Transcript Quantum Networks & Multi

Lecture Note 5
Multi-Particle Entanglement & It’s
Application in Quantum Networks
Jian-Wei Pan
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Polarization Entangled Photons
1
| H 1 | H  2  | V 1 | V  2 
2
1

| H 1 | V  2  | V 1 | H  2 
|  12 
2
|   12 
[P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995).]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Post Selection
 H  V 1   H  V 2

 H 2'  iV1'    H1'  iV2' 

 H1' H 2'  V1'V2'   i  VH 1'   HV 2' 
 H1' H2' V1'V2' 
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
3-Photon
Initial State:
1
( H aVb  Va H b )  ( H a' Vb'  Va' H b' )
2
Four-fold coincidence
HT H1H2V3 or HTVV
1 2 H3
Final State:
1
2
Jian-Wei Pan
H T ( H 1 H 2V3  V1V2 H 3 )
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Bell’s Inequality and Violation of Local Realism
[J. S. Bell, Physics 1, 195 (1964)]
Bell’s inequality states that certain statistical correlations
predicted by QM for measurements on two-particle ensembles
cannotbe understood within a realistic picture based on local
properties of each individual particle.
LR prediction:
S MAX  2
QM prediction: S MAX  2 2
An unstatisfactory feature!
In the derivation of BI such a local realistic and thus
classical picture can explain perfect correlations and is only
in conflict with statistical prediction of quantum mechanics.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
conflict with local realism
Consider a three-photon GHZ state written in  z basis
 123
1

 H1 H 2 H3  V1 V2 V3
2

1 
 0
H    , V    denotes the  eigenstate of  z respectively
 0
1 
Linear polarizatoin basis
1
x : H ' 
 H  V ,
2
1
V' 
 H  V .
2
Jian-Wei Pan
circular polarization basis
1
y : R 
 H  i V ,
2
1
L 
 H  i V .
2
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
conflict with local realism
 1 y 2 y 3 x :  123
 1 y 2 x 3 y :  123
 1x 2 y 3 y :  123
1
  R1 L2 H 3'  L1 R2 H 3'  R1 R2V3'  L1 L2V3' 
2
1
'
'
  R1 H 2' L3  L1 H 2' R3  RV
R

LV
1 2 3
1 2 L3 
2
1 '
  H1 R2 L3  H1' L2 R3  V1' R2 R3  V1' L2 L3 
2
therefore state  123 is the eigenstate of operators
 1 y 2 y 3 x、 1 y 2 x 3 y、 1x 2 y 3 y with value -1
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
conflict with local realism
EPR reality criterion: the individual value of any
local operator is predetermined. There exists an
element of local reality Six corresponding to
operator  ix  i  1, 2,3 .
All six of the elements of reality
Six and Siy have to be there, each with
the values +1 and –1!
S1 y S 2 y S3 x  1,
S1 y S 2 x S3 y  1,
S1x S2 y S3 y  1.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
What outcomes are possible?
Consider measurement of 45°
linear polarization basis
local realism
S1x S2 x S3 x  S1x ( S1 y ) 2 S2 x ( S2 y ) 2 S3 x ( S3 y ) 2
 ( S1x S2 y S3 y )( S1 y S2 x S3 y )( S1 y S2 y S3 y )
 1
Possible outcomes:
' ' '
1 2 3
'
1
' '
2 3
' '
1 2
'
3
'
1
'
2
V V V , H H V , H V H ,V H H
Jian-Wei Pan
Physikalisches Insitut
'
3
07.06.2006 Uni-Heidelberg
What outcomes are possible?
quantum physics
 123
1
  H1' H 2' H 3'  H1'V2'V3'  V1' H 2' V3'  V1'V2' H 3' 
2
 S1x S2 x S3 x  1!
Possible outcomes:
H1' H2' H3' , H1'V2'V3' , V1' H2' V3' , V1'V2' H3'
Whenever local realism predicts a specific result
definitely to occur for a measurement for one of the
photons based on the results for the other two,
quantum physics definitely predicts the opposite result!
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Experimental Results
J.-W. Pan et al., Nature (London) 403, 515 (2000)
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
An improved 3- & 4-photon source

H
12
1
 
H
2
34
V
1
H 1 H

2
1
  H 4V
2


V
1
2
4
 H
2
H
3
 H
H
1
4
H
3
H
V
2
H
4
1
3
V
V
2
V
3
V
1
V
3
V
4
V
2
4

V
3

[J.-W. Pan et al., Rev. Lett. 86, 4435 (2001) ]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Advanced Quantum Cryptography
• Quantum secret sharing
A procedure for splitting a message into several parts
so that no subset of parts is sufficient to read the
message, but the entire set is.
[M. Hillery et al., Phys. Rev. A 59, 1829 (1999).]
• Third-Man Quantum Cryptography
A procedure that the third man, a controller, can
control whether the users, say Alice and Bob, can
communicate in a secure way while he has no access
whatsoever on the content of the communication
between Alice and Bob.
[M. Żukowski et al., Acta Phys. Pol. 93, 187 (1998).]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Schemes for QSS and TQC

abc

1
H

2
H
a
b
H
c
V
a
V
b
V
c

1
 H  V ,
2
1
 H i V 
y 
2
x 
A xxx measurement

abc

1
2
 x
  x
Jian-Wei Pan
Physikalisches Insitut
x
 a
a
x
b
b
 x
 x
a
x
x
a
b
b
x
 c
x 
 c
07.06.2006 Uni-Heidelberg
Quantum Secret Sharing
 



1
1
2
1
2
1
x
  x
b
 x
a
 a
y
b
 y
a
y
b
x
 c
  y
a
y
b
 y
a
y
 a
x
b
 y
a
x
b
y
 c
  y
a
x
b
 y
a
x
b
y 
 a
y
b
 x
a
y
b
y
 c
  x
a
y
b
 x
a
y
b
y 
y


2
y


2
x


2
x
b
 c
a
x
 x
x 
x
 a
2
2
x


2
b
x
a
b
 c
b
x 
 c
 c
 c
xxx, xyy, yxy, yyx
xyx, yxx, xxy, xyx
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Setup
A ultra-stable high intensity source:
2 four-fold coincidence per second!
100 times brighter!
stable for a few months!
[Z. Zhao et al., Phy. Rev. Lett. 91 180401 (2003).
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Result for QSS
From 327 579 bits of
raw key with a QBER
of 12.9%, after
security check and
error reduction, Alice
and Bob jointly
generate 87 666 bits
cured key with
Charlie with a QBER
of 0.3%.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Third-Man’s Control
• If all of them randomly select the
base to measure the polarization. Any
two of them can create a coding by
being told the other one’s
measurement result.
• If Charlie rejects to tell them his
selection or just does not make any
selection then Alice and Bob can get
nothing useful for the cryptography.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Result for TQC
With the permission of Charlie, after security check and error
reduction Alice can generate a 87 666 bits cured key with Bob, with the
same QBER. Otherwise, Without knowing Charlie's results, the only
thing Alice and Bob can do is to randomly guess Charlie's results and
continue the same encoding and error reduction procedure. In our
experiment, after performing twice error reductions, the QBER
remains 49.999%.
[Y.-A. Chen et al., Phy. Rev. Lett. 95, 200502 (2005) ]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
conflict with local realism in
4-photon case
1
 
H

2
1
V
2
V
3
H
4
V
1
H
2
H
3
V
4

 x x x x ,  x y x y ,  x x y y   x y y x
 y y y y ,  y x y x ,  y y x x   y x x y
1
1
H

V
、
V
'



H V
2
2
1
1
 y: R   H  i V 、L   H  i V
2
2
 x: H ' 
 x x x x :
 
1
 H'


H' H' H'  H' H' V' V'
2 2
 H' V' H' V'  H' V' V' H'
 V' H' H' V'  V' H' V' H'
 V' V' H' H'  V' V' V' V'
Jian-Wei Pan
Physikalisches Insitut

07.06.2006 Uni-Heidelberg
Violation of Local Realism
[Z. Zhao et al., Phy. Rev. Lett. 91 180401 (2003).
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
A new protocol for 3-Photon
H
1
V
  HHH
1
   HH
123
 VVV
23
123
 VV
23


[J. G. Rarity and P. R. Tapster,
Phys. Rev. A 59, R35 (1999).]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
5-Photon
H
1
V
1
  HH
23
 VV
23
  HH
Five-fold Coincidence:

 HHHHH
Jian-Wei Pan
12345
 VVVVV
12345
45
 VV
45


Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Encoding operation for simple
quantum error correction
| 12345  | 1 |  2345

1 
[|  12 ( | H  3 | H  4 | H  5   | V  3 | V  4 | V  5 )
2
 |   12 ( | H  3 | H  4 | H  5   | V  3 | V  4 | V  5 )
 |   12 ( | V  3 | V  4 | V  5   | H  3 | H  4 | H  5 )
 |   12 ( | V  3 | V  4 | V  5   | H  3 | H  4 | H  5 )]
This implies that a joint Bell measurement on photons 1 and 2 would
project the state of photons 3, 4 and 5 into one of the four
corresponding states, which can be used for either one bit-flip error or
phase-shift error correction in quantum communication.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Quantum State Sharing &
Open-destination Teleportation
 H 1 V
  HHH
1
345
  VVV
345
If we perform a +45degree measurement on
photons 4 and 5, then
photon 3 is left in the
state of photon 1.
In a similar manner the
initial state of photon 1
can also be teleported
either onto photon 4 or
photon 5.
[A. Karlsson, et al., Phys. Rev. A 58, 4394 (1998) ]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Further Demonstration
•
In contrast to the original teleportation scheme, after the encoding
operation the destination of teleportation is left open until we
perform a polarization measurement on two of the remaining three
photons.
•
Even though photons 3, 4 and 5 are far away from each other, one
can still choose which particle should act as the output where the
initial state of photon 1 is transferred to. This is why we have called
the encoding-decoding procedure as open-destination teleportation.
•
It is therefore a generalization of standard teleportation, when no
prior agreement on the final destination of the teleportation is
necessary.
•
It is also a generalization of Quantum State Sharing. No subset of
parts is sufficient to decode the state, but the entire set is. It
broadens the scope of quantum information networks allowing
quantum communication between multiple nodes, while providing
security against malicious parties in the network as well as node and
channel failures.
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Setup for Five-photon GHZ Entanglement
[Z. Zhao et al., Nature
430, 54 (2004). ]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
CNOT operation for twoindependent photons
+/-?

H  V
H/V?
   HV
 VH
2
PBS1

  HH
T(H ); R(V )

34
  H   V

5
   H   V 

  HH
  VV
      H  V   H  V 
   HH
  VV
      H  V   H  V 
  H   V   H    V  H   V 
PBS2
T(H V ); R(H V )
23
V
4
  VV
23
Jian-Wei Pan
2
H
4
5
23
23
Conditional 3 at H+V
Conditional 4 at H
23
4
23
5
5
4
4
5
Physikalisches Insitut
2
5
4
5
5
5
5
5
07.06.2006 Uni-Heidelberg
CNOT Gate
A full Bell state
Measurement
for 100%
Teleportation
Control
Target
Control’
Target’
H
H
H
V
H
V
H
H
V
H
V
H
V
V
V
V
Jian-Wei Pan
[T. B. Pittman, PRA 64,062311(2001)]
[S. Gasparoni et al., Phys. Rev. Lett. 93, 020504
(2004);
Z. Zhao et al., Phys. Rev. Lett. 94, 0304501 (2005).]
Physikalisches Insitut
07.06.2006 Uni-Heidelberg
Most recently... 6-Photon
[Q. Zhang et al., In
preparation for Science ]
Jian-Wei Pan
Physikalisches Insitut
07.06.2006 Uni-Heidelberg