Transcript Quantum information with photons and atoms: from tomography to error correction C.

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
Jalani Fox
Mirco Siercke
Samansa Maneshi
TBA: Rob Adamson
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
CH
CV
Density matrix: 2x2=4 coefficients
CHH
CVH
CHV
CVV
Measure
intensity of horizontal
intensity of vertical
intensity of 45 o
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
HWP
QWP
HWP
Detector A
PBS
QWP
SPDC source
"Black Box" 50/50
Beamsplitter
QWP
HWP
QWP
PBS
HWP
Detector B
Argon Ion Laser
Two waveplates per
photon for state analysis
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 BellState filter.
“Measuring” the superoperator
Coincidencences
Output DM
}
}
}
}
16
input
states
Input
HH
HV
etc.
VV
16 analyzer settings
VH
“Measuring” the superoperator
Input
Superoperator
Output DM
HH
HV
VV
VH
etc.
Input
Output
“Measuring” the superoperator
Input
Superoperator
Output DM
HH
HV
VV
VH
etc.
Input
Output
Testing the superoperator
LL = input state
Predicted
Nphotons = 297 ± 14
Testing the superoperator
LL = input state
Predicted
Nphotons = 297 ± 14
Detector A
HWP
QWP
HWP
PBS
QWP
"Black Box" 50/50
Beamsplitter
BBO two-crystal
downconversion
source.
QWP
HWP
Argon Ion Laser
QWP
PBS
HWP
Detector B
Observed
Nphotons = 314
So, How's Our Singlet State Filter?
Bell singlet state:  = (HV-VH)/√2
Observed  
Model of real-world beamsplitter
Singlet
filter
multi-layer dielectric
AR coating
45° “unpolarized” 50/50
dielectric beamsplitter
at 702 nm (CVI Laser)


birefringent element
+
singlet-state filter
+
birefringent element
Model beamsplitter predicitons
Singlet
filter

Best Fit: 1 = 0.76 π
2 = 0.80 π

Predicted
Comparison to measured Superop
Observed
Predicted
Performing a quantum computation
in a decoherence-free subspace
The Deutsch-Jozsa algorithm:
A
0
1
Oracle
A
H
x
H
y y f(x)
x
f (0)  f (1)
H
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
1
1000
2
3
4
0100
Computational basis
1st qubit
0010
00
01
10
0001
11
2nd 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
01
11 11
10 ei 10
00
ei 01
e i 2 z 2 z
But after oracle, only qubit 1 is needed for calculation.
Encode this logical qubit in
either DFS: (00,11) or (01,10).
Modified Deutsch-Jozsa Quantum Circuit
0
H
x
1
H
y y f(x) 
x
H

Experimental Setup
1
Random Noise
2
1
3
4
23
2
Preparation
Oracle
4
3
3/4
B
Phase Shifter
PBS
Detector
 / 2 Waveplate
Mirror
Optional swap for
choice of encoding
D
4/3
A
C
DJ without noise -- raw data
Original encoding
DFS Encoding
C
B
C
B
B
C
B
C
C
Constant function
B
Balanced function
Implementation of D-J in presence of noise
Original Encoding
Normalized intensity
DFS Encoding
C
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
B
C
B
B
C
B
C
Detectors A and C
Detectors B and D
0
10
20
30
40
Different settings of oracle in time(s)
50
60
C
Constant function
B
Balanced function
Tomography in Optical Lattices
Part I: measuring state populations in a lattice…
Houston, we have separation!
Quantum state reconstruction
p
p
p
=x
t
x
Initial phasespace distribution
x
Wait…
x
Shift…
p
Q(0,0) = Pg
W(0,0) =  (-1)n Pn
(More recently: direct
density-matrix reconstruction)
x
Measure ground
state population
Quasi-Q (Pg versus shift) for a 2-state
lattice with 80% in upper state.
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Exp't:"W" or [Pg-Pe](x,p)
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
W(x,p) for 80% excitation
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Photo - JPEG decompressor
are needed to see this picture.
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
1
00
2
01
3
10
4
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
p
p
t
t
x
Wait…
p
x
x
Shift…
Initial phasespace distribution
Q(0,0) = Pg
x
x
Measure ground
state population
W(0,0) =  (-1)n Pn
Q(x,p) for a coherent H.O. state
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are needed to see this picture.
Theory for 80/20 mix of e and g
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