Weak measurements:
from the 3-box problem to Hardy's
Paradox to the which-path debate
• The 3-box problem
• Another case where airtight classical reasoning yields seemingly
contradictory information
• Experimental consequences of this information
• Actual experiment!
• Weak measurements shed light on Hardy's Paradox as well
• "Weak probabilities" obey all the constraints we expected in that example.
• There is no contradiction, because if negative "probabilities" are accepted, all
the joint probabilities can be constructed.
• Which-path experiments
• Recall the debate: is momentum disturbed or not?
How could
one determine whether or not momentum had changed?
• Weak measurement predictions share some of the properties claimed by
Scully et al. and some of those claimed by Walls et al.
• More negative probabilities needed...
Let's Make a Quantum Deal!
Let's Make a Quantum Deal!
• Conclusion
• These "probabilities" are real, measurable things...
• ...but what in the world are they?
25 Nov 2003
(some material thanks to Kevin Resch,
Reza Mir, Howard Wiseman,...)
Recall principle of weak
measurements...
Initial State of Pointer
Final Pointer Readout
Hint=gApx
x
System-pointer
coupling
By using a pointer with a big uncertainty, one can
prevent entanglement ("collapse").
x
By the same token, no single event
provides much information...
Initial State of Pointer
Final Pointer Readout
But after many trials, the centre can be determined
to arbitrarily good precision...
x
x
x
x
Predicting the past...
A+B
B+C
What are the odds that the particle
was in a given box (e.g., box B)?
It had to be in B, with 100% certainty.
Consider some redefinitions...
In QM, there's no difference between a box and any other state
(e.g., a superposition of boxes).
What if A is really X + Y and C is really X - Y?
A+B
= X+B+Y
X
Y
B+C=
X+B-Y
A redefinition of the redefinition...
So: the very same logic leads us to conclude the
particle was definitely in box X.
X + B'
= X+B+Y
X
Y
X + C' =
X+B-Y
What does this mean?
Then we conclude that if you prepare in (X + Y) + B
and postselect in (X - Y) + B, you know the particle was
in B.
But this is the same as preparing (B + Y) + X and
postselecting (B - Y) + X, which means you also know
the particle was in X.
If P(B) = 1 and P(X) = 1, where was the particle really?
But back up: is there any physical sense in which this is true?
What if you try to observe where the particle is?
The 3-box problem: weak msmts
Prepare a particle in a symmetric superposition of
three boxes: A+B+C.
Look to find it in this other superposition:
A+B-C.
Ask: between preparation and detection, what was
the probability that it was in A? B? C?
Aw 
f Ai
f i
PA = < |A><A| >wk = (1/3) / (1/3) = 1
PB = < |B><B| >wk = (1/3) / (1/3) = 1
PC = < |C><C|>wk = (-1/3) / (1/3) = -1.
Questions:
were these postselected particles really all in A and all in B?
can this negative "weak probability" be observed?
[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]
Remember that test charge...
ee-
e-
e-
Aharonov's N shutters
PRA 67, 42107 ('03)
The implementation –
A 3-path interferometer
(Resch et al., quant-ph/0310091)
Diode Laser
Spatial Filter: 25um PH, a 5cm and a 1” lens
l/2
GP A
BS1, PBS
l/2
MS, fA
GP B
BS2, PBS
GP C
BS4,
50/50
MS, fC l/2
BS3, 50/50
PD
CCD
Camera
Screen
The pointer...
• Use transverse position of each photon as pointer
• Weak measurements can be performed by tilting a
glass optical flat, where effective
Hint  g A A p x
Mode A
q
Flat
gt
cf. Ritchie et al., PRL 68, 1107 ('91).
The position of each photon is uncertain to within the beam waist...
a small shift does not provide any photon with distinguishing info.
But after many photons arrive, the shift of the beam may be measured.
A negative weak value
Intensity (arbitrary units)
Perform weak msmt
on rail C.
Post-select either A,
B, C, or A+B–C.
Compare "pointer
states" (vertical
profiles).
1.4
1.2
A+B–C
(neg. shift!)
Rail C
(pos. shift)
1
0.8
0.6
Rails A and B (no shift)
0.4
220
200
180
[There exists a natural optical explanation for
this classical effect – this is left as an exercise!]
160
140
Pixel Number
120
100
Post-selected state displacement
(Units of RMS Width)
Data for PA, PB, and PC...
2
Rails A
and B
1
0
Rail C
-1
WEAK
STRONG
STRONG
-2
-3
-2
-1
0
1
2
Displacement of Individual Rail
(Units of RMS Width)
3
Is the particle "really" in
2 places at once?
• If PA and PB are both 1, what is PAB?
• For AAV’s approach, one would need an
interaction of the form
Hint  g A A B B p x
OR: STUDY CORRELATIONS OF PA & PB...
- if PA and PB always move together, then
the uncertainty in their difference never changes.
- if PA and PB both move, but never together,
then D(PA - PB) must increase.
Practical Measurement of PAB
Use two pointers (the two transverse directions)
and couple to both A and B; then use their
correlations to draw conclusions about PAB.
Hint  g A A A p x  g B B B p y
We have shown that the real part of PABW
can be extracted from such correlation
measurements:
Re PABW  
2 xy
g Ag Bt
2
- Re(P
*
AW
BBW )
Non-repeatable data which happen
to look the way we want them to...
anticorrelated
particle model
exact calculation
no correlations
(PAB = 1)
And a final note...
The result should have been obvious...
|A><A| |B><B|
= |A><A|B><B|
is identically zero because
A and B are orthogonal.
Even in a weak-measurement sense, a particle
can never be found in two orthogonal states at
the same time.
Hardy's Paradox
C+
D+
D-
BS2+
C-
But … if they were
both in, they should
have annihilated!
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
D- e+ was in
D+D- both were in?
Outcome Prob
D+ and C- 1/16
D- and C+ 1/16
C+ and C- 9/16
D+ and D- 1/16
Explosion 4/16
But what can we say about where the particles
were or weren't, once D+ & D– 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.
PROBLEM SOLVED!(?)
Quantum Eraser
(Scully, Englert, Walther)
Suppose we perform a which-path measurement using a
microscopic pointer, z.B., 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?
Which-path measurements destroy
interference (modify p-distrib!)
How is complementarity enforced?
The fringe pattern (momentum distribution) is clearly changed –
yet every moment of the momentum distribution remains the same.
The debate since then...
Why the ambiguity?
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...
Convoluted implementation...
Glass plate in focal
plane measures
P(pi) weakly (shifting
photons along y)
Half-half-waveplate
in image plane measures
path strongly
CCD in Fourier plane measures
<y> for each position x; this
determines <P(pi)>wk for each
final momentum pf.
Calibration of the weak
measurement
A few distributions P(pi | pf)
EXPERIMENT
THEORY
(finite width due to finite
width of measuring plate)
Note: not delta-functions; i.e., momentum may have changed.
Of course, these "probabilities" aren't always positive, etc etc...
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.
CONCLUSIONS
• Weak-measurement theory can predict the output of meas-urements
without specific reference to the measurement technique.
• They are consistent with the surprising but seemingly airtight
conclusions classical logic yields for the 3-box problem and for
Hardy's Paradox.
• They also shed light on tunneling times, on the debate over whichpath measurements, and so forth.
• Of course, they are merely a new way of describing predictions
already implicit in QM anyway.
• And the price to pay is accepting very strange (negative, complex,
too big, too small) weak values for observables (inc. probabilities).
Some references
Tunneling times et cetera:
Hauge and Støvneng, Rev. Mod. Phys. 61, 917 (1989)
Büttiker and Landauer, PRL 49, 1739 (1982)
Büttiker, Phys. Rev. B 27, 6178 (1983)
Steinberg, Kwiat, & Chiao, PRL 71, 708 (1993)
Steinberg, PRL 74, 2405 (1995)
Weak measurements:
Aharonov & Vaidman, PRA 41, 11 (1991)
Which-path debate:
Aharonov et al, PRL 60, 1351 (1988)
Scully et al, Nature 351, 111(1991)
Ritchie, Story, & Hulet, PRL 66, 1107 (1991)
Storey et al, Nature 367 (1994) etc
Wiseman, PRA 65, 032111
Wiseman & Harrison, N 377,584 (1995) Brunner et al., quant-ph/0306108
Wiseman, PLA 311, 285 (2003)
Resch and Steinberg, quant-ph/0310113
Hardy's Paradox:
Hardy, PRL 68, 2981 (1992)
Aharonov et al, PLA 301, 130 (2001).
The 3-box problem:
Aharonov et al, J Phys A 24, 2315 ('91);
PRA 67, 42107 ('03)
Resch, Lundeen, & Steinberg, quant-ph/0310091