Experimental Issues in Quantum Measurement Being a quantum physicist is like being an alcoholic....the first step is to admit you have a problem. Today,

Experimental Issues in Quantum Measurement

Being a quantum physicist is like being an alcoholic.

...the first step is to admit you have a problem.

• Today, 7.10.03: OVERVIEW– a survey of some important situations in q. msmt. theory

(“why bother coming to these lectures?”)

• 13.10.03: SOME TECHNICAL BACKGROUND– introduction to “standard” quantum measurement theory (measurement postulate, collapse, von Neumann msmts, density matrices and entanglement,...)

• 20.10.03: THE QUANTUM ERASER– Bohr-Einstein debates– Scully, Englert, Walther: complementarity vs uncertainty– Two-photon experiments– Alternate pictures (collapse vs correlations)

• 27.10.03–: OTHER MODERN EXPERIMENTS...

FIRST TOPIC: Interruptions

MAKE THEM!

I’m going too fast.I’m going too slow.You want to correct my grammar.You disagree with something I said.I seem to disagree with something I’ve said.You have a question about something I’ve said.You have a question about something

completely unrelated.

VALID REASONS TO INTERRUPT ME:

Some referencesI will not be following any particular textbook, but for obviousreasons, will draw disproportionately from experiments I myselfhave worked on...

Appropriate references will be provide as the lectures progress.

General references on quantum mechanics:Your favorite QM text + Shankar’s Principles of QM

Background on the quantum measurement problem:Wheeler & Zurek’s Quantum Theory and MeasurementBell’s Speakable and Unspeakable in Quantum Mechanics

My general perspective on these issues:References on my web page,

http://www.physics.utoronto.ca/~aephraim/aephraim.html

{where slides from these lectures will be too, eventually}“Speakable and Unspeakable, Past and Future”

http://lanl.arxiv.org/abs/quant-ph/0302003

The Copenhagen Viewpoint (Toronto description of)

Bohr, Heisenberg:We must only discuss the outcomes of measurements.An experiment described to measure wave properties

will measure wave properties.An experiment described to measure particle properties

will measure particle properties.

In an experiment which measures wave properties, a questionabout particle properties is not a question about the outcome of real measurements – it is “not a properquestion.”

Wave and particle descriptions are “complementary” – they cannever both be observed in a single experiment.

The Bohr-Einstein debates

Two-slit interference:the prototypical wavephenomenon.

Each particle seems to “gothrough both slits”; we can’task which one it came from.

Inserting “Welcher Weg” detectorsdestroys our ignorance –and thus the interference.

“Heisenberg microscope”: photons which allow you to lookat the particle bounce off it, disturbing its momentum.

Feynman’s Rules for interference

If two or more indistinguishable processes can lead to thesame final event (particle could go through either slit andstill get to the same spot on the screen), then add their complexamplitudes and square, to find the probability:

P = |A1+A2 |2 ≈ |eikL1 + eikL2 |2 ≈ 1 + cos k(L1-L2)

If multiple distinguishable processes occur, find the realprobability of each, and then add:

P = |A1|2 + |A2|2 ≈ |eikL1 |2 + |eikL2 |2 ≈ 1

If there is any way – even in principle – to tell which processoccurred, then there can be no interference (if you knew which slit the particle came from, you’d see a 1-slit pattern) !

The quantum eraser

spin-up ()particles

Waveplate: flips the spin of particlespassing slit 2, without affectinglinear momentum.

Still no interference – because we could check the spin ofthe particle, and discover which slit it had traversed.

Must there be a disturbance?Bohr:

Measurement of X disturbs P; et ceteraMeasurement means amplification of a quantum phenomenon

by interaction with some “large” (classical) deviceMsmt involves some uncontrollable, irreversible disturbanceWe must treat the measuring device classically.

Wigner: Why must we? What will happen to us if we don’t?

Scully, Englert, Walther: Complementarity is more fundamental than uncertainty.We can use information to destroy interference,

without disturbing the momentum.

Storey, Tan, Collett, Walls: No. Any such measurement always disturbs the momentum.

Wiseman (+ Toronto experiment): They’re both right.And we can measure how much the momentum is disturbed.

RECALL:

Spin-projections along different axes are “incompatible”(can’t be measured simultaneously -- like X & P)

If you find Sz = +1 (spin ), and then measure Sx,Sx = +1 (spin ) and Sx= –1 (spin ) are equally likely.

Then if you find one of those, and become equally likely.

Bohr & Heisenberg tell us we must choose:we can know Sz, but give up all knowledge of Sx...or know Sx and give up all knowledge of Sz...

The EPR “Paradox” –– and superluminal signalling?

Particle with 0 angularmomentum decays...

here implies... ... here...but here implies...

... here.

EPR: we could measure Sx on particle 1, but simultaneouslyknow what we would have undoubtedly gotten if we had measured Sz; aren’t these both real?

Copenhagen: no wave function has both those properties defined –and the wave function is all you can possibly know.EPR are cheating, discussing measurements they didn’t do.

Some important lessons

One of the more subtle ones:You can extract very limited information from a single particle.In fact, to duplicate the particle, you must destroy it –

information in QM is never gained or lost.

The first one (only 30 years... or maybe 50, or 70+):QUANTUM MECHANICS IS NOT LOCAL(i.e.: it is not always possible to describe what happens inVienna without simultaneously taking into account whatis going on in Toronto– even for times so short that evenat the speed of light, no signal could have connected the two.)

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John Bell

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NO CLONING!(...and yet, recent “quantum cloning” experiments...)

“Distinguishing the indistinguishable”• Non-orthogonal quantum states cannot be distinguished with certainty.

• This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications.

• Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do.

H-polarized photon 45o-polarized photon

If it gets through an H polarizer,...it could still have been 45,and it’s too late to tell.

If it gets through a 45 polarizer,same story.

BUT: a clever measurement can tell with certainty, 25% of the time.BUT BUT: a non-standard quantum measurement can do better!

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” ( = everything in your textbook).

"I don't know"

"Definitely 3"

"Definitely 2"

"Definitely 1"

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.

Quantum CAT scansIf you measure momentum P... you don’t know anything about X.If you measure position X... you don’t know anything about P.

But in real life, don’t I know something about each?Don’t I also know that if a car left this morning and is already

in Budapest, it’s going faster than if it’s still on Währingerstr.?

Wigner function: W(x,p) is like the probability for a particle to beat x and have momentum p.

Its integrals correctly predict P(x), P(p), and everything else you want.

Of course, you must study a large ensemble of particles to getso much information: “quantum state tomography”

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momentumposition

Pro

bab

ilit

y

P(0,0) < 0 ?!

The Wigner quasiprobability function for an atom trapped in a light wave

Predicting the past ?

Standard recipe of quantum mechanics:1. Prepare a state |i> (by measuring a particle to be in that state; see 4)2. Let Schrödinger do his magic: |i> |f>=U(t) |i>, deterministically3. Upon a measurement, |f> some result |n> , randomly4. Forget |i>, and return to step 2, starting with |n> as new state.

Aharonov’s objection (as I read it):No one has ever seen any evidence for step 3 as a real process;

we don’t even know how to define a measurement.Step 2 is time-reversible, like classical mechanics.Why must I describe the particle, between two measurements (1 & 4)

based on the result of the first, propagated forward,rather than on that of the latter, propagated backward?

Predicting the past...

A+B

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

B+C

A+B

Pick a box, any box...

A+B+C

A+B–CWe’ll see that applying similar logic here lets us conclude:

P(A) = 100%P(B) = 100%and then, necessarily: P(C) = –100% (?!)

....and that real measurements agree (somehow!)

special |i >

a|0> + b|1> + c|2> a|0> + b|1> – c|2>

Measurement as a tool: KLM...INPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

particular |f >

Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since.Experiments by Franson et al., White et al., Zeilinger et al...

MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...

Summary: the kinds of thingswe’ll cover...

• Why does one thing happen and not another?• When is a quantum measurement?

• Does a measurement necessarily disturb the system, and how?

• What can we say about an observable before we measure it?

• Does a wave function describe a single particle, or only an ensemble?

• Is a wave function a complete description of a single particle?

• Can we predict the past?