Experimental Issues in
Quantum Measurement
•
Today, 7.10.03: OVERVIEW
– a survey of some important situations in q. msmt. theory
(“why bother coming to these lectures?”)
Being a quantum physicist is like being an alcoholic.
•
13.10.03: SOME TECHNICAL BACKGROUND
– introduction to “standard” quantum measurement theory
(measurement postulate, collapse, von Neumann msmts,
density
matrices
and is
entanglement,...)
...the first
step
to admit you have a problem.
•
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!
VALID REASONS TO INTERRUPT ME:
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.
Some references
I will not be following any particular textbook, but for obvious
reasons, will draw disproportionately from experiments I myself
have 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 Measurement
Bell’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 question
about particle properties is not a question about the
outcome of real measurements – it is “not a proper
question.”
Wave and particle descriptions are “complementary” – they can
never both be observed in a single experiment.
The Bohr-Einstein debates
Inserting
Weg” detectors
Two-slit“Welcher
interference:
destroys
our ignorance
the prototypical
wave –
and
thus the interference.
phenomenon.
Each particle seems to “go
through both slits”; we can’t
ask which one it came from.
“Heisenberg microscope”: photons which allow you to look
at the particle bounce off it, disturbing its momentum.
Feynman’s Rules for interference
If two or more indistinguishable processes can lead to the
same final event (particle could go through either slit and
still get to the same spot on the screen), then add their complex
amplitudes 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 real
probability 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 process
occurred, 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 particles
passing slit 2, without affecting
linear momentum.
Still no interference – because we could check the spin of
the particle, and discover which slit it had traversed.
Must there be a disturbance?
Bohr:
Measurement of X disturbs P; et cetera
Measurement means amplification of a quantum phenomenon
by interaction with some “large” (classical) device
Msmt involves some uncontrollable, irreversible disturbance
We 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.
The EPR “Paradox”
–– and superluminal signalling?
Particle with 0 angular
RECALL:
momentum decays...
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,
 here implies...
...  here...
Sx = +1 (spin ) and Sx= –1
(spin
) are
equally likely.
but  here
implies...
here.
Then...if
you
find one of those,  and  become equally likely.
EPR: we could measure Sx on particle 1, but simultaneously
know
what we would
have
Bohr
& Heisenberg
tell us
weundoubtedly
must choose:gotten
if wewe
hadcan
measured
Sz;but
aren’t
real? of Sx...
know Sz,
givethese
up allboth
knowledge
Copenhagen:or
noknow
waveSx
function
hasup
both
properties
defined –
and give
all those
knowledge
of Sz...
and the wave function is all you can possibly know.
EPR are cheating, discussing measurements they didn’t do.
Some important lessons
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 in
Vienna without simultaneously taking into account what
is going on in Toronto– even for times so short that even
John Bell at the speed of light, no signal could have connected the two.)
Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e .
Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e .
Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e .
Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e .
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.
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.
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.
H-polarized photon
45o-polarized photon
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!
"Definitely 3"
"Definitely 2"
"Definitely 1"
"I don't know"
The correct state was identified 55% of the time-Much better than the 33% maximum for
“standard measurements” ( = everything in your textbook).
" 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)
Problem:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
The bomb must be there... yet
my photon never interacted with it.
Quantum CAT scans
If 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 be
at 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 get
so much information: “quantum state tomography”
The Wigner quasiprobability function
for an atom trapped in a light wave
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
P(0,0) < 0 ?!
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>, deterministically
3. Upon a measurement, |f>  some result |n> , randomly
4. 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
B+C
What are the odds that the particle
was in a given box (e.g., box B)?
Pick a box, any box...
A+B+C
A+B–C
We’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!)
Measurement as a tool: KLM...
INPUT STATE
a|0> + b|1> + c|2>
OUTPUT STATE
a|0> + b|1> – c|2>
MAGIC MIRROR:
Acts differently if there are 2 photons or only 1.
TRIGGER (postselection)
ANCILLA
In other words, can be a “transistor,” or “switch,”
or “quantum
special
|i > logic gate”...
particular | >
f
Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since.
Experiments by Franson et al., White et al., Zeilinger et al...
Summary: the kinds of things
we’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?