Revealing%20anyonic%20statistics%20by%20multiphoton%20entanglement

1
Revealing anyonic statistics by multiphoton
entanglement
Jiannis K. Pachos Witlef Wieczorek Christian
Schmid Nikolai Kiesel Reinhold Pohlner Harald
Weinfurter
arXiv0710.0895
QEC07, USC, December 2007
2
Overview

• In condensed matter anyons appear as ground or
  excited states of two dimensional systems
• Superconducting electrons in a strong magnetic
  field (Fractional Quantum Hall Effect)
• Lattice systems (Kitaevs toric code/hexagonal
  lattice, Wens models, Ioffes model,
  Freedman-Nayak-Shtengel model)
• Energy gap protects anyons for applications
• Relatively large systems to have many anyons and
  move them in arbitrary paths

3
Overview

• Close the gap between theory and experiment.
• Anyonic statistics is a property of (highly
  entangled) states.
• Obtain state by cooling, adiabatically ,
  dynamically.
• Use linear optics!
• Employ the toric code model.
• One plaquette one anyon and path of another.
• No Hamiltonian employ decoherence-free system.
• How to generate, manipulate, measure anyons?
• Criteria for implementing Topological Order and
  TQC
• One plaquette topological entropy
• Non-abelian statistics and philosophy

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Anyons
Anyons have non-trivial statistics.
3D
2D
Consider as composite particles of fluxes and
charges. Then phase is like the Aharonov-Bohm
effect.
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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)
• Scalability (Large system, many anyons, desired
  braiding)
• Low decoherence (Temperature, impurities, anyon
  identification)

I. Cirac, S. Simon, private communication
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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)
• Scalability (Large system, many anyons, desired
  braiding)
• Low decoherence (Temperature, impurities, anyon
  identification)

7
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)
• Scalability (Large system, many anyons, desired
  braiding)
• Low decoherence (Temperature, impurities, anyon
  identification)

Ocneanu rigidity algebraic structure is rigid to
small perturbations of the Hamiltonian
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Toric Code ECC
Consider the lattice Hamiltonian
p
s
s
p
p
p
Spins on the vertices. 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.
s
s
p
X1
s
X4
X2
X3
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Toric Code ECC
Consider the lattice Hamiltonian
p
s
s
p
p
p
Good quantum numbers
s
s
p

• eigenvalues of XXXX and ZZZZ terms are 1 and -1
• Also Hamiltonian exactly solvable

X1
s
X4
X2
X3
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Toric Code ECC
Consider the lattice Hamiltonian
p
s
s
p
Indeed, the ground state is
p
p
s
s
p
The 000gt state is a ground state of the ZZZZ
term and the (IXXXX) term projects that state to
the ground state of the XXXX term.
X1
s
X4
X2
F. Verstraete, et al., PRL, 96, 220601 (2006)
X3
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Toric Code ECC

• Excitations are produced by Z or X rotations of
  one spin.
• 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.

p
Z
s
s
p
p
p
s
s
X
p
X1
s
X4
X2
X3
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One Plaquette
It is possible to demonstrate the anyonic
properties with one s plaquette only. The
Hamiltonian
1
2
The ground state
s
4
3
GHZ state!
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
1
Energy of ground state
2
s
4
Energy of excited state
3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
1
Note that the second anyon from the Z rotation is
outside the system. Move an X anyon around the Z
one. What we really want is the path that it
traces and this can be spanned on the spins
1,2,3,4.
2
s
4
3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
1
Assume there is an X anyon outside the system.
With successive X rotations it can be transported
around the plaquette.
2
s
4
3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
X1
1
Assume there is an X anyon outside the system.
With successive X rotations it can be transported
around the plaquette.
2
s
4
3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
X1
1
Assume there is an X anyon outside the system.
With successive X rotations it can be transported
around the plaquette.
X2
2
s
4
3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
X1
1
Assume there is an X anyon outside the system.
With successive X rotations it can be transported
around the plaquette.
X2
2
s
4
3
X3
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One Plaquette
One can demonstrate the anyonic statistics with
only this plaquette. First create excitation with
Z rotation at one spin
Z1
X1
1
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
X2
2
s
X4
4
3
X3
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One Plaquette
After a complete rotation of an X anyon around a
Z anyon (two successive exchanges) the resulting
state gets a phase (a minus sign) hence
ANYONS with statistical angle A property we used
is that A crucial point is that these properties
can be demonstrated without the Hamiltonian!!! An
interference experiment can reveal the presence
of the phase factor.
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Extended System
5
From the formula
Z1
X1
1
6
X2
2
s
X4
4
the extended state is given by
3
X3
7
9
8
The two X, Z anyonic state is given by
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Interference Process
Create state
With half Z rotation on spin 1, , one can
create the superposition between a Z anyon and
the vacuum
for . Then the X anyon is rotated
around it
Then we make the inverse half Z rotation
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Interference Process
That we obtained the state is due to the
minus sign produced from the anyonic statistics.
If it wasnt there then we would have returned
to the vacuum state . Distinguish between
states has even number
of 1s. has odd number of 1s.
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Experiment
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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, arXiv0710.0895
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Experiment
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 
Measurements and manipulations are repeated over
all modes.
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Experiment State identification
Consider correlations
Visibility gt 64
Fidelity
Witness for genuine 4-qubit GHZ entanglement
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Experiment Fusion Rules
Fusion rules
Generation of anyon
Fusion exe1 (invariance of vacuum state)
Fusion ex1e (Invariance of anyon state)
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Experiment Statistics
Interference
- loop around empty plaquette
- loop around occupied plaquette
- interference of the two processes
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Experiment
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Properties and Applications

• Invariance of vacuum w.r.t. to closed paths
• Useful for
• quantum error correction,
• topological quantum memory,
• quantum anonymous broadcasting
• Implement Hamiltonian, larger systems

A
C
B
arXiv0710.0895 J.K.P., Annals of Physics 2007,
IJQI 2007