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Some recent experiments on weak measurements and quantum state generation. Aephraim Steinberg Univ. Toronto (presently @ Institut d'Optique, Orsay). Let's Make a Quantum Deal!. Let's Make a Quantum Deal!. Let's Make a Quantum Deal!. OUTLINE. The 3-box problem

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some recent experiments on weak measurements and quantum state generation
Some recent experimentson weak measurementsand quantum state generation

Aephraim Steinberg

Univ. Toronto

(presently @ Institut

d'Optique, Orsay)


Let's Make a Quantum Deal!

Let's Make a Quantum Deal!

Let's Make a Quantum Deal!

  • The 3-box problem
    • Overture: an alternative introduction to
    • retrodiction, the 3-box problem, and weak measurements
    • Experimental results
    • Nonlocality?
  • Hardy's Paradox and retrodiction
    • Retrodiction is claimed to lead to a paradox in QM
    • "Weak probabilities" seem to "resolve" the "paradox"?
    • Experiment now possible, thanks to 2-photon "switch"
  • Which-path experiments (collab. w/ Howard Wiseman)
    • Old debate (Scully vs. Walls, e.g.):
    • When which-path measurements destroy interference, must
    • momentum necessarily be disturbed?
    • Weak values allow one to discuss this momentum shift, and
      • reconcile some claims of both sides
    • (Negative values essential, once more...)
  • And now for something completely different
    • Non-deterministic generation of |0,3> + |3,0>
        • "maximally path-entangled states"
    • Phase super-resolution (Heisenberg limit?)

U of T quantum optics & laser cooling group:

PDFs:Morgan Mitchell Marcelo Martinelli (back Brazil)

Optics: Kevin Resch(Zeilinger) Jeff Lundeen

Krister Shalm Masoud Mohseni (Lidar)

Reza Mir[real world(?)] Rob Adamson

Karen Saucke (back )

Atom Traps: Jalani Fox Stefan Myrskog (Thywissen)

Ana Jofre (NIST?) Mirco Siercke

Samansa Maneshi Salvatore Maone ( real world)

Chris Ellenor

Some of our theory collaborators:

Daniel Lidar, János Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman

recall principle of weak measurements




Recall principle of weak measurements...

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").

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...







By the same token, no single event provides much information...
predicting the past




Predicting the past...

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

A + B

= X+B+Y



B + C =


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 redefinition of the redefinition

X + B'

= X+B+Y



X + C' =


A redefinition of the redefinition...

So: the very same logic leads us to conclude the

particle was definitely in box X.

what does this mean
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
The 3-box problem: weak msmts

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.

Prepare a particle in a symmetric superposition of

three boxes: A+B+C.

Look to find it in this other superposition:


Ask: between preparation and detection, what was

the probability that it was in A? B? C?


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)]

an application n shutters
An "application": N shutters

Aharonov et al., PRA 67, 42107 ('03)

the implementation a 3 path interferometer resch et al phys lett a 324 125 04
The implementation – A 3-path interferometer(Resch et al., Phys Lett A 324, 125('04))

Diode Laser

Spatial Filter: 25um PH, a 5cm and a 1” lens





MS, fA



BS3, 50/50



BS4, 50/50


MS, fC




the pointer

Use transverse position of each photon as pointer

  • Weak measurements can be performed by tilting a glass optical flat, where effective


Mode A



The pointer...

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


(neg. shift!)

Rail C

(pos. shift)

Rails A and B (no shift)

A negative weak value

Perform weak msmt

on rail C.

Post-select either A,

B, C, or A+B–C.

Compare "pointer

states" (vertical


[There exists a natural optical explanation for

this classical effect – this is left as an exercise!]

data for p a p b and p c
Data for PA, PB, and PC...

Rails A

and B

Rail C




is the particle really in 2 places at once
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


- 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 p ab
Practical Measurement of PAB

Resch &Steinberg, PRL 92,130402 ('04)

Use two pointers (the two transverse directions)

and couple to both A and B; then use their

correlations to draw conclusions about PAB.

We have shown that the real part of PABW can be extracted from such correlation


non repeatable data which happen to look the way we want them to


particle model

exact calculation

no correlations

(PAB = 1)

Non-repeatable data which happen to look the way we want them to...
and a final note
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.

quantum seeing in the dark
" Quantum seeing in the dark "





The bomb must be there... yet

my photon never interacted with it.

(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)


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

what do you mean interaction free
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


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
Hardy's Paradox
















D+ e- was in

D- e+ was in

D+D- ?

But … if they were

both in, they should

have annihilated!

what does this mean24
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"


K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).

2LO- PUMP = 






PUMP - 2 x LO


2x LO




Experimental Setup

Det. V (D+)

Det. H (D-)









Diode Laser









H Pol DC

V Pol DC

407 nm Pump

but what can we say about where the particles were or weren t once d d fire
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]





Upcoming experiment: demonstrate that "weak

measurements" (à la Aharonov + Vaidman) will

bear out these predictions.


An experimental implementation of Hardy’s Paradox is now possible.

  • A single-photon level switch allows photons to interact with a high efficiency.
  • A polarization based system is now running.
  • Once some stability problems solved, we will look at the results of weak measurements in Hardy’s Paradox.
which path controversy scully englert walther vs the world
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?

how is complementarity enforced
How is complementarity enforced?

The fringe pattern (momentum distribution) is clearly changed –

yet every moment of the momentum distribution remains the same.

weak measurements to the rescue
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
Convoluted implementation...

Glass plate in focal

plane measures

P(pi) weakly (shifting

photons along y)


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.

a few distributions p p i p f
A few distributions P(pi | pf)



(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
The distribution of the integrated momentum-transfer



Note: the distribution

extends well beyond h/d.

On the other hand, all its moments

are (at least in theory, so far) 0.


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 which-path 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).
highly number entangled states low noon experiment
Highly number-entangled states("low-noon" experiment) .

Morgan W. Mitchell et al., to appear

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.

A number of proposals for producing these states have been made,

but so far none has been observed for n>2.... until now!

(But cf. related work in Vienna)

practical schemes
Practical schemes?

[See for example

H. Lee et al., Phys. Rev. A 65, 030101 (2002);

J. Fiurásek, Phys. Rev. A 65, 053818 (2002)]




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!

Important factorisation:

trick 1
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...

|0> + e |2> + O(e2)

 |3> + O(2) + O( 2)

+ terms with <3 photons

|0> +  |1> + O(2)

trick 2
Trick #2


Yes, it's that easy! If you see three photons

out one port, then they all went out that port.

How to combine three non-orthogonal photons into one spatial mode?

trick 3
Trick #3

(or nothing)

(or nothing)

(or <2 photons)

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


it works
It works!





  • Three-box paradox implemented
  • Some more work possible on nonlocal observables
  • Hardy's paradox implemented
  • Setting up to perform the joint weak measurements
  • Wiseman's proposal re which-path measurements carried out
  • Paper in preparation
  • What to do next? (Suggestions welcome!)
  • 3-photon entangled state produced.
  • What next? (Probably new sources required.)
  • Other things I didn't have time to tell you about:
  • Process tomography working in both photonic and atomic systems.
  • Next steps: adaptive error correction (bang-bang, DFS,...)
  • Optimal (POVM) discrimination of non-orthogonal states
  • Using decoherence-free-subspaces for optical implementations of q. algorithms
  • BEC project .... plans to probe tunneling atoms in the forbidden region
  • Coherent control of quantum chaos in optical lattices
  • Tunneling-induced coherence " " "
some references
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)

Aharonov et al, PRL 60, 1351 (1988)

Ritchie, Story, & Hulet, PRL 66, 1107 (1991)

Wiseman, PRA 65, 032111

Brunner et al., quant-ph/0306108

Resch and Steinberg, quant-ph/0310113

Which-path debate:

Scully et al, Nature 351, 111(1991)

Storey et al, Nature 367 (1994) etc

Wiseman & Harrison, N 377,584 (1995)

Wiseman, PLA 311, 285 (2003)

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