Transcript Manipulating and Measuring the Quantum State of Photons and Atoms

Manipulating and Measuring the Quantuum State of
Photons and Atoms
Aephraim M. Steinberg
Centre for Q. Info. & Q. Control
Institute for Optical Sciences
Dept. of Physics, U. of Toronto
CANADA
The 3 quantum computer scientists:
see nothing (must avoid "collapse"!)
hear nothing (same story)
say nothing (if any one admits this thing
is never going to work,
that's the end of our
funding!)
QUEST 05, Santa Fe
DRAMATIS PERSONAE
Toronto quantum optics & cold atoms group:
Postdocs: Morgan Mitchell ( Barcelona)
Matt Partlow
Optics: Jeff Lundeen
Kevin Resch(Zeilinger 
Masoud Mohseni (Lidar)
)
Lynden(Krister) Shalm
Rob Adamson
QuickTime™
and a TIFF (Uncompressed) decompressor are needed to see this picture.
Atoms: Jalani Fox
Ana Jofre(NIST)
Samansa Maneshi
Stefan Myrskog (Thywissen)
Mirco Siercke
Chris Ellenor
Some theory collaborators:
...Daniel Lidar, Pete Turner, János Bergou, Mark Hillery, Paul Brumer, Howard Wiseman,...
There are many types of
measurement!
A few to keep in mind:
• Projective measurement (or von Neumann); postselection
• Quantum state “tomography” (reconstruction of , W, etc)
• standard, adaptive, ...
• incomplete?
• in the presence of inaccessible information
• Quantum process “tomography” (CP map from 
)
• standard, ancilla-assisted, “direct”,...
• POVMs
• “Direct” measurement of functions of 
• “Interaction-free” measurements
• “Weak measurements” (various senses)
• Aharonov/Vaidman application to postselection
OUTLINE
Tomography – characterizing quantum
states & processes... brief review
Entangled photon pairs
2-photon process tomography
Direct measurement of purity
Generating entanglement by postselection
Characterizing states with “inaccessible” info
Motional states of atoms in optical lattices
Process tomography
Pulse echo
Inverted states, negative Wigner functions,...
Bonus topic if you don’t interrupt me enough:
Weak measurements and “paradoxes”
(which-path debate; Hardy’s paradox)
0
Quantum tomography: what & why?
1.
2.
3.
4.
5.
Characterize unknown quantum states & processes
Compare experimentally designed states & processes to design goals
Extract quantities such as fidelity / purity / tangle
Have enough information to extract any quantities defined in the future!
• or, for instance, show that no Bell-inequality could be violated
Learn about imperfections / errors in order to figure out how to
• improve the design to reduce imperfections
• optimize quantum-error correction protocols for the system
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, VV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.
Wigner function of an ion in the
excited state
Liebfried, Meekhof, King, Monroe, Itano, Wineland, PRL 77, 4281 (96)
Some density matrices...
Much work on reconstruction of optical density matrices in the Kwiat
group; theory advances due to Hradil & others, James & others, etc...;
now a routine tool for characterizing new states, for testing gates or
purification protocols, for testing hypothetical Bell Inequalities, etc...
Spin state of Cs atoms (F=4),
Polarisation state of 3 photons
in two bases
(GHZ state)
Klose, Smith, Jessen, PRL 86 (21) 4721 (01)
Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)
QPT of QFT
Weinstein et al., J. Chem. Phys. 121, 6117 (2004)
To the trained eye, this is a Fourier transform...
From those superoperators, one can extract Kraus operator
amplitudes, and their structure helps diagnose the process.
Ancilla-assisted process tomography
Proposed in 2000-01 (Leung; D’Ariano & Lo Presti; Dür & Cirac):
one member of an maximally-entangled pair could be collapsed to
any given state by a measurement on the other; replace multiple state
preparations with coincidence measurement.
Altepeter et al, PRL 90, 193601 (03)
2-qbit state tomography with the entangled
input is equivalent to 1-qbit process
tomography using 4 different inputs
(and both require 16 measurements).
1-qbit processes represented
as deformations of Bloch sphere
...some unphysical results with
engineered decoherence.
1
Quantum tomography experiments
on photons, and how to avoid them
Two-photon Process Tomography
[Mitchell et al., PRL 91, 120402 (2003)]
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 interf. (if photons indistinguishable).
If the photons begin in a symmetric state, no coincidences.
{Exchange effect; cf. behaviour of fermions in analogous setup!}
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
of a Bell-state filter
Coincidencences
Output DM
}
}
}
}
16
input
states
Input
HH
HV
etc.
VV
16 analyzer settings
VH
[Mitchell et al., PRL 91, 120402 (2003)]
“Measuring” the superoperator
Input
Superoperator
Output DM
HH
HV
VV
VH
etc.
Input
Output
Superoperator provides information
needed to correct & diagnose operation
Measured superoperator,
in Bell-state basis:
The ideal filter would have a
single peak.
Leading Kraus operator allows
us to determine unitary error.
Superoperator after transformation
to correct polarisation rotations:
Dominated by a single peak;
residuals allow us to estimate
degree of decoherence and
other errors.
(Experimental demonstration delayed for technical reasons;
now, after improved rebuild of system, first addressing some other questions...)
Some vague thoughts...
(1) QPT is incredibly expensive (16n msmts for n qbits)
(2) Both density matrices and superoperators we measure typically
are very sparse... a lot of time is wasted measuring coherences
between populations which are zero.
(a) If aiming for constant errors, can save time by making a
rough msmt of a given rate first and then deciding how
long to acquire data on that point.
(b) Could also measure populations first, and then avoid
wasting time on coherences which would close to 0.
(c) Even if r has only a few significant eigenvalues, is there
a way to quickly figure out in which basis to measure?
(3) If one wants to know some derived quantity, are there short-cuts?
(a) Direct (joint) measurements of polynomial functions
(b) Optimize counting procedure based on a given cost function
(c) Adaptive search
E.g.: suppose you would like to find a DFS within a larger
Hilbert space, but need not characterize the rest.
A sample error model:
the "Sometimes-Swap" gate
Consider an optical system with
stray reflections – occasionally a
photon-swap occurs accidentally:
Two subspaces are
decoherence-free:
1D:
3D:
Experimental implementation: a slightly misaligned beam-splitter
(coupling to transverse modes which act as environment)
TQEC goal: let the machine identify an optimal subspace in which
to compute, with no prior knowledge of the error model.
random
tomography
purity of best 2D
DFS found
Some strategies for a DFS search
(simulation; experiment underway)
# of inputs tested
standard
tomography
adaptive
tomography
# of input states used
Best known
2-D DFS
(average
purity).
averages
Our adaptive algorithm always
identifies a DFS after testing 9 input
states, while standard tomography
routinely requires 16 (complete QPT).
Surprise: in the absence of noise, simulations show that essentially
any 2 input states suffice to identify the DFS (required max-lik to
work). Project to revisit: add noise, do experiment, study scaling,...
Polynomial Functions of a
Density Matrix
(T. A. Brun, e-print: quant-ph/0401067)
• Often, only want to look at a single figure of merit of a state (i.e.
tangle, purity, etc…)
• Would be nice to have a method to measure these properties without
needing to carry out full QST.
• Todd Brun showed that mth degree polynomial functions of a density
matrix fm() can be determined by measuring a single joint observable
involving m identical copies of the state.
Linear Purity of a Quantum State
• For a pure state, P=1
• For a maximally mixed state, P=(1/n)
• Quadratic  2-particle msmt needed
Measuring the purity of a qubit
• Need two identical copies of the state
• Make a joint measurement on the two copies.
• In Bell basis, projection onto the singlet state
P = 1 – 2  –  – 
Singlet-state probability can be
measured by a singlet-state filter (HOM)
HOM as Singlet State Filter
Pure State on either side = 100% visibility
HH H
H HH
H
+
Mixed State = 50% visibility
HV H H
V V
H V
+
HOM Visibility = Purity
Experimentally Measuring the Purity
of a Qubit
•Use Type 1 spontaneous parametric downconversion to prepare two
identical copies of a quantum state
•Vary the purity of the state
•Use a HOM to project onto the singlet
•Compare results to QST
 /2
Single Photon
Detector
Quartz
Slab
Type 1 SPDC
Crystal

Singlet
Filter
 /2

Coincidence
Circuit
Quartz
Slab
Single Photon
Detector
Results For a Pure State
Measuring +45 +45
Prepared the state |+45>
3500
Measured Purity
from Singlet State
Measurement
P=0.92±0.02
Counts per 30 s
3000
2500
2000
1500
1000
500
0
0
50
100
150
200
Delay (um )
Measured Purity
from QST
P=0.99±0.01
250
300
350
Preparing a Mixed State
Can a birefringent delay decohere polarization (when we trace over timing info) ?
[cf. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, and P. G. Kwiat, Phys. Rev. Lett., 90, 193601 ]
Case 1: Same birefringence in each arm
 /2
H

Visibility = (90±2) %
V
V
 /2
H

100% interference
Case 2: Opposite birefringence in each arm
H and V Completely Decohered Due to Birefringence
1800
1600

H
1400
V
H
 /2

V
25% interference
Counts per 30s
 /2
1200
1000
800
600
Visibility = (21±2) %
400
200
0
0
50
The HOM isn’t actually insensitive to timing information.
100
150
200
250
Delay (um)
300
350
400
450
Not a singlet filter, but an
“Antisymmetry Filter”
• The HOM is not merely a polarisation singlet-state filter
• Problem:
• Used a degree of freedom of the photon as our bath instead of some
external environment
• The HOM is sensitive to all degrees of freedom of the photons
• The HOM acts as an antisymmetry filter on the entire photon state
• Y Kim and W. P. Grice, Phys. Rev. A 68, 062305 (2003)
• S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003)
Preparing a Mixed State
Randomly rotate the half-waveplates to produce |45> and |-45>
|45>
Preliminary results
 /2

No Birefringence, Even Mixture of +45/+45 and +45/-45
3500
3000

|45> or |-45>
Currently setting up LCD waveplates which
will allow us to introduce a random phase shift
between orthogonal polarizations to produce a
variable degree of coherence
Could produce a “better”
maximally mixed state by using
four photons. Similar to Paul
Kwiat’s work on Remote State
Preparation.
Counts per 30 s
 /2
2500
2000
1500
1000
Visibility = (45±2) %
500
0
0
50
100
150
200
250
Delay (um )
 /2
Coincidence
Circuit

 /2

300
350
2
When the distinguishable isn’t…
Highly number-entangled states
("low-noon" experiment).
M.W. Mitchell et al., Nature 429, 161 (2004);
and cf. P. Walther et al., Nature 429, 158 (2004).
The single-photon superposition state |1,0> + |0,1>,
which may be regarded as an entangled state of two
fields, is the workhorse of classical interferometry.
The output of a Hong-Ou-Mandel interferometer is |2,0> + |0,2>.
States such as |n,0> + |0,n> ("high-noon" states, for n large) have
been proposed for high-resolution interferometry – related to
"spin-squeezed" states.
Multi-photon entangled states are the resource required for
KLM-like efficient-linear-optical-quantum-computation schemes.
A number of proposals for producing these states have been made,
but so far none has been observed for n>2.... until now!
Practical schemes?
[See for example
H. Lee et al., Phys. Rev. A 65, 030101 (2002);
J. Fiurásek,
Phys. Rev. A 65, 053818 (2002)]
˘
Important factorisation:
+
=
A "noon" state
A really odd beast: one 0o photon,
one 120o photon, and one 240o photon...
but of course, you can't tell them apart,
let alone combine them into one mode!
The germ of the KLM idea
INPUT STATE
a|0> + b|1> + c|2>
ANCILLA
|1>
OUTPUT STATE
a'|0> + b'|1> + c'|2>
TRIGGER (postselection)
|1>
In particular: with a similar but somewhat more complicated
setup, one can engineer
a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;
effectively a huge self-phase modulation (p per photon).
More surprisingly, one can efficiently use this for scalable QC.
KLM Nature 409, 46, (2001); Cf. experiments by Franson et al., White et al., ...
Trick #1
Okay, we don't even have single-photon sources.
But we can produce pairs of photons in down-conversion, and
very weak coherent states from a laser, such that if we detect
three photons, we can be pretty sure we got only one from the
laser and only two from the down-conversion...
SPDC
|0> + e |2> + O(e2)
laser
|0> +  |1> + O(2)
e |3> + O(3) + O(e2)
+ terms with <3 photons
Trick #2
How to combine three non-orthogonal photons into one spatial mode?
"mode-mashing"
Yes, it's that easy! If you see three photons
out one port, then they all went out that port.
Trick #3
But how do you get the two down-converted photons to be at 120o to each other?
More post-selected (non-unitary) operations: if a 45o photon gets through a
polarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be
anywhere...
(or nothing)
(or nothing)
(or <2 photons)
The basic optical scheme
+ e i3
Dark ports
PBS
DC
photons
HWP
to
analyzer
PP
Phase
shifter
QWP
Ti:sa
It works!
Singles:
Coincidences:
Triple
coincidences:
Triples (bg
subtracted):
Generating / measuring other states
With perfect detectors and perfect single-photon sources, such schemes
can easily be generalized.
With one or the other (and typically some feedback), many states may be
synthesized by iteratively adding or subtracting photons, and in some cases
implementing appropriate unitaries.
Postselection has also been used to generate GHZ, W, and cluster states
(to various degrees of fidelity).
Photon subtraction can be used to generate non-gaussian states.
Postselection is also the heart of KLM and competing schemes, and can
be used to implement arbitrary unitaries, and hence to entangle anything.
“Continuous” photon subtraction (& counting) can be used, even with
inefficient detectors, to reconstruct the entire photon-number distribution.
Fundamentally Indistinguishable
vs.
Experimentally Indistinguishable
But what if when we combine our photons,
there is some residual distinguishing information:
some (fs) time difference, some small spectral
difference, some chirp, ...?
This will clearly degrade the state – but how do
we characterize this if all we can measure is
polarisation?
LeftArnold RightDanny
OR –Arnold&Danny ?
Quantum State Tomography
Indistinguishable
Photon Hilbert Space
2
H
,0V , 1H ,1V , 0 H , 2V
 HH
, HV  VH , VV


?
Distinguishable Photon
Hilbert Space
 H1H 2 , V1H 2 , H1V2 , V1V2

Yu. I. Bogdanov, et al
Phys. Rev. Lett. 93, 230503 (2004)
If we’re not sure whether or not the particles are distinguishable,
do we work in 3-dimensional or 4-dimensional Hilbert space?
If the latter, can we make all the necessary measurements, given
that we don’t know how to tell the particles apart ?
The Partial Density Matrix
The answer: there are only 10 linearly independent parameters which
are invariant under permutations of the particles. One example:
  HH, HH
 HV VH, HH

 HH, HV VH  HV VH, HV VH

 HV VH,VV
  HH,VV

Inaccessible

VV , HH 

VV , HV VH 

VV ,VV 


information



 HV VH, HV VH 
Inaccessible
information
The sections of the density matrix labelled
inaccessible correspond to information about
the ordering of photons with respect to
inaccessible degrees of freedom.
Experimental Apparatus
Experimental Results
No Distinguishing Info
Distinguishing Info
When distinguishing
information is introduced the
HV-VH component increases
without affecting the state in
the symmetric space
HH + VV
Mixture of
45–45 and –4545
More Photons…
If you have a collection of spins, what
are the permutation-blind observables
that describe the system?
They correspond to measurements
of angular momentum operators
J and mj ... for N photons, J runs to N/2
So the total number of operators accessible to measurement is
N /2
Number of ordering - blind ops   2 j  1   N  3 N  2  N  1 / 6
2
j
Total # of projectors  4 N
Total # of projectors onto symmetric states   N  1
2
3
Tomography in optical lattices,
and steps towards control...
Tomography in Optical Lattices
[Myrkog et al., quant-ph/0312210
Kanem et al., quant-ph/0506140]
Rb atom trapped in one of the quantum levels
of a periodic potential formed by standing
light field (30GHz detuning, 10s of mK depth)
Complete characterisation of
process on arbitrary inputs?
Towards QPT:
Some definitions / remarks
• "Qbit" = two vibrational states of atom in a well of a 1D lattice
• Control parameter = spatial shifts of lattice (coherently couple
states), achieved by phase-shifting optical beams (via AO)
• Initialisation: prepare |0> by letting all higher states escape
• Ensemble: 1D lattice contains 1000 "pancakes", each with
thousands of (essentially) non-interacting atoms.
No coherence between wells; tunneling is a decoherence mech.
• Measurement in logical basis: direct, by preferential tunneling
under gravity
• Measurement of coherence/oscillations: shift and then measure.
• Typical experiment:
• Initialise |0>
• Prepare some other superposition or mixture (use shifts, shakes, and delays)
• Allow atoms to oscillate in well
• Let something happen on its own, or try to do something
• Reconstruct state by probing oscillations (delay + shift +measure)
First task: measuring state
populations
Time-resolved quantum states
Recapturing atoms after setting
them
into
final
vs midterm,
bothoscillation...
adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
...or failing to recapture them
if you're too impatient
final vs midterm, both adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
Oscillations in lattice wells
(Direct probe of centre-of-mass oscillations in 1mm wells;
can be thought of as Ramsey fringes or Raman pump-probe exp’t.)
Quantum state reconstruction
p
p
t
Dx
x
Wait…
x
Shift…
p
Dx
x
Measure ground
state population
Q(0,0) = 1p Pg
W(0,0) = 1p S (-1)n Pn
(former for HO only; latter requires only symmetry)
Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96)
& Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96)
Husimi distribution of coherent state
Atomic state measurement
(for a 2-state lattice, with c0|0> + c1|1>)
initial state
displaced
delayed & displaced
left in
ground band
tunnels out
during adiabatic
lowering
(escaped during
preparation)
|c0|2
|c1|2
|c0 + c1 |2
|c0 + i c1 |2
Extracting a superoperator:
prepare a complete set of input states and measure each output
Likely sources of decoherence/dephasing:
Real photon scattering (100 ms; shouldn't be relevant in 150 ms period)
Inter-well tunneling (10s of ms; would love to see it)
Beam inhomogeneities (expected several ms, but are probably wrong)
Parametric heating (unlikely; no change in diagonals)
Other
Towards bang-bang error-correction:
pulse echo indicates T2 ≈ 1 ms...
comparing oscillations for shift-backs
applied after time t
2
Free-induction-decay signal for comparison
1.5
1/(1+2)
echo after “bang” at 800 ms
1
echo after “bang” at 1200 ms
0.5
echo after “bang” at 1600 ms
0
00
(bang!)
50
500
ms
100
1000
ms
150
1500
ms
200
2000
ms
250
t(10us)
decay of coherence introduced by echo pulses
themselves (since they are not perfect p-pulses)
Cf. Hannover experiment
Far smaller echo, but far better signal-to-noise ("classical" measurement of <X>)
Much shorter coherence time, but roughly same number of periods
– dominated by anharmonicity, irrelevant in our case.
Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000).
A better "bang" pulse for QEC?
position shift
(previous slides)
time
double shift
(similar to a momentum shift)
initial state
T = 900 ms
A = –60°
t=0
measurement
t
initial state
T = 900 ms
A = –60°
pulse
variable hold
delay = t
t=0
measurement
t
Under several (not quite valid) approximations, the double-shift is a
momentum displacement.
We expected a momentum shift to be at least as good as a position shift.
In practice: we want to test the idea of letting learning algorithms
search for the best pulse shape on their own, and this is a first step.
le shift-back
e
1
Echo from compound pulse
Pulseamplitude
900 us for
after
stateshift-back
preparation,
Echo
a single
vs.
a pulse (shift-back,
shift) at 900 us
and track delay,
oscillations
0.9
single-shift echo
(≈10% of initial oscillations)
0.8
0.7
0.6
double-shift echo
(≈30% of initial oscillations)
0.5
0.4
0.3
0
200
400
600
800
1000 1200 1400 1600
time ( microseconds)
Future: More parameters; find best pulse.
Step 2 (optional): figure out why it works!
Also: optimize # of pulses (given imperfection of each)
A pleasant surprise from
tomography…
To characterize processes such as our echo pulses, we extract the
completely positive map or “superoperator,” shown here in the
Choi-matrix representation:
(
)
0.109
0.003+0.007 i
-0.006+0.028 i
0.14+0.037 i
0.003-0.007 i
0.259
0.018+0.024 i
0.011-0.034 i
-0.006-0.028 i
0.018-0.024 i
0.414
-0.019-0.037 i
0.14-0.037 i
0.011+0.034 i
-0.019+0.037 i
0.202
Upper left-hand quadrant
indicates output density matrix
expected for a ground-state input
Ironic fact: when performing tomography, none of our inputs was
a very pure ground state, so in this extraction, we never saw Pe > 55%
or so, though this predicts 70% – upon observing this superoperator,
we went back and confirmed that our echo can create 70% inversion!
Data:"W-like" [Pg-Pe](x,p) for
a mostly-excited incoherent mixture
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Why does our echo decay?
Finite bath memory time:
So far, our atoms are free to move in the directions transverse to
our lattice. In 1 ms, they move far enough to see the oscillation
frequency change by about 10%... which is about 1 kHz, and hence
enough to dephase them.
What if we try “bang-bang”?
(Repeat pulses before the bath gets amnesia; trade-off since each pulse
is imperfect.)
Some coherence out to > 3 ms now...
How to tell how much of the
coherence is from the initial state?
Future: • Tailor phase & amplitude of successive pulses to cancel
out spurious coherence
• Study optimal number of pulses for given total time.
(Slow gaussian decay down to exponential?)
And now for something completely
different (?)
Can we talk about what goes on behind closed doors?
The Rub
What does that really mean?
Hint=gApx
System-pointer
coupling
By using a pointer with a big uncertainty (relative to the
strength of the measurement interaction), one can
obtain information, without creating entanglement
between system and apparatus (effective "collapse").
What will that look like?
A Gedankenexperiment...
ee-
e-
e-
" Quantum seeing in the dark "
(AKA: The Elitzur-Vaidman bomb experiment)
A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)
P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996)
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
The bomb must be there... yet
my photon never interacted with it.
What do you mean, interaction-free?
Measurement, by definition, makes some quantity certain.
This may change the state, and (as we know so well), disturb conjugate variables.
How can we measure where the bomb is without disturbing its momentum (for
example)?
But if we disturbed its momentum, where did the momentum go? What exactly
did the bomb interact with, if not our particle?
It destroyed the relative phase between two parts of the particle's wave function.
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
… if they4/16
were
Explosion
both in, they should
have annihilated!
What does this mean?
Common conclusion:
We've got to be careful about how we interpret these
"interaction-free measurements."
You're not always free to reason classically about what would
have happened if you had measured something other than what
you actually did.
(Do we really have to buy this?)
How to make the experiment
possible: The "Switch"
LO

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).

PUMP
2


LO
Coinc.
Counts
PUMP - 2 x LO
2 x LO
PUMP
+
2LO- PUMP = p
=
Experimental Setup
Det. V (D+) Det. H (D-)
50-50
BS2
CC
PBS
PBS
GaN
Diode Laser
DC BS
50-50
BS1
(W)
CC
V
H
Switch
DC BS
But what can we say about where the particles
were or weren't, once D+ & D– fire?
[Y. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, quant-ph/0104062]
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.
PROBLEM SOLVED!(?)
Two-Particle Weak Measurements
Problem: For two-particle weak measurements we need a strong nonlinearity to
implement a Von Neuman measurement interaction (Hint=gPÂ1Â2).
Solution: Do two single-particle weak measurements and study correlations →
• If Pointer1 and Pointer2 always move together, then the uncertainty in their
difference never changes.
• If Pointer1 and Pointer 2 both move, but never together, then
Δ(Pointer1-Pointer2) must increase.
Pointer Polarization Correlations for Â1Â2weak
D- Polarizer Angle (rad.)
D+ Polarizer Angle (rad.)
Weak Measurement for a
Polarization Pointer (N particles):
Spin Lowering Operator
Lundeen & Resch, Phys. Lett. A 334 (2005) 337–344
Resch & Steinberg, PRL 92,130402 (2004)
Weak Measurements in Hardy’s Paradox
Ideal Weak Values
N(I-)
N(O)
N(I+)
N(O+)
0
1
1
1
-1
0
1
0
Experimental Weak Values
N(I-)
N(O)
N(I+)
0.243±0.068
0.663±0.083
0.882±0.015
N(O+)
0.721±0.074
-0.758±0.083
0.087±0.021
•0.925±0.024
-0.039±0.023
Which-path controversy
(Scully, Englert, Walther vs the world?)
Suppose we perform a which-path measurement using a
microscopic pointer, e.g., a single photon deposited into
a cavity. Is this really irreversible, as Bohr would have all
measurements? Is it sufficient to destroy interference? Can
the information be “erased,” restoring interference?
Scully et al, Nature 351, 111(1991)
The debate since then...
Storey, Tan, Collett, & Walls proved that all WWMs must disturb
the momentum of any momentum eigenstate.
But Scully, Englert, and Walther were right in that every moment
of the momentum distribution of the two-slit wavefunction was
unchanged by their proposed WWM.
Wiseman and Harrison argued that aside from considering different
initial conditions, the two sides had different definitions of momentum
transfer (probability versus amplitude, roughly).
Shouldn’t one be able to measure some momentum transfer kernel,
regardless of the choice of initial state?
Typically, only by starting in momentum eigenstates.
Weak measurements
to the rescue!
To find the probability of a given momentum transfer,
measure the weak probability of each possible initial
momentum, conditioned on the final momentum
observed at the screen...
The distribution of the integrated
momentum-transfer
EXPERIMENT
THEORY
Note: the distribution
extends well beyond h/d.
On the other hand, all its moments
are (at least in theory, so far) 0.
Some concluding remarks/questions...
1. Quantum process tomography can be useful for
characterizing and "correcting" quantum systems
2. It taught us how to “invert” the c-o-m oscillation of atoms
3. What other quantities can one extract from superop’s?
4. How much control is possible with a single knob
(translating our lattice, e.g.), in the presence of strong
dephasing? How to find optimal processes?
5. It’s really expensive! How much will feedback help us do
more efficient tomography? In what circumstances can one
simply avoid tomography altogether?
6. “Effective” decoherence is very subtle when the
“environment” is a degree of freedom of the system itself
7. State-tomography ideas can be generalized to a situation
where experimentally indistinguishable particles may or
may not have some degree of distinguishing information
8. Weak measurements on subensembles are very strange...
but perhaps less strange that the paradoxes they resolve?