From Quantum Tomography to Quantum Error Correction:
playing games with the information in atoms and photons
Aephraim Steinberg
Dept. of Physics, University of Toronto
Acknowledgments
U of T quantum optics & laser cooling group:
PDFs: Morgan Mitchell
Marcelo Martinelli
Optics: Kevin Resch ( Vienna) Jeff Lundeen
Chris Ellenor ( Korea) Masoud Mohseni ( Lidar)
Reza Mir
Rob Adamson
Atom Traps: Stefan Myrskog
Ana Jofre
Samansa Maneshi
Jalani Fox
Mirco Siercke
Salvatore Maone ( real world)
Theory friends:
Daniel Lidar, Janos Bergou, Mark Hillery, 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.
• Can there be nonlinear optics with <1 photon?
- Using our "photon switch" to test Hardy's Paradox.
• Quantum process tomography on entangled photon pairs
- E.g., quality control for Bell-state filters.
- Input data for tailored Quantum Error Correction.
• Quantum tomography (state and process) on center-of-mass
states of atoms in optical lattices.
• Summary / Coming attractions…
(Optimal discrimination of non-orthogonal states…
Tunneling-induced coherence between lattice sites…
Coherent control of quantum chaos…
Quantum computation in the presence of noise…)
Quantum Information
What's so great about it?
Quantum Information
What's so great about it?
If a classical computer takes input |n> to output |f(n)>,
an analogous quantum computer takes a state
|n>|0> and maps it to |n>|f(n)> (unitary, reversible).
By superposition, such a computer takes
n |n>|0> to n |n>|f(n)>; it calculates f(n)
for every possible input simultaneously.
A clever measurement may determine some global
property of f(n) even though the computer has
only run once...
A not-clever measurement "collapses" n to some
random value, and yields f(that value).
The rub: any interaction with the environment
leads to "decoherence," which can be thought
of as continual unintentional measurement of n.
The Rub
What makes a quantum computer?
What makes a computer quantum?
If a quantum "bit" is described by two numbers:
|> = c0|0> + c 1|1>,
then n quantum bits are described by 2n coeff's:
|> = c00..0|00..0>+c 00..1|00..1>+...c 11..1|11..1>;
this is exponentially more information than the 2n coefficients it
would take to describe n independent (e.g., classical) bits.
It is also exponentially sensitive to decoherence.
Photons are ideal carriers of quantum information-- they
can be easily produced, manipulated, and detected, and
don't interact significantly with the environment. They
are already used to transmit quantum-cryptographic
information through fibres under Lake Geneva, and soon
through the air up to satellites.
Unfortunately, they don't interact with each other very much
either! How to make a logic gate?
PART 1:
Can we build a two-photon switch?
Photons don't interact
(good for transmission; bad for computation)
Nonlinear optics: photon-photon interactions
Generally exceedingly weak.
Potential solutions:
Cavity QED
Better materials (1010 times better?)
Measurement as nonlinearity (KLM)
Novel effects (slow light, EIT, etc)
Interferometrically-enhanced nonlinearity
Entangled photon pairs
(spontaneous parametric down-conversion)
The time-reverse of second-harmonic generation.
A purely quantum process (cf. parametric amplification)
Each energy is uncertain, yet their sum is precisely defined.
Each emission time is uncertain, yet they are simultaneous.
Is SPDC really the time-reverse of SHG?
(And if so, then why doesn't it exist in classical e&m?)
The probability of 2 photons upconverting in a typical
nonlinear crystal is roughly 10-10 (as is the probability
of 1 photon spontaneously down-converting).
Quantum Interference
Suppression/Enhancement
of Spontaneous Down-Conversion
(57% visibility)
PART 1a:
Applications of 2-photon switch
N.B.: Does not work on Fock states!
Have demonstrated controlled-phase operation.
Have shown theoretically that a polarisation version could
be used for Bell-state determination (and, e.g., dense
coding)… but not for projective Bell measurements.
Present "application," however, is to a novel test of QM.
"Interaction-Free Measurements"
(AKA: The Elitzur-Vaidman bomb experiment)
Problem:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
Hardy’s Paradox
C+
D+
D-
BS2+
C-
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
Outcome Prob
D+
e- was
D+ and
C- in
1/16
D- e+ was in
D- and C+ 1/16
C+ and ?C- 9/16
D+DD+ and D- 1/16
But
…
Explosion
4/16
Hardy's Paradox: Setup
Det. A
Det. B
CC
PBS
50-50
BS2
GaN
Diode Laser
50-50
BS1
V
DC BS
H
Switch (W)
CC
DC BS
Cf. Torgerson et al., Phys. Lett. A. 204, 323 (1995)
Conclusions when both "dark" detectors
fire
Probabilities e- in
e- out
e+ in
0
1
1
e+ out
1
-1
0
1
0
Upcoming experiment: demonstrate that "weak
measurements" (à la Aharonov + Vaidman) will
bear out these predictions.
The Real Problem
• The danger of errors grows exponentially with the size
of the quantum system.
• Without error-correction techniques, quantum
computation would be a pipe dream.
• A major goal is to learn to completely characterize the
evolution (and decoherence) of physical quantum
systems in order to design and adapt error-control
systems.
• The tools are "quantum state tomography" and "quantum
process tomography": full characterisation of the density
matrix or Wigner function, and of the "$uperoperator"
which describes its time-evolution.
PART 2:
State and process tomography
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 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.
Part 2a:
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
r
r
+
t
t
How often will both detectors fire together?
r2+t2 = 0; total destructive interference.
If the photons begin in a symmetric state, no coincidences.
The only antisymmetric state is the singlet state
|HV> – |VH>, in which each photon is
unpolarized but the two are orthogonal.
This interferometer is a "Bell-state filter," needed
for quantum teleportation and other applications.
Our Goal: use process tomography to test this 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
1/2 -1/2
-1/2 1/2
Observed  -, but a different maximally entangled state:
Model of real-world
beamsplitter
Singlet
filter
multi-layer dielectric
AR coating
45° “unpolarized” 50/50
dielectric beamsplitter
at 702 nm (CVI Laser)
1

birefringent element
+
singlet-state filter
+
birefringent element
Comparison to ideal filter
Measured superoperator,
in Bell-state basis:
Superoperator after transformation
to correct polarisation rotations:
A singlet-state filter would have
a single peak, indicating the one
transmitted state.
Dominated by a single peak;
residuals allow us to estimate
degree of decoherence and
other errors.
Part 2b:
Tomography in Optical Lattices
Part I: measuring state populations in a lattice…
Time-resolved quantum states
Setup for lattice with adjustable position & velocity
Atoms oscillating
final vs midterm, both adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
Also
Atoms
oscillating
final vs midterm, both adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
Oscillations in lattice wells
Ground-state population vs. time bet. translations
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Quantum state reconstruction
p
p
p
t
t
x
Wait…
p
xx
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.
Exp't:"W" or [Pg-Pe](x,p)
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are needed to see this picture.
W(x,p) for 80% excitation
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are needed to see this picture.
Atom superoperators
sitting in lattice, quietly
decohering…
QuickTime™ and a
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are needed to see this picture.
QuickTime™ and a
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are needed to see this picture.
being shaken back and
forth resonantly
Initial Bloch sphere
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are needed to see this picture.
Coming attractions…
State Discrimination
• Non-orthogonal quantum states cannot be distinguished
with certainty.
• This is one of the central features of quantum information
which leads to secure (eavesdrop-proof) communications.
• Crucial element: we must learn how to distinguish quantum
states as well as possible -- and we must know how well
a potential eavesdropper could do.
H-polarized photon
45o-polarized photon
Theory: how to distinguish nonorthogonal states optimally
Step 1:
Repeat the letters "POVM" over and over.
Step 2:
Ask Janos, Mark, and Yuqing for help.
The view from the laboratory:
A measurement of a two-state system can only
yield two possible results.
If the measurement isn't guaranteed to succeed, there
are three possible results: (1), (2), and ("I don't know").
Therefore, to discriminate between two non-orth.
states, we need to use an expanded (3D or more)
system. To distinguish 3 states, we need 4D or more.
A test case
Consider these three non-orthogonal states:
Projective measurements can distinguish these states
with certainty no more than 1/3 of the time.
(No more than one member of an orthonormal basis is orthogonal
to two of the above states, so only one pair may be ruled out.)
But a unitary transformation in a 4D space produces:
…and these states can be distinguished with certainty
up to 55% of the time
Experimental layout
(ancilla)
Success!
"Definitely 3"
"Definitely 2"
"Definitely 1"
"I don't know"
The correct state was identified 55% of the time-Much better than the 33% maximum for standard measurements.
SUMMARY
• Quantum interference allows huge enhancements of
optical nonlinearities. Useful for quantum computation?
• Two-photon switch useful for studies of quantum weirdness
(Hardy's paradox, weak measurement,…)
• Two-photon process tomography useful for characterizing
(e.g.) Bell-state filters.
Next round of experiments on tailored quantum error correction
(w/ D. Lidar et al.).
• Wigner-function and Superoperator reconstruction also underway
in optical lattices, a strong candidate system for quantum computation. Characterisation and control of decoherence expected soon.
• Other work: Implementation of a quantum algorithm in the
presence of noise; Optimal discrimination of non-orthogonal states;
Tunneling-induced coherence; et cetera…