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Quantum Mechanics:
Interpretation and Philosophy
Significant content from: “Quantum Mechanics and
Experience” by David Z. Albert, Harvard University
Press (1992).
Main Concepts:
-- complementarity
-- the uncertainty principle
-- superposition
-- collapse of the wavefunction
-- the measurement problem
Assume electrons have a “color”
and a “hardness” property (which
we can accurately measure).
b
Color
measurement
h
w
Hardness
Assume the “color” of an electron
can be either black (b) or white (w).
Assume the “hardness” of an electron
can be either hard (h) or soft (s).
(discrete values -- quantization)
measurement
s
ˆ
A
ˆ
Hardness: B
ˆB
ˆ -B
ˆA
ˆ 0
A
Color:
Lets do something easy:
e-
e-
say we measure that
an electron is white
then if we measure its color again, we will find it is white
(same for black, same for “hardness” properties)
(our measuring device is reliable)
Are the color and hardness properties
of an electron correlated?
50%
No: 50% of white
electrons are hard
w
h
Hardness
s
knowing the color of an electron tells
us nothing about its hardness (and no
additional information helps)
50%
Now assume a white electron emerges from the
soft aperture of a hardness box (50% will do so).
e50%
Let us measure the color
e-
50%
somehow the hardness box has tampered with the color value !
Can we build a
“color and hardness” box?
This box would need to
consist of a hardness
box and a color box.
But the hardness
measurement
scrambles the color,
and vice versa.
w+h
w+s
Color
and
Hardness
b+s
b+h
To say “the color of this electron is now such-andsuch and the hardness of this electron is now suchand-such” seems to be fundamentally beyond our
means (uncertainly principle: color and hardness are
incompatible physical properties)
the “black box” is just a fancy mirror
that makes the two paths coincide
If we feed an s (or an h)
electron in, it emerges
along the h and s path,
unchanged.
Use a w electron and measure hardness.
Find 50% h and 50% s
w
Use a h electron and measure color.
Find 50% w and 50% b
h
Use a w electron and measure color.
This device is just a fancy
hardness box (it is a hardness
box with a few harmless
mirrors), and we know a
hardness measurement
scrambles the color.
w
Use a w electron and measure color.
Find 100% w !!!
w
Let us add a movable wall that absorbs electrons
w
Slide the wall into place:
1.) 50% reduction in the number of
electrons emerging along the h and s path
2.) Of the electrons that emerge, their color is
now scrambled: 50% w and 50% b.
What can possibly be going on ?
Consider an electron which passes through the
apparatus when the sliding wall is out.
Does it take route h ? No, because h
electrons have 50/50 b/w color statistics.
Does it take route s ? No, same reason. w
Can it somehow have taken both routes ? No: if we
look to see where the electron is inside the apparatus, we find that
50% of the time it is on route h, and 50% of the time it is on route s.
We never find two electrons inside, or two halves of a single, split
electron, or anything like that. There isn’t any sense in which the
electron seems to be taking both routes.
Can it have taken neither route? No: if we put sliding walls
in place to block both routes, nothing gets through at all.
But these are all the logical possibilities that we have any
notion whatever of how to entertain !
w
What can these electrons be doing? Electrons would
seem to have modes of being available to them which are
quite unlike what we know how to think about.
The name of this new mode (which is just a name for something
we don’t understand) is superposition.
What we say about an initially white electron which is now
passing through our apparatus (with the wall out) is that it’s not
on h and not on s and not on both and not on neither, but, rather,
that it’s in a superposition of being on h and being on s. And what
this means (other than “none of the above”) we don’t know.
We know, by experiment, that electrons emerge from the
hard aperture of a hardness box if and only if they’re hard
electrons when they enter that box.
When a white electron is fed into a hardness box, it emerges
neither through the hard aperture nor through the soft one nor
through both nor through neither. So, it follows that a white electron
can’t be a hard one, or a soft one, or (somehow) both, or neither. To
say that an electron is white must be just the same as to say that
it’s in a superposition of being hard and soft.
Why can’t we say “the color of this electron is now such-and-such and
the hardness of this electron is now such-and-such”? It isn’t at all a
matter of our being unable to simultaneously know what the color and
the hardness of a certain electron is (that is: it isn’t a matter of
ignorance). It’s deeper than that.
Why is quantum mechanics a probabilistic theory?
If it could ever be said of a white electron that a measurement of its
hardness will with certainty produce the outcome (say) “soft”, that
would be inconsistent with what we know (that it’s in a
superposition of hard and soft)
On the other hand, our experience is that every hardness
measurement whatsoever either comes out “hard” or it
comes out “soft”.
So apparently the outcome of a hardness measurement on a white
electron has got to be a matter of probability !
Elitzer-Vaidman bomb testing problem
w
w
We can tell whether or not the wall is in path s
Elitzer-Vaidman bomb testing problem
w
w
Assume the wall is actually a very sensitive bomb, so
sensitive that if an electron hits it, it will explode.
How can we detect the presence of the bomb
without setting it off ??
Elitzer-Vaidman bomb testing problem
50% of the time (compared to
the bomb not being there) no
particle emerges along h and
s. The bomb explodes.
25% of the time, we get a w
electron out at h and s. We
learn nothing (same result as
bomb being absent)
w
25% of the time, we get a b electron out at h and s.
We have detected the presence of the bomb
without touching it !!!!
(interaction-free measurement)
Use a w electron and measure color.
Find 100% w
w
Actually, this device does nothing,
so of course the color doesn’t
change. This device doesn’t
measure hardness, because when
the electron exits the device, we
don’t know which path it followed!
We erased the hardness
information by recombining the
paths.
In this context the bomb acts as a measures
device – it tells us which path (hence which
hardness value) the electron took.
incoming
electron
ready
hard
soft
e-
hardness measurement
outgoing
electron
e-
Niels Bohr (Nobel Prize 1922):
Quantum mechanics does not describe how nature is
but rather articulates what we can say about nature.
“We must only discuss the outcomes of measurements.”
Expressed in modern language, this means that quantum
mechanics is a science of knowledge, of information.
Suppose we accept that collapse happens
(since this is consistent with our experience that
measurements yield definitive outcomes).
We could then ask, when does collapse occur?
Eugene Wigner (Nobel Prize 1963) proposed an answer in 1961:
Collapse occurs precisely at the level of consciousness,
and perhaps, moreover, consciousness is always the agent
that brings it about.
(this doesn’t really help)
When does collapse occur?
Suppose that Alice has a theory about collapse:
collapse happens immediately after the electron
exits the measurement box.
And suppose that Bob has another theory about
collapse:
collapse happens later, for example when a
human retina or optic nerve or brain gets
involved.
Can we decide who is right empirically, that is, by performing some experiment?

Can we decide who is right empirically, that is, by performing some experiment?
Here’s how to start: Feed a black electron into a hardness device
and give it enough time to pass through. If Alice is right, the state of
the system is now
either
or
hard 
m
soft 
m
e
h
e
s
(with 50% prob.)
(with 50% prob.)
whereas if Bob is right, the state of the system is currently


hard  
m
1
2
e
h
soft 
m
1
2
e
s
so all we need to do is to figure out a way to distinguish,
by means of a measurement, these two cases: In one
case the pointer points in a particular (but as yet
unknown) direction, and in the other case the pointer isn’t
pointing in any particular direction at all.
What if we measure the position of the tip of the pointer? That is,
lets measure where the pointer is pointing. This won’t work.
If Alice is right, of course we will find a 50/50 chance of finding the
pointer “pointing-at-hard” vs. “pointing-at-soft”. This is because,
according to Alice, the pointer is already in one of those two states.
But if Bob is right, then a measurement of the position of
the tip of the pointer will have a 50% change of collapsing
the wavefunction of the pointer onto the “pointing-at-hard”
state, and 50% change of collapsing it to “pointing-atsoft”.
Therefore the probability of any given outcome of a measurement of
the position of the pointer will be the same for both these theories;
and so this isn’t the sort of measurement we are looking for.
What if we measure the hardness of the electron? This won’t work.
What if we measure the color of the electron? This won’t work.
These arguments establish that different conjectures about precisely
where and precisely when collapse occurs cannot be empirically
distinguished from one another.
And so the best we can do at present is to try to think of precisely
where and precisely when collapses might possibly occur (that is,
without contradicting what we do know to be true by experiment). But
it turns out to be hard to do even that.
Schroedinger’s Cat
Wigner’s Friend
Quantum Cryptography:
how to communicate without eavesdropping
(from S.J. Lomonaco, Jr., http://www/csee.ubmc.edu/~lomonaco)
Key idea: How to determine if Eve is listening to Alice and Bob’s conversation?
Quantum Cryptography:
how to communicate without eavesdropping
(from S.J. Lomonaco, Jr., http://www/csee.ubmc.edu/~lomonaco)
Eve
Bob
Alice
Key idea: How to determine if Eve is listening to Alice and Bob’s conversation?
 ˆ
ih
 H
t

This is our hardness
value (hard/soft)
This is our color
value (black/white)
Quantum Teleportation
Classical information can be copied any number of
times, but perfect copying of quantum information is
impossible, a result known as the no-cloning theorem,
which was proved in 1982 by Wootters and Zurek of
Los Alamos National Laboratory.
(If we could clone a quantum state, we could use the
clones to violate Heisenberg’s Uncertainty Principle.)
Quantum Teleportation
Quantum mechanics seems to make such a teleportation
scheme impossible in principle. Heisenberg’s uncertainty
principle rules that one cannot know both the precise position
of an object and its momentum at the same time.
Thus, one cannot perform a perfect scan of the object to be
teleported; the location or velocity of every atom and electron
would be subject to errors.
(In Star Trek the “Heisenberg Compensator” somehow
miraculously overcomes this difficulty.)
However, in 1993 physicists discovered a theoretical way to
use quantum mechanics itself for teleportation. They used
entanglement to circumvent the limitations imposed by
Heisenberg’s uncertainty principle without violating it.
Entangled Pair
Occasionally a single UV photon is converted into two photons of lower
energy, one vertically polarized and one horizontally polarized.
crystal
laser beam
(photon)
If these two photons emerge along the cone intersection (green areas),
neither of them has a definite polarization. Why? Indistinguishability
(7th postulate, photons are bosons).
Feynman(1965 Nobel Prize)’s Rules for interference
If two or more indistinguishable processes can lead to the
same final event (particle could take either path), 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 path the particle
came from, you’d see a randomized color) !
Entangled Pair
 (1,2) 
1
1
c


1
2
2
2
1c 2
Photon 1 does not have a definite polarization, but its polarization is
complementary to that of photon 2.
Measurement of the polarization of photon 1 will collapse photon 2 into
the opposite polarization (action at a distance).
Quantum Teleportation
No cloning allowed, so how can we teleport a photon?
Answer: we will use an entangled pair of photons (A and B) to teleport a
photon X from Alice to Bob.
Quick summary: Alice begins with photon X but does not know what its
properties are (because if she measured it, its properties might change!).
She teleports it to Bob, which destroys her “copy”. Bob ends up with
photon X.
Quantum Teleportation
Quantum Teleportation
Quantum Teleportation
h and s
h
black
box
s
h
Hardness
track
s
sliding
wall
general outline:
-David Z. Albert book chapter 1
-collapse postulate
-Schroedinger’s cat
-bomb detection