Aephraim M. Steinberg Centre for Q. Info. & Q. Control

Institute for Optical SciencesDept. of Physics, U. of Toronto

Real-World Quantum Measurements:Fun With Photons and Atoms

CAP 2006, Brock University

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DRAMATIS PERSONÆToronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( ICFO)

Matt Partlow An-Ning Zhang Optics: Rob Adamson Kevin Resch(Zeilinger )

Lynden(Krister) Shalm Masoud Mohseni (Lidar)Xingxing Xing Jeff Lundeen (Walmsley)

Atoms: Jalani Fox (...Hinds) Stefan Myrskog (Thywissen)Ana Jofre(Helmerson) Mirco SierckeSamansa Maneshi Chris Ellenor Rockson Chang Chao Zhuang

Current ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan DarabiSome helpful theorists: Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...

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!)

Quantum Computer Scientists

OUTLINEThe grand unified theory of physics talks:

“Never underestimate the pleasure peopleget from being told something they already know.”

Beyond the standard model:“If you don’t have time to explain something

well, you might as well explain lots of things poorly.”

OUTLINEMeasurement: this is not your father’s observable!

• {Forget about projection / von Neumann}

• “Generalized” quantum measurement• Weak measurement (postselected quantum systems)

• “Interaction-free” measurement, ...• Quantum state & process tomography• Measurement as a novel interaction (quantum logic)• Quantum-enhanced measurement• Tomography given incomplete information

• {and many more}

Distinguishing the indistinguishable...

1

How to distinguish non-orthogonal states optimally

Use generalized (POVM) quantum measurements. [see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys.

Rev. A 66, 032315 (2002).]

The view from the laboratory:A measurement of a two-state system can onlyyield two possible results.

If the measurement isn't guaranteed to succeed, thereare 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.

H-polarized photon 45o-polarized photon

vs.

The geometric picture

Two non-orthogonal vectorsThe same vectors rotated so theirprojections onto x-y are orthogonal

(The z-axis is “inconclusive”)

90o

1

2

1

But a unitary transformation in a 4D space produces:

…and these states can be distinguished with certaintyup to 55% of the time

A test caseConsider these three non-orthogonal states:

Projective measurements can distinguish these stateswith certainty no more than 1/3 of the time.

(No more than one member of an orthonormal basis is orthogonal to two of the above states, so only one pair may be ruled out.)

Experimental schematic

(ancilla)

A 14-path interferometer for arbitrary 2-qubit unitaries...

Success!

The correct state was identified 55% of the time--Much better than the 33% maximum for standard measurements.

"I don't know"

"Definitely 3"

"Definitely 2""Definitely 1"

M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, 200403 (2004)

Can we talk about what goes on behind closed doors?

(“Postselection” is the big new buzzword in QIP...but how should one describe post-selected states?)

Conditional measurements(Aharonov, Albert, and Vaidman)

Prepare a particle in |i> …try to "measure" some observable A…postselect the particle to be in |f>

Does <A> depend more on i or f, or equally on both?Clever answer: both, as Schrödinger time-reversible.

Conventional answer: i, because of collapse.

ii ffMeasurement of A

Reconciliation: measure A "weakly."Poor resolution, but little disturbance.

…. can be quite odd …

ifiAf

A =w

AAV, PRL 60, 1351 ('88)

Predicting the past...

A+B

What are the odds that the particlewas in a given box (e.g., box B)?

B+C

A+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

B + C =X+B-Y

X Y

A redefinition of the redefinition...

X + B'= X+B+Y

X + C' =X+B-Y

X Y

So: the very same logic leads us to conclude theparticle was definitely in box X.

The Rub

A (von Neumann) Quantum Measurement of A

Well-resolved statesSystem and pointer become entangled

Decoherence / "collapse"Large back-action

Initial State of Pointer

x x

Hint=gApx

System-pointercoupling

Final Pointer Readout

A Weak Measurement of A

Poor resolution on each shot. Negligible back-action (system & pointer separable)

Mean pointer shift is given by <A>wk.

Hint=gApx

System-pointercouplingx

Initial State of Pointer

x

Final Pointer Readout

Need not lie within spectrum of A, or even be real...

The 3-box problem: weak msmtsPrepare 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?

Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed?

PA = < |A><A| >wk = (1/3) / (1/3) = 1PB = < |B><B| >wk = (1/3) / (1/3) = 1

PC = < |C><C|>wk = (-1/3) / (1/3) = 1.ifiAf

A =w

[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

A Gedankenexperiment...

e-

e-

e-

e-

0.4

0.6

0.8

1

1.2

1.4

100120140160180200220

Intensity (arbitrary units)

Pixel Number

Rails A and B (no shift)

Rail C(pos. shift)

A+B–C(neg. shift!)

A negative weak value for Prob(C)Perform weak msmt on rail C.

Post-select either A, B, C, or A+B–C.

Compare "pointer states" (vertical profiles).

K.J. Resch, J.S. Lundeen, and A.M. Steinberg, Phys. Lett. A 324, 125 (2004).

Seeing without looking

2a

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QuickTime™ and aTIFF (Uncompressed) decompressor

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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.

BS1-

e-

BS2-

O-

C-D-

I-

BS1+

BS2+

I+

e+

O+

D+C+

W

Outcome ProbD+ 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(for Elitzur-Vaidman “interaction-free measurements”)

D- –> e+ was in

D+D- –> ?

But … if they wereboth in, they shouldhave annihilated!

D+ –> e- was in

Probabilities e- in e- oute+ 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?

In fact, this is precisely what Aharonov et al.’s weak measurementformalism predicts for any sufficiently gentle attempt to “observe”these probabilities...

N(I-) N(O)

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

1 0

0.039±0.0230.925±0.024

0.087±0.0210.758±0.0830.721±0.074N(O+)

0.882±0.0150.663±0.0830.243±0.068N(I+)

N(O)N(I-)

Experimental Weak Values (“Probabilities”?)

Ideal Weak Values

Weak Measurements in Hardy’s Paradox

Quantum tomography: what & why?

3

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

Quantum InformationWhat's so great about it?

What makes a computer quantum?

If a quantum "bit" is described by two numbers: |Ψ> = c0|0> + c1|1>,

then n quantum bits are described by 2n coeff's:|Ψ> = c00..0|00..0>+c00..1|00..1>+...c11..1|11..1>;

this is exponentially more information than the 2n coefficients it would take to describe n independent (e.g., classical) bits.

It is also exponentially sensitive to decoherence.

Photons are ideal carriers of quantum information theycan be easily produced, manipulated, and detected, anddon't interact significantly with the environment. Theyare already used to transmit quantumcryptographicinformation through fibres under Lake Geneva, and soonthrough the air up to satellites.

Unfortunately, they don't interact with each other very mucheither! How to make a logic gate?

across the Danube

(...Another talk, or more!)

We need to understand the nature of quantum information itself.

How to characterize and compare quantum states? How to most fully describe their evolution in a given system? How to manipulate them?

The danger of errors & decoherence grows exponentially with system size.The only hope for QI is quantum error correction.We must learn how to measure what the system is doing, and then correct it.

(One partial answer...)

Density matrices and superoperatorsOne 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.

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...

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Wigner function of atoms’ vibrational quantum state in optical lattice

J.F. Kanem, S. Maneshi, S.H. Myrskog, and A.M. Steinberg, J. Opt. B. 7, S705 (2005)

Superoperator after transformationto correct polarisation rotations:

Residuals allow us to estimatedegree of decoherence andother errors.

Superoperator provides informationneeded to correct & diagnose operation

Measured superoperator,in Bell-state basis:

Leading Kraus operator allowsus to determine unitary error.

(expt) (predicted)

M.W. Mitchell, C.W. Ellenor, S. Schneider, and A.M. Steinberg, Phys. Rev. Lett. 91 , 120402 (2003)

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

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

The status of quantum cryptography

Measurement as a tool:Post-selective operations for the construction of

novel (and possibly useful) entangled states...

4a

Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)

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

Important factorisation:

=+

A "noon" stateA 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!

States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states.

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> + e |2> + O(e2)

|0> + a |1> + O(a2)

ea |3> + O(a3) + O(e2) + terms with <3 photons

* But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

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

Postselective nonlinearityPostselective nonlinearity

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

"mode-mashing"

-

+ei3φ

HWP

QWPPhaseshifter

PBS

DCphotons PP

Dark ports

Ti:sa

to analyzer

The basic optical scheme

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)

Complete characterisationwhen you have incomplete information

4b 4b

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?

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

€

rHH ,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:

(For n photons, the # of parameters scales as n3, rather than 4n)

R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, quant-ph/0601134

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

The moral of the story1. Post-selected systems often exhibit surprising behaviour which

can be probed using weak measurements.

2. Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states.

3. POVMs, or generalized quantum measurements, are in some ways more powerful than textbook-style projectors

4. Quantum process tomography may be useful for characterizing and "correcting" quantum systems (ensemble measurements).

5. A modified sort of tomography is possible on “practically indistinguishable” particles

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Predicted Wigner-Poincaréfunction for a variety of“triphoton states” we arestarting to produce: