Title slide
Quantum information with photons and atoms: from tomography to error correction
C. W. Ellenor, M. Mohseni, S.H. Myrskog, J.K. Fox,
J. S. Lundeen, K. J. Resch,
M. W. Mitchell, and
Aephraim M. Steinberg
Dept. of Physics, University of Toronto
PQE 2003

Acknowledgments
U of T quantum optics & laser cooling group:
PDF: Morgan Mitchell
Optics: Kevin Resch (

Wien) Jeff Lundeen
Chris Ellenor (

Korea) Masoud Mohseni
Reza Mir (

Lidar )
Atom Traps: Stefan Myrskog
Ana Jofre
Salvatore Maone
TBA: Rob Adamson
Jalani Fox
Mirco Siercke
Samansa Maneshi
Theory friends: Daniel Lidar, Janos Bergou, John Sipe, Paul Brumer, Howard Wiseman

OUTLINE
• Introduction:
Photons and atoms are promising for QI.
Need for real-world process characterisation and tailored error correction.
No time to say more.
• Quantum process tomography on entangled photon pairs
- E.g., quality control for Bell-state filters.
- Input data for tailored Quantum Error Correction.
• An experimental application of decoherence-free subspaces in a quantum computation.
• Quantum state (and process?) tomography on center-of-mass states of atoms in optical lattices.
• Coming attractions…

Density matrices and superoperators
One photon: H or V.
State: two coefficients
( )
C
V
Density matrix: 2x2=4 coefficients ( )
C
HV
C
VV
Measure intensity of horizontal intensity of vertical intensity of 45o intensity of RH circular.
Propagator (superoperator): 4x4 = 16 coefficients.
Two photons: HH, HV, VH, HV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.

Two-photon Process Tomography
Two waveplates per photon for state preparation
Detector A
HWP
QWP
HWP
QWP
PBS
SPDC source
"Black Box" 50/50
Beamsplitter
Argon Ion Laser
QWP
HWP
QWP
HWP
Two waveplates per photon for state analysis
PBS
Detector B

Hong-Ou-Mandel Interference
> 85% visibility for HH and VV polarizations
HOM acts as a filter for the Bell state:
  = (HV-VH)/√2
Goal: Use Quantum Process Tomography to find the superoperator which takes  in
  out
Characterize the action (and imperfections) of the Bell-
State filter.

16 input states
“Measuring” the superoperator
Coincidencences
Output DM Input
}
}
}
HH
HV etc.
VV
16 analyzer settings
VH

“Measuring” the superoperator
Input Output DM
Superoperator
HH
HV
VV
VH
Input
Output etc.

“Measuring” the superoperator
Input Output DM
Superoperator
HH
HV
VV
VH
Input
Output etc.

Testing the superoperator
LL = input state
Predicted N photons
= 297 ± 14

Testing the superoperator
LL = input state
Predicted N photons
= 297 ± 14
BBO two-crystal downconversion source.
Argon Ion Laser
Detector A
HWP
QWP
HWP
QWP
PBS
"Black Box" 50/50
Beamsplitter
QWP
HWP
QWP
HWP
PBS
Detector B
Observed N photons
= 314

So, How's Our Singlet State Filter?
Bell singlet state:   = (HV-VH)/√2
Observed   

Model of real-world beamsplitter multi-layer dielectric
AR coating

45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser)
 birefringent element
+ singlet-state filter
+ birefringent element
Singlet filter

Model beamsplitter predicitons

Best Fit: 
1

2
= 0.76 π
= 0.80 π

Singlet filter
Predicted

Comparison to measured Superop
Observed

Performing a quantum computation in a decoherence-free subspace
The Deutsch-Jozsa algorithm:
0
1
A
H
H x
Oracle y y  x f(x)
H
A f ( 0 )
 f ( 1 )
0

1
2
We use a four-rail representation of our two physical qubits and encode the logical states 00, 01, 10 and 11 by a photon traveling down one of four optical rails numbered 1, 2, 3 and 4, respectively.
Photon number basis Computational basis
1 st qubit 1
2
3
4
1000
0100
0010
0001
00
01
10
11
2 nd qubit

Error model and decoherence-free subspaces
Consider a source of dephasing which acts symmetrically on states 01 and 10 (rails 2 and 3)…
00 00
01 e i

01
11 11
10 e i

10 e i

2 z

2 z

But after oracle, only qubit 1 is needed for calculation .
Modified Deutsch-Jozsa Quantum Circuit
0
1
H x
H y x y  f(x)

H


DJ experimental setup
Experimental Setup
1
2
3
4
Random Noise
1
23
2
Preparation
Oracle
4
3
3/4
Optional swap for choice of encoding
Phase Shifter
PBS
Detector

/ 2 Waveplate
Mirror
4/3
A
B
D
C

DJ without noise -- raw data
C
DFS Encoding
B C B B
Original encoding
C B C
C
B
Constant function
Balanced function

DJ with noise-- results
0.75
0.7
0.65
0.6
0.55
0.95
1
0.9
0.85
0.8
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
C
Implementation of D-J in presence of noise
DFS Encoding
B C B B
Original Encoding
C B C
10 20 30 40
Different settings of oracle in time(s)
50 60
Detecto rs A and C
Detecto rs B and D
C
B
Constant function
Balanced function

Tomography in Optical Lattices
Part I: measuring state populations in a lattice…

Houston, we have separation!

Quantum state reconstruction p p
 t p
 =  x x x
Initial phasespace distribution
Wait… Shift… p x
Q( 0,0 ) = P g
W(0,0) =

(-1)n Pn
(More recently: direct density-matrix reconstruction)
Measure ground state population x

Quasi-Q (Pg versus shift) for a 2-state lattice with 80% in upper state.
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Exp't:"W" or [P g
-P e
](x,p)
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W(x,p) for 80% excitation
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Coming attractions
• A "two-photon switch": using quantum enhancement of two-photon nonlinearities for
- Hardy's Paradox ( and weak measurements )
- Bell-state determination and quantum dense coding(?)
• Optimal state discrimination/filtering ( w/ Bergou, Hillery )
• The quantum 3-box problem ( and weak measurements )
• Process tomography in the optical lattice
- applying tomography to tailored Q. error correction
• Welcher-Weg experiments ( and weak measurements, w/ Wiseman )
• Coherent control in optical lattices ( w/ Brumer )
• Exchange-effect enhancement of 2x1-photon absorption
( w/ Sipe, after Franson )
• Tunneling-induced coherence in optical lattices
• Transient anomalous momentum distributions ( w/ Muga )
• Probing tunneling atoms ( and weak measurements )
… et cetera

Schematic diagram of D-J interferometer
1 2 3 4
Oracle
3
4
1
2
00
01
10
11
1 2 3 4
“Click” at either det. 1 or det. 2 (i.e., qubit 1 low) indicates a constant function; each looks at an interferometer comparing the two halves of the oracle.
Interfering 1 with 4 and 2 with 3 is as effective as interfering
1 with 3 and 2 with 4 -- but insensitive to this decoherence model .

Quantum state reconstruction p
 t
 t x
Wait… p
 x x
Shift…
Initial phasespace distribution
Q(0,0) = Pg
W(0,0) =

(-1)n Pn
Measure ground state population

Q(x,p) for a coherent H.O. state
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Theory for 80/20 mix of e and g
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