Revealing anyonic statistics by
multiphoton entanglement
Jiannis K. Pachos
Witlef Wieczorek
Christian Schmid
Nikolai Kiesel
Reinhold Pohlner
Harald Weinfurter
arXiv:0710.0895
QEC07, USC, December 2007
Criteria for Topo Order and TQC (CM)
[G. Brennen and J.K.P., Proc. Roy. Soc. A, 2007]
•Initialization (creation of highly entangled state with TO)
•Dynamically, adiabatically, cooling, (possibly H)
•Addressability (anyon generation, manipulation)
•Trapping, adiabatic transport, pair creation &
annihilation.
•Measurement (Topological entropy of ground state,
interference of anyons) W  0  Stopo  0
•Scalability (Large system, many anyons, desired braiding)
•Low decoherence (Temperature, impurities, anyon
identification)
[I. Cirac, S. Simon, private communication]
Toric Code: ECC
Consider the lattice Hamiltonian
p
H    Z1 Z 2 Z 3 Z 4   X 1 X 2 X 3 X 4
p
s
s
s
Spins on the vertices.
s
s
Two different types of
plaquettes, p and s, which
support ZZZZ or XXXX
interactions respectively.
The four spin interactions involve
spins at the same plaquette.
p
p
p
p
X1
X4
s
X3
X2
Toric Code: ECC
Consider the lattice Hamiltonian
p
H    Z1 Z 2 Z 3 Z 4   X 1 X 2 X 3 X 4
p
Indeed, the ground state is:
 
s
s
s
The |00…0> state is a ground state of
the ZZZZ term and the (I+XXXX) term
projects that state to the ground state
of the XXXX term.
[F. Verstraete, et al., PRL, 96, 220601 (2006)]
p
p
p
1
(  X 1 X 2 X 3 X 4 ) s 00...0
2
s
s
s
p
X1
X4
s
X3
X2
Toric Code: ECC
• Excitations are produced by Z or X
rotations of one spin.
p
Z
s
• These rotations anticommute
with the X- or Z-part of the
Hamiltonian, respectively.
• Z excitations on s plaquettes.
• X excitations on p plaquettes.
• X and Z excitations behave as anyons
with respect to each other.
s
p
p
p
s
s
X
p
X1
X4
s
X3
X2
One Plaquette
It is possible to demonstrate the anyonic properties with
one s plaquette only. The Hamiltonian:
H   X1 X 2 X 3 X 4
 Z1Z 2  Z 2 Z 3  Z 3 Z 4  Z 4 Z1
The ground state:
1
 
(  X 1 X 2 X 3 X 4 ) 010 2 030 4
2
1

( 010 2 030 4  11121314 )
2
1
2
s
4
3
GHZ state!
One Plaquette
One can demonstrate the anyonic statistics with only this
plaquette. First create excitation with Z rotation at one
spin:
Z1
1
Z  Z1  
( 01020304  11121314 )
2
Assume there is an X anyon
outside the system. With
successive X rotations it can be
transported around the plaquette.
1
2
s
Z
3
4
One Plaquette
One can demonstrate the anyonic statistics with only this
plaquette. First create excitation with Z rotation at one
spin:
Z1
1
X1
Z  Z1  
2
( 01020304  11121314 )
Assume there is an X anyon
outside the system. With
successive X rotations it can be
transported around the plaquette.
1
2
s
Z
3
4
One Plaquette
One can demonstrate the anyonic statistics with only this
plaquette. First create excitation with Z rotation at one
spin:
Z1
1
X1
Z  Z1  
2
( 01020304  11121314 )
Assume there is an X anyon
outside the system. With
successive X rotations it can be
transported around the plaquette.
1
X2
2
s
Z
3
4
One Plaquette
One can demonstrate the anyonic statistics with only this
plaquette. First create excitation with Z rotation at one
spin:
Z1
1
X1
Z  Z1  
2
( 01020304  11121314 )
Assume there is an X anyon
outside the system. With
successive X rotations it can be
transported around the plaquette.
1
X2
2
s
Z
X3
3
4
One Plaquette
One can demonstrate the anyonic statistics with only this
plaquette. First create excitation with Z rotation at one
spin:
Z1
1
X1
Z  Z1  
2
( 01020304  11121314 )
Assume there is an X anyon
outside the system. With
successive X rotations it can be
transported around the plaquette.
The final state is given by:
Final  X 4 X 3 X 2 X 1 Z 
1
X2
2
s
Z
X3
3
1

( 010 2 030 4  11121314 )   Initial
2
4
X4
Interference Process
Create state
1
 
( 01020304  11121314 )
2
1/ 2
With half Z rotation on spin 1, Z1 , one can create
the superposition between a Z anyon and the vacuum:
e
 i
1/ 2
1
Z
  (  i Z )/ 2
for   3 / 4 . Then the X anyon is rotated around it:
X 4 X 3 X 2 X1 (   i Z ) / 2  (   i Z ) / 2
Then we make the inverse half Z rotation
i
e Z1
1/ 2
(  i  )/ 2  
Experiment
Experiment
Qubit states 0 and 1 are
encoded in the polarization,
V and H, of four photonic
modes.
[J.K.P., W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, H. Weinfurter,
arXiv:0710.0895]
Qubit states 0 and 1 are
encoded in the polarization,
V and H, of four photonic
modes.
The states that come from
this setup are of the form:
Counts
Experiment
Vacuum
 
1
( 0000  1111 )
2
  a GHZ  b EPR  EPR 
a( HHHH  VVVV ) 
Anyon
Measurements and
manipulations are repeated
over all modes.
Counts
b( VHVH  HVVH  VHHV  HVHV )
Z 
1
( 0000  1111 )
2
Experiment: State identification
Vacuum
Consider correlations:

Z    cos( 4 )
Correlations

tr(cos 
tr (cos  x  sin  y ) 4     cos( 4 )
x
 sin  y ) 4 Z
Visibility > 64%
x-displacement: -7° and +2°
Fidelity:
2
*
1 16
*
1 16
Witness for genuine
4-qubit GHZ entanglement:
WGHZ4
1
 F 0
2
Anyon
Correlations
F | a1 |  | a16 |  a a  a a  73%
2


y-displacement ~ EPRxEPR
Experiment: Fusion Rules
Fusion rules
Generation of anyon:
Fusion exe=1:
(invariance of vacuum state)
Fusion ex1=e:
(Invariance of anyon state)
Experiment: Statistics
Interference
- loop around empty
plaquette:
- loop around occupied
plaquette:
- interference of the
two processes:
Properties and Applications
• Invariance of vacuum w.r.t. to closed paths:
XXXX   
Zi Z j   
4 qubit GHZ stabilizers
Z  Zi 
Useful for:
• quantum error correction,
• topological quantum memory,
• quantum anonymous broadcasting
Implement Hamiltonian, larger systems…
A
C
B
[arXiv:0710.0895; J.K.P., Annals of Physics 2007, IJQI 2007]