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!)

Aephraim M. Steinberg Centre for Q. Info. & Q. ControlInstitute for Optical SciencesDept. of Physics, U. of Toronto

CANADA

Manipulating and Measuring the Quantuum State of Photons and Atoms

QUEST 05, Santa Fe

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

DRAMATIS PERSONAE

Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( Barcelona)

Matt Partlow

Optics: Jeff Lundeen Kevin Resch(Zeilinger )

Lynden(Krister) Shalm Masoud Mohseni (Lidar)

Rob Adamson

Atoms: Jalani Fox Stefan Myrskog (Thywissen)

Ana Jofre(NIST) Mirco Siercke

Samansa Maneshi Chris Ellenor

Some theory collaborators: ...Daniel Lidar, Pete Turner, János Bergou, Mark Hillery, Paul Brumer, Howard Wiseman,...

There are many types of measurement!

• 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

A few to keep in mind:

OUTLINETomography – characterizing quantum

states & processes... brief review

Motional states of atoms in optical latticesProcess tomography Pulse echoInverted states, negative Wigner functions,...

Entangled photon pairs 2-photon process tomographyDirect measurement of purityGenerating entanglement by postselectionCharacterizing states with “inaccessible” info

Bonus topic if you don’t interrupt me enough:Weak measurements and “paradoxes”

(which-path debate; Hardy’s paradox)

Quantum tomography: what & why?

0

1. Characterize unknown quantum states & processes2. Compare experimentally designed states & processes to design goals3. Extract quantities such as fidelity / purity / tangle4. Have enough information to extract any quantities defined in the future!

• or, for instance, show that no Bell-inequality could be violated5. 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 ()CHCV

( )CHH

CHV

CVH

CVV

Density matrix: 2x2=4 coefficientsMeasure

intensity of horizontalintensity of verticalintensity of 45ointensity 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...

Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)Klose, Smith, Jessen, PRL 86 (21) 4721 (01)

Polarisation state of 3 photons(GHZ state)

Spin state of Cs atoms (F=4),in two bases

Much work on reconstruction of optical density matrices in the Kwiatgroup; theory advances due to Hradil & others, James & others, etc...;now a routine tool for characterizing new states, for testing gates orpurification protocols, for testing hypothetical Bell Inequalities, etc...

QPT of QFTWeinstein et al., J. Chem. Phys. 121, 6117 (2004)

To the trained eye, this is a Fourier transform...

From those superoperators, one can extract Kraus operatoramplitudes, and their structure helps diagnose the process.

Ancilla-assisted process tomography

Altepeter et al, PRL 90, 193601 (03)

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 statepreparations with coincidence measurement.

2-qbit state tomography with the entangledinput is equivalent to 1-qbit processtomography using 4 different inputs(and both require 16 measurements).

1-qbit processes representedas deformations of Bloch sphere...some unphysical results withengineered decoherence.

Quantum tomography experimentson photons, and how to avoid them

1

HWP

HWP

HWP

HWP

QWP

QWPQWP

QWPPBS

PBS

Argon Ion Laser

Beamsplitter"Black Box" 50/50

Detector B

Detector ATwo waveplates per photonfor state preparation

Two waveplates per photon for state analysis

SPDC source

Two-photon Process Tomography [Mitchell et al., PRL 91, 120402 (2003)]

Hong-Ou-Mandel Interference

How often will both detectors fire together?

r

r

tt

+

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 isunpolarized but the two are orthogonal.

This interferometer is a "Bell-state filter," neededfor quantum teleportation and other applications.

Our Goal: use process tomography to test this filter.

“Measuring” the superoperatorof a Bell-state filter

}

Output DM Input

HH

HV

VV

VH

}

}}

etc.

16 analyzer settings

16 input states

Coincidencences

[Mitchell et al., PRL 91, 120402 (2003)]

“Measuring” the superoperator

Input Output DM

HH

HV

VV

VH

etc.

Superoperator

Input Output

Superoperator after transformationto correct polarisation rotations:

Dominated by a single peak;residuals allow us to estimatedegree of decoherence andother errors.

Superoperator provides informationneeded to correct & diagnose operationMeasured superoperator,in Bell-state basis:

The ideal filter would have a single peak.Leading Kraus operator allowsus to determine unitary error.

(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 howlong to acquire data on that point.

(b) Could also measure populations first, and then avoidwasting time on coherences which would close to 0.

(c) Even if r has only a few significant eigenvalues, is therea 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 largerHilbert space, but need not characterize the rest.

A sample error model:the "Sometimes-Swap" gate

Consider an optical system withstray reflections – occasionally aphoton-swap occurs accidentally:

Two subspaces aredecoherence-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 whichto compute, with no prior knowledge of the error model.

Some strategies for a DFS search (simulation; experiment underway)

randomtomography

adaptivetomography

Best known2-D DFS (averagepurity).

# of input states used

averages

Our adaptive algorithm always identifies a DFS after testing 9 input states, while standard tomography routinely requires 16 (complete QPT).

standardtomography

# of inputs tested

puri

ty o

f be

st 2

D

DF

S f

ound

Surprise: in the absence of noise, simulations show that essentiallyany 2 input states suffice to identify the DFS (required max-lik towork). Project to revisit: add noise, do experiment, study scaling,...

• 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.

(T. A. Brun, e-print: quant-ph/0401067)

Polynomial Functions of a Polynomial Functions of a Density MatrixDensity Matrix

• 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

HOM as Singlet State Filter

+

Pure State on either side = 100% visibility

HH

HH

HH

H

HOM Visibility = Purity

Mixed State = 50% visibility

+V

HHH

HV V

V

Linear Purity of a Quantum State

P = 1 – 2 – –

Singlet-state probability can be measured by a singlet-state filter (HOM)

•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

Coincidence Circuit

Single PhotonDetector

Single PhotonDetector

€

λ /2

€

λ /2

Type 1 SPDC Crystal

Singlet Filter

QuartzSlab

QuartzSlab

Experimentally Measuring the Purity of a Qubit

Prepared the state |+45>

Measured Purity from Singlet State MeasurementP=0.92±0.02

Measured Purity from QSTP=0.99±0.01

Measuring +45 +45

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350

Delay (um)

Counts per 30 s

Results For a Pure 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 ]

€

λ /2

€

λ /2

Case 1: Same birefringence in each arm

V

V

H

H

100% interference

Visibility = (90±2) %

€

λ /2

€

λ /2

Case 2: Opposite birefringence in each arm

V

V

H

H

25% interference

H and V Completely Decohered Due to Birefringence

0

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200 250 300 350 400 450

Delay (um)

Counts per 30s

Visibility = (21±2) %

Preparing a Mixed State

The HOM isn’t actually insensitive to timing information.

• 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)

Not a singlet filter, but an “Antisymmetry Filter”

Randomly rotate the half-waveplates to produce |45> and |-45>

No Birefringence, Even Mixture of +45/+45 and +45/-45

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350

Delay (um)

Counts per 30 s

€

λ /2

€

λ /2

|45>

|45> or |-45>

Preliminary results

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.

Coincidence Circuit

€

λ /2

€

λ /2

Visibility = (45±2) %

Preparing a Mixed State

When the distinguishable isn’t…

2

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

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 forKLM-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!

M.W. Mitchell et al., Nature 429, 161 (2004);and cf. P. Walther et al., Nature 429, 158 (2004).

[See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002);J. Fiurásek, Phys. Rev. A 65, 053818 (2002)]˘

Practical schemes?

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!

|1>

a|0> + b|1> + c|2> a'|0> + b'|1> + c'|2>

The germ of the KLM ideaINPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

|1>

In particular: with a similar but somewhat more complicatedsetup, one can engineer

a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;effectively a huge self-phase modulation ( 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, andvery weak coherent states from a laser, such that if we detectthree photons, we can be pretty sure we got only one from thelaser and only two from the down-conversion...

SPDC

laser

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

|0> + |1> + O(2)

|3> + O(3) + O(2) + terms with <3 photons

Trick #2

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

Yes, it's that easy! If you see three photonsout one port, then they all went out that port.

"mode-mashing"

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 apolarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be anywhere...

(or nothing)

(or <2 photons)

(or nothing)

-

+ei3φ

HWP

QWP

Phaseshifter

PBS

DCphotons PP

Dark ports

Ti:sa

to analyzer

The basic optical scheme

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

Generating / measuring other states

With perfect detectors and perfect single-photon sources, such schemescan easily be generalized.

With one or the other (and typically some feedback), many states may besynthesized by iteratively adding or subtracting photons, and in some casesimplementing 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 canbe used to implement arbitrary unitaries, and hence to entangle anything.

“Continuous” photon subtraction (& counting) can be used, even withinefficient detectors, to reconstruct the entire photon-number distribution.

Fundamentally Indistinguishablevs.

Experimentally Indistinguishable

But what if when we combine our photons,there is some residual distinguishing information:some (fs) time difference, some small spectraldifference, some chirp, ...?

This will clearly degrade the state – but how dowe characterize this if all we can measure ispolarisation?

LeftArnold RightDanny

OR –Arnold&Danny ?

Quantum State Tomography

{ }21212121 ,,, VVVHHVHH

Distinguishable Photon Hilbert Space

{ }, ,HH HV VH VV+

{ }2 ,0 , 1 ,1 , 0 ,2H V H V H V

Indistinguishable Photon Hilbert Space

??Yu. I. Bogdanov, et alPhys. 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, giventhat we don’t know how to tell the particles apart ?

The sections of the density matrix labelled inaccessible correspond to information about the ordering of photons with respect to inaccessible degrees of freedom.

The Partial Density Matrix

Inaccessible

information

Inaccessible

information

€

HH ,HH ρ HV +VH ,HH ρVV ,HH

ρ HH ,HV +VH ρ HV +VH ,HV +VH ρVV ,HV +VH

ρ HH ,VV ρ HV +VH ,VV ρVV ,VV

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

ρ HV −VH ,HV −VH( )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example:

Experimental Apparatus

When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space

Experimental ResultsNo Distinguishing Info Distinguishing Info

HH + VVMixture of45–45 and –4545

More Photons…

So the total number of operators accessible to measurement is

( ) ( )( )( )

( )2

2/2

1states symmetric onto projectors of # Total

4projectors of # Total

6/12312ops blind-ordering ofNumber

+=

=

+++=+= ∑

N

NNNj

N

N

j

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

Tomography in optical lattices, and steps towards control...

3

Rb atom trapped in one of the quantum levelsof a periodic potential formed by standing

light field (30GHz detuning, 10s of K depth)

Tomography in Optical Lattices

[Myrkog et al., quant-ph/0312210Kanem et al., quant-ph/0506140]

Complete characterisation ofprocess 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 oscillation...

...or failing to recapture themif you're too impatient

Oscillations in lattice wells(Direct probe of centre-of-mass oscillations in 1m wells;can be thought of as Ramsey fringes or Raman pump-probe exp’t.)

x

p

ωt

Wait…

Quantum state reconstruction

x

p

Shift…

x

Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96)& Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96)

Measure groundstate population

x

p

x

(former for HO only; latter requires only symmetry)

Q(0,0) = Pg1

W(0,0) = (-1)n Pn1

Husimi distribution of coherent state

Atomic state measurement(for a 2-state lattice, with c0|0> + c1|1>)

left inground band

tunnels outduring adiabaticlowering

(escaped duringpreparation)

initial state displaced delayed & displaced

|c0|2 |c0 + c1 |2 |c0 + i c1 |2

|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 s 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

0

0.5

1

1.5

2

0 50 100 150 200 250

comparing oscillations for shift-backs applied after time t

1/(1+2)

t(10us)

0 500 s 1000 s 1500 s 2000 s

Towards bang-bang error-correction:pulse echo indicates T2 ≈ 1 ms...

Free-induction-decay signal for comparison

echo after “bang” at 800 ms

echo after “bang” at 1200 ms

echo after “bang” at 1600 ms

decay of coherence introduced by echo pulses themselves (since they are not perfect -pulses)

(bang!)

Cf. Hannover experiment

Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000).

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.

A better "bang" pulse for QEC?

Under several (not quite valid) approximations, the double-shift is amomentum 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 algorithmssearch for the best pulse shape on their own, and this is a first step.

A = –60°

t

t = 0

T = 900 s

measurement

time

position shift (previous slides)

initial state

A = –60°

t

variable holddelay = t = 0

T = 900 s

pulse

measurement

double shift (similar to a momentum shift)

initial state

time ( microseconds)

single-shift echo(≈10% of initial oscillations)

double-shift echo (≈30% of initial oscillations)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200 1400 1600

Echo amplitude for a single shift-back vs. a pulse (shift-back, delay, shift) at 900 us

Single shift-backpulse

ground state ratio

Echo from compound pulse

Future: More parameters; find best pulse.Step 2 (optional): figure out why it works!Also: optimize # of pulses (given imper-fection of each)

Pulse 900 us after state preparation,and track oscillations

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

i0.202

A pleasant surprise from tomography…

To characterize processes such as our echo pulses, we extract theTo characterize processes such as our echo pulses, we extract thecompletely positive map or “superoperator,” shown here in thecompletely positive map or “superoperator,” shown here in theChoi-matrix representation:Choi-matrix representation:

( )Upper left-hand quadrant indicates output density matrixexpected for a ground-state input

Ironic fact: when performing tomography, none of our inputs wasa 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!

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

Data:"W-like" [Pg-Pe](x,p) for a mostly-excited incoherent mixture

Why does our echo decay?

Finite bath memory time:

So far, our atoms are free to move in the directions transverse toour lattice. In 1 ms, they move far enough to see the oscillationfrequency change by about 10%... which is about 1 kHz, and henceenough to dephase them.

What if we try “bang-bang”?(Repeat pulses before the bath gets amnesia; trade-off since each pulseis 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 cancelout spurious coherence

• Study optimal number of pulses for given total time.(Slow gaussian decay down to exponential?)

Can we talk about what goes on behind closed doors?

And now for something completely different (?)

The Rub

By using a pointer with a big uncertainty (relative to thestrength of the measurement interaction), one canobtain information, without creating entanglement between system and apparatus (effective "collapse").

Hint=gApx

System-pointercoupling

What does that really mean?

What will that look like?

A Gedankenexperiment...

e-

e-

e-

e-

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 thatthey will not blow up when triggered.

Is there any way to identify the working bombs (orsome of them) without blowing them up?

" 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)

BS1

BS2

DC

Bomb absent:Only detector C fires

Bomb present:"boom!" 1/2 C 1/4 D 1/4

The bomb must be there... yetmy 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 exactlydid the bomb interact with, if not our particle?

It destroyed the relative phase between two parts of the particle's wave function.

BS1-

e-

BS2-

O-

C-D-

I-

BS1+

BS2+

I+

e+

O+

D+C+

W

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

Hardy's Paradox

D- e+ was in

D+D- ?

But … if they wereboth in, they shouldhave annihilated!

D+ e- was in

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"

PUMP

ω

ω

Coinc.Counts2ω

PUMP - 2 x LO

LO

ω

ωLO

+

PUMP 2 x LO

=

2LO- PUMP =

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

GaNDiode Laser

PBS

Det. H (D-)Det. V (D+)

DC BS DC BS

50-50BS1

50-50BS2

Switch H

V

CC

CC

PBS

BS1-

BS2-

O-

C-

BS1+

BS2+

I+

e+ e-

I-O+

D+C+ D-

WBS1-

BS2-

O-

C-

BS1+

BS2+

I+

e+ e-

I-O+

D+C+ D-

W

Experimental Setup

(W)

Probabilities e- in e- out

e+ in 1

e+ out 0

1 0

0 1

1 1

But what can we say about where the particles were or weren't, once D+ & D– fire?

Upcoming experiment: demonstrate that "weakmeasurements" (à la Aharonov + Vaidman) willbear out these predictions.

[Y. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, quant-ph/0104062]

PROBLEM SOLVED!(?)

Problem: For two-particle weak measurements we need a strong nonlinearity to implement a Von Neuman measurement interaction (Hint=gPÂ1Â2).

Two-Particle Weak Measurements

Pointer Polarization Correlations for Â1Â2weak

D+ Polarizer Angle (rad.)

D- P

olar

izer

Ang

le (

rad.

)

Weak Measurement for a Polarization Pointer (N particles):

Resch & Steinberg, PRL 92,130402 (2004)

Lundeen & Resch, Phys. Lett. A 334 (2005) 337–344

• 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.

Spin Lowering Operator

Solution: Do two single-particle weak measurements and study correlations →

N(I-) N(O)

N(I+) 0 1 1N(O+) 1 -1 0

1 0

Weak Measurements in Hardy’s Paradox

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

Experimental Weak Values

Ideal Weak Values

Which-path controversy(Scully, Englert, Walther vs the world?)

Suppose we perform a which-path measurement using amicroscopic pointer, e.g., a single photon deposited intoa cavity. Is this really irreversible, as Bohr would have allmeasurements? Is it sufficient to destroy interference? Canthe 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 momentof the momentum distribution of the two-slit wavefunction wasunchanged by their proposed WWM.

Wiseman and Harrison argued that aside from considering differentinitial conditions, the two sides had different definitions of momentumtransfer (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 initialmomentum, 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 momentsare (at least in theory, so far) 0.

Some concluding remarks/questions...1. Quantum process tomography can be useful for

characterizing and "correcting" quantum systems2. It taught us how to “invert” the c-o-m oscillation of atoms3. 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?