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Wringing John Bell
 vocabulary
 the EPR paradox
 Bell’s theorem
 Bell’s assumptions
 what does it mean?
Guy Blaylock Williams College 4/17/15
Blaylock - Williams College 4/17/15
Characteristics of a
Garden Variety Classical Scientific
Theory
locality – actions at one location do not immediately have
any effect at a separate location.
(counter) factual definiteness – any measurable quality of a
physical system has a single well-defined value when it is
measured (factual) or before it is measured (counterfactual).
determinism – complete knowledge of the current state of a
physical system is sufficient to determine the future state of
the system.
Blaylock - Williams College 4/17/15
Characteristics of Orthodox QM
Orthodox QM obeys none of these characteristics…
• When a wave function collapses, it collapses everywhere
at once. (nonlocal)
• When a physical system is in a state of superposition, it is
not in a single well-defined state. (indefinite)
• When a system collapses to a single final state among
multiple possibilities of a superposition, it does so
randomly. (indeterminate)
Blaylock - Williams College 4/17/15
EPR à la Bohm (almost)
Consider a pair of photons produced with the same polarization.
Measure the polarization of one. The polarization of the other must
always turn out to be the same (in QM this is the“twin state”).
There are several sources that do this:
1. atoms
2. downconverters
3. subatomic decays/annihilation
down conversion
crystal
polaroid filter
Blaylock - Williams College 4/17/15
Quantum Twin State
QM explains the EPR
experiment using the
quantum twin state:
ytwin =
1
(V1V2 +H1H 2 )
2
• Prior to measurement, the two-photon system is not in one
definite state; it’s in a superposition of V1V2 and H1H2.
• When a measurement is made, both photon polarizations
collapse, nonlocally.
• The final choice, V1V2 or H1H2, is determined randomly.
Blaylock - Williams College 4/17/15
EPR
logic
• One could determine the polarization
of photon 2 simply by looking at
photon 1, without disturbing photon 2.
Similarly, one could determine the
polarization of photon 1 without
disturbing it.
• If one can determine certain parameters (such as polarizations in
Bohm’s EPR) without interfering with the system, those parameters
must be‘real’.
• If a theory is to be considered complete, it should predict all real
parameters, including the polarizations in Bohm’s EPR experiment.
• QM does not predict the polarizations.
QM is not complete!
Blaylock - Williams College 4/17/15
OriginalEPR
statement
caveat of EPR
“…one would not arrive at our conclusion if … [the values
of the second system] depend upon the process of
measurement carried out on the first system”
“No reasonable definition of reality
could be expected to permit this.”
Blaylock - Williams College 4/17/15
newspapers
Why can’t the photons just be
generated with some definite
polarization, like two
newspapers sent to different
places?
Blaylock - Williams College 4/17/15
Bell’s Theorem
1964 - John S. Bell publishes
“ON THE EINSTEIN PODOLSKY
ROSEN PARADOX”
Physics 1 (1964) p.195-200.
Reprinted in Speakable and Unspeakable in QM
Exploring the correlations between different measurements leads to
new constraints based on common sense (Bell inequalities).
e.g. What if we measured polarizations at arbitrary angles 1, 2?
QM makes predictions about the correlations of polarizations that
are different from the predictions of ‘common sense’ theories!
Blaylock - Williams College 4/17/15
QM prediction
What should we expect from Quantum Mechanics?
What is the probability of getting the same
measurement (i.e. both transmitted or both
absorbed)?
Prob( M1(1) = M2(2) ) = cos2(2 - 1)
down conversion
crystal
Blaylock - Williams College 4/17/15
1
2
Amplitude filtering
For a wave
impinging
on a filter
at an
arbitrary
angle…

Acos(
…the amplitude that passes through is Acos ( 
The probability that a photon passes through is cos2 (.
Blaylock - Williams College 4/17/15
Common Sense
What should we expect from Common Sense?
This is where Bell comes in.
Blaylock - Williams College 4/17/15
Prob( M1() = M2() ) is 100% coincidence
Arbitrary angle
A series of photon pairs will show a sequence of both being
absorbed, or both transmitted, never one absorbed and one
transmitted.
F : A T T A A T A T A T T T A T A A A A T A T T A T
F : A T T A A T A T A T T T A T A A A A T A T T A T
Prob( M1() = M2() ) = 0% coincidence
For between 0 and 90o,
the coincidence is between 100% and 0%
In particular, let  be some angle such that
Prob( M1() = M2() ) = 75% coincidence; mismatch 25%
Blaylock - Williams College 4/17/15
Common
Sense
Prediction
Apply this
“common sense”
to several
different cases:

avg mismatch 25%
F : A T T A A T A T A T T T A T A A A A T A T T A T
F : A T T T A T A T A A T A A T T A T A T A T T T T

avg mismatch 25%
F : A A A A A T A T T T T T A T A T T A T A T A T T
F : A T T A A T A T A T T T A T A A A A T A T T A T
Bell’s Inequality

avg mismatch  50%
F : A A A A A T A T T T T T A T A T T A T A T A T T
F : A T T T A T A T A A T A A T T A T A T A T T T T
Blaylock - Williams College 4/17/15
QM disagrees!
QM for 30/60o
For = 30o, coincidence is 75%,
mismatch 25%
(Remember cos2(30o) = 0.75)
For 
QM says the coincidence should be:
cos2(30o + 30o) = cos2(60o) = 25%
mismatch = 75%, certainly not less than 50%
Blaylock - Williams College 4/17/15
Experiment vindicates QM
1972 -- John Clauser (Berkeley) performs a Bell measurement
using mercury vapor atoms that produce twin state photons.
QM wins but the experiment does not rule out slower than light
speed interactions.
1982 -- Alain Aspect performs an experiment with extremely
fast acousto-optical switches to demonstrate faster-than-light
effects.
1997 -- Nicolas Gisin uses Swiss telecom network optical fiber
and a downconverter to demonstrate quantum effects over a
distance of 7 miles.
…and many more.
Blaylock - Williams College 4/17/15
Why is
is Bell’s
Bell’sinequality
inequalityviolated?
violated?
Why

avg mismatch 25%
F : A T T A A T A T A T T T A T A A A A T A T T A T
F : A T T T A T A T A A T A A T T A T A T A T T T T

avg mismatch 25%
F : A A A A A T A T T T T T A T A T T A T A T A T T
F : A T T A A T A T A T T T A T A A A A T A T T A T
Assume that rotating F2 from  to
does not affect what happens at F1.
locality!

avg mismatch  50%
F : A A A A A T A T T T T T A T A T T A T A T A T T
F : A T T T A T A T A A T A A T T A T A T A T T T T
Blaylock - Williams College 4/17/15
…the other assumption
1.

The two photons always yield the same polarization.
easily verified by experiment
2.

There exists an angle , such that mismatch = 25%.
easily verified by experiment
3.
The mismatch for  is the same as for
 (i.e. rotational symmetry)
easily verified by experiment

4.
The mismatch rate between and  is still 25% even when we
don’t make the measurement for 
Counterfactual definiteness (CFD).
QM says there is more than one possibility for each measurement.
The  sequence that disagrees with  by 25% is not the same as
the  sequence that disagrees with  by 25%.
Blaylock - Williams College 4/17/15
Common Sense Prediction

avg mismatch 25%
F : A T T A A T A T A T T T A T A A A A T A T T A T
F : A T T T A T A T A A T A A T T A T A T A T T T T

avg mismatch 25%
F : A A A A A T A T T T T T A T A T T A T A T A T T
F : A T T A A T A T A T T T A T A A A A T A T T A T
In QM the two sequences for and 
don’t need to be the same!
Blaylock - Williams College 4/17/15
Conclusions
• The universe is nonlocal or non-CFD, or both.
• There are interpretations of QM that follow each.
• Whatever you decide, the world is
Blaylock - Williams College 4/17/15
underlying realities
Orthodox/Copenhagen - (Bohr, Heisenberg)
non-CFD, non-deterministic & non-local
Bohm’s interpretation - (Bell, Bohm, deBroglie)
definite, deterministic & non-local
Pilot waves direct the particles non-locally.
Many Worlds - (DeWitt, Everett)
local, deterministic & indefinite
No collapse; every possibility exists as a part of the superposition.
Agnostic - (many contemporary scientists)
Who knows, who cares.
Makes no sense to ask what is going on outside of observation.
Blaylock - Williams College 4/17/15
More on Definiteness
• CFD might seem to imply‘realism’. This is
probably what EPR were trying to say with their
‘elements of reality’.
• The reverse is not true. Realism does not imply
CFD. If one considered a photon wave function
(which may be a superposition) to be real, it still
would not imply a definite polarization.
• Definiteness does not imply determinism. The
definite characteristic could evolve randomly.
• BTW, if a theory is local it must also be
deterministic. (deduce from EPR expt.)
Blaylock - Williams College 4/17/15
Blaylock - Williams College 4/17/15
History of the Worlds
1957 Hugh Everett writes a thesis on the “relative
state” interpretation of QM
[Hugh Everett III, “Relative State’ Formulation of Quantum Mechanics”,
Rev. Mod. Phys. 29, 454-462 (1957)]
Bryce DeWitt popularizes, embellishes and
somewhat misrepresents the concept in the “many
worlds” interpretation
[Bryce S. DeWitt , “Quantum mechanics and Reality”, Physics Today 23,
30-35. (1970)]
The essence of Everett’s many
worlds interpretation is the
same as orthodox QM except
that collapse does not happen.
Superpositions persist.
Blaylock - Williams College 4/17/15
“…every quantum
transition taking place
on every star, in every
galaxy, in every remote
corner of the universe is
splitting our local world
on earth in myriads of
copies of itself.”
polarizing filters
A wave that moves only in a
plane is called plane-polarized
or linearly polarized.
A vertical filter allows a
vertically polarized
wave to pass, …
but blocks a horizontally
polarized wave,…
Blaylock - Williams College 4/17/15
and let’s the vertical part
of a 45o wave through.
vertically
polarized
polarized photons
For light waves, the plane of oscillation defines the polarization.
A photon’s polarization is determined by whether it does or does not
pass through a polarizing filter.
A photon that passes
through a vertical filter is
“vertically polarized”.
It will pass through
any number of other
vertical filters.
…but it will not
make it through a
horizontal filter.
Think of photon polarization as a binary quantity.
A polarizer provides a way of measuring it.
Blaylock - Williams College 4/17/15
Blaylock - Williams College 4/17/15