quantum computer – Building blocks
Consider the cluster:
U
|  1 |  2 |  3 |  4 
| 0  0   | 0 1   |1  0   |1 1 
Measure particles 2,3 in the base
If the outcomes are
|  |  
2,3
| 00
1,4
 | 01 1,4  |10
1,4

 |11 1,4  |  
2,3
| 0 
1,4
 |1 
1,4

This is equivalent to the operation on the encoded qubits:
     CPhase | 
2
     | 0 2 | 0 3  | 0
2
       CPhase | 0 | 0
|1  |1 | 0  |1 |1   | 0
|
2
3
2
3
1
These kind of operations
generates entanglement.
2
3
2
3
3
2
 | 0 2 |1 3  |1 2 | 0 3  |1 2 |1
|  3  |1 2 | 
3


3
4
28
3

One way Quantum computer – Experiment
Accounting for all possible 2 pair generated
in PDC which are super imposed on a PBS:
| cluster  | H
|V
1
1
|H
|V
2
2
|H
|H
3
3
|H
|H
4
4
| H
|V
1
1
|H
|V
2
2
|V
|V
3
3
|V
|V
4

4
This state is equivalent to the four qubit
linear cluster under the local unitary
operation:
1  I 2  I 3   4
1
2
3
2
1
3
4
4
29
One way Quantum computer – Experiment
Single qubit rotation was presented using three qubit linear cluster
1
2
3



4
 out  Rx    Rz    in
The rotated photon is left on photon 4.
Photon 4 was characterized using QST.
Theory
Exp.
Input state:
 in  


 
, ,0
2 4

2
fidelity   0.86  0.03 ,  0.85  0.04 ,  0.83  0.03
30
One way Quantum computer – Experiment
2
1
3
4
CPhase gate was presented using:
Measure photons 2,3 in

2
Theory
Exp.
 3   out   1   4  CPhase  in
 in  
2
 3   out 
1
0
2
1
 4 11 
4

The twp photon density matrix was
reconstructed using QST.
fidelity   0.84  0.03
S   2.47  0.08  2
31
One way Quantum computer – Summary
outputs
inputs
We can simulate any network computation using the appropriate cluster and
measurements!
So far the largest cluster state generated 6 (photons) 8 (ions).
32
References:
“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).
33
34
Summary
• Multipartite entanglement characteriation
• Multipartite entanglement properties.
• Bell theory without inequalities.
• One way quantum computer.
35
Experiment:
1
|    (| H  a | V  b  | V  a | H  b )  (| H ' a | V ' b  | V ' a | H ' b )
2
1
| V  b  | V  3 
2
1
1
| V a 
| V  a  | H  a  | H  b  | H  b  | H  3 
2
2
| H  a | H  T | V  b 
Restricting for 4 fold coincidence:
|   | H  T | H '1 | H  2 | V ' 3  | V '1 | V  2 | H ' 3  
| H ' T | H 1 | H  2 | V ' 3  | V 1 | V ' 2 | H  3 
| GHZ  | H  T | H 1 | H  2 | V  3  | V 1 | V  2 | H  3 
36
| H
|    | H    | V 
  
2
2
1
|   cos   | H   ei sin   | V 
Measurement Operator
1

0
 0 i 
y y  

i
0


|M
| L
Polarization base
1 0 
z z  

0

1


0
x x  
1
| R
| P
|V 
| H ,| V 
| P 
1
1
| H  | V   ,| M   | H  | V  
2
2
| L 
1
1
| H   i | V   ,| R  | H   i | V  
2
2
37
Two random polarized photon can be entangled using a Bell state projection:
1
2
For the case that two photons emerge from
| 
1
| H1  | H 2  | V1  | V2  
2
38
|   
1
| H1  | V2  | V1  | H 2  
2
1
2
39
Delay
line
Delay
line
(a)
40
Delay
line
(b)
41
1
2
EPR
3
4
EPR
5
6
EPR
42
43
44
45
46