NEW COVER SLIDE qinfo with p

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title
From Quantum Tomography to Quantum Error
Correction playing games with the information
in atoms and photons
Aephraim Steinberg Dept. of Physics, University
of Toronto
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Acknowledgments
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 Jalani
Fox Ana Jofre Mirco Siercke Samansa
Maneshi Salvatore Maone (? real world) Theory
friends Daniel Lidar, Janos Bergou, Mark
Hillery, John Sipe, Paul Brumer, Howard Wiseman
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OUTLINE
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 lt1 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)

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Quantum Information
What's so great about it?
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Quantum Information
What's so great about it?
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The Rub
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What makes a quantum computer?
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What makes a computer quantum?
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Conventional Answers

• Computers are made from Silicon, not photons.
• Maybe trapped atoms/ions have some of the
  advantages of photons without the disadvantages.
• Maybe SQUIDs or quantum dots or something else
  will prove the right technology instead.
• Maybe using quantum measurement and postselection
  as an "effective interaction" will save the day
  for optics.
• Maybe photons can be made to interact better
  after all

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PART 1Can 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
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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.
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Is SPDC really the time-reverse of SHG?
(And if so, then why doesn't it exist in
classical em?)
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).
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Quantum Interference
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Type-II down-conversion
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2-photon "Switch" experiment
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Suppression/Enhancementof Spontaneous
Down-Conversion
(57 visibility)
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Switchiness ("Nonlinearity")
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Switch cartoon
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PART 1aApplications 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.
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"Interaction-Free Measurements"
(AKA The Elitzur-Vaidman bomb experiment)
Problem Consider a collection of bombs so
sensitive that a collision with any single
particle (photon, electron, etc.) is guarranteed
to trigger it. Suppose that certain of the bombs
are defective, but differ in their behaviour in
no way other than that they will not blow up when
triggered. Is there any way to identify the
working bombs (or some of them) without blowing
them up?
Bomb absent Only detector C fires
Bomb present "boom!" 1/2 C 1/4
D 1/4
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Hardy Cartoon
Hardys Paradox
D- e was in DD- ? But
D e- was in
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Hardy's Paradox Setup
Det. A
Det. B
CC
50-50 BS2
PBS
50-50 BS1
GaN Diode Laser
CC
V
H
DC BS
DC BS

Switch (W)
Cf. Torgerson et al., Phys. Lett. A. 204, 323
(1995)
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Conclusions when both "dark" detectors fire
0
1
1
-1
Upcoming experiment demonstrate that
"weak measurements" (à la Aharonov Vaidman)
will bear out these predictions.
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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.

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PART 2State and process tomography
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Density matrices and superoperators
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Part 2aTwo-photon Process Tomography
Two waveplates per photon for state preparation
Detector A
HWP
HWP
PBS
QWP
QWP
SPDC source
QWP
QWP
PBS
HWP
HWP
Detector B
Argon Ion Laser
Two waveplates per photon for state analysis
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Hong-Ou-Mandel Interference
How often will both detectors fire together?
r2t2 0 total destructive interference. If the
photons begin in a symmetric state, no
coincidences. The only antisymmetric state is the
singlet state HVgt VHgt, 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.
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Measuring the superoperator
Coincidencences
Output DM Input

HH



16 input states
HV
etc.
VV
16 analyzer settings
VH
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Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
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Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
Observed
Nphotons 314
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So, How's Our Singlet State Filter?
Bell singlet state ?? (HV-VH)/v2
Observed ? ??, but a different maximally
entangled state
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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
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Comparison to ideal filter
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Part 2bTomography in Optical Lattices
Part I measuring state populations in a lattice
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Time-resolved quantum states
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Lattice experimental setup
Setup for lattice with adjustable position
velocity
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Atoms oscillating
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Also Atoms oscillating
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Oscillations in lattice wells
Ground-state population vs. time bet. translations
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Quantum state reconstruction
Wait
Shift
Initial phase- space distribution
Measure ground state population
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Q(x,p) for a coherent H.O. state?
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Quasi-Q for a mostly-excited statein a 2-state
lattice
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Theory for 80/20 mix of e and g
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Exp't"W" or Pg-Pe(x,p)
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W(x,p) for 80 excitation
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Atom superoperators
sitting in lattice, quietly decohering
being shaken back and forth resonantly
Initial Bloch sphere
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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.

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Theory how to distinguish non-orthogonal 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.
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A test case
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Experimental layout
(ancilla)
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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.
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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
  comp-utation. 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