EPR and quantum entanglement

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EPR and quantum entanglement
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Spin
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Quantum physics of spin

• Spin is an intrinsic angular momentum of a
  particle such as an electron, etc. (N.B. If
  the spin of a particle is all we are interested
  in, we might just call the particle itself a
  spin.)
• The spin of a particle is related to its magnetic
  properties.
• How to visualize The spin of a particle is a
  little arrow pointing along some direction in
  space (indicates rotation axis).

• We cannot just measure the spin. We can only
  measure the spin along some specified direction.
• Examples We can measure the spin along X, Y or
  Z axes.
• In any measurement, we can get only two results
  ? or ? .

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spin up (?)
two possible measurement outcomes
spin
spin down (?)
measuring device for spin along Z axis
spin right (?)
two possible measurement outcomes
spin
spin left (?)
measuring device for spin along X axis
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Uncertainty Principle for Spin

• Different spin directions are complementary
  variables (analogous to particle position x and
  momentum p)
• If we know spin along Z axis, we know nothing
  about spin along X axis, and vice versa.
• Quantum theory Particle cannot have definite Z
  and X spins at the same time.

result of X spin measurement cant be predicted
P ½
spin with known Z value
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Uncertainty Principle for Spin

• Different spin directions are complementary
  variables (analogous to particle position x and
  momentum p)
• If we know spin along Z axis, we know nothing
  about spin along X axis, and vice versa.
• Quantum theory Particle cannot have definite Z
  and X spins at the same time.

result of Z spin measurement cant be predicted
P ½
spin with known X value
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The EPR argument
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EPR criterion of reality
If, without in any way disturbing a system, we
can predict with certainty . . . the value of a
physical quantity, then there exists an element
of physical reality corresponding to this
physical quantity. Einstein, Podolsky and Rosen
(1935)

• If we can know the value of Q without touching
  the system in any way, then Q must be physically
  real (i.e., have a determinate value) even before
  we measure it.
• If quantum mechanics says that the value of Q is
  indeterminate before we measure it, then quantum
  mechanics is incomplete.

system inside
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An entangled spin state

• Singlet state is an entangled state of two
  spins (EPR state)
• Can be created experimentally (e.g., two
  electrons in He)
• EPR state of a spin pair has total spin zero
• For each spin, any measurement result has P ½
• If we make a measurement on the L spin and find
  some result, then we know for certain that the R
  spin is opposite.

Suppose we get this result for L (P½) ...
... then we know the R spin is this.
N.B. In this diagram, L and R spins could be
extremely far apart!
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Central argument of EPR
Create an EPR pair (singlet state) of two spins.
Make measurements on L only.
If we measure Z on L, then we will find out the Z
value of R.
L
R
If we measure X on L, then we will find out the X
value of R.
L
R
We can find out the value of X and Z for the R
spin without touching R in any way. Therefore,
both Z and X must have a predetermined value! EPR
conclusion Quantum mechanics is an incomplete
theory.
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Local realism (cartoon version)

• Interaction of particles allows them to cooperate
  later on.
• After interaction, each particle acts separately,
  equipped with a set of instructions (and maybe
  some local randomness).
• All cooperation between the particles must be
  encoded into the instruction sets of the
  particles.
• If the particles can cooperate in the outcomes of
  many different measurements, then the outcomes of
  these measurements must already exist in the
  instruction sets of the particles.

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Bohrs response
Be careful! What does EPR mean by without in
any way disturbing the system?

• These two experiments are complementary you
  cannot do both of them. Therefore...
• Choosing to do one over the other changes the
  situation for the R spin, even if there are no
  forces acting between L and R.

EPR argument breaks down because their criterion
of reality is ambiguous when applied to quantum
systems.
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From our point of view we now see that the
wording of the above-mentioned criterion of
physical reality proposed by Einstein, Podolsky
and Rosen contains an ambiguity as regards the
meaning of the expression without in any way
disturbing the system. Of course there is no
question of a mechanical disturbance of the
system under investigation during the last
critical stage of the measuring procedure. But
even at this stage there is essentially the
question of an influence on the very conditions
which define the possible types of predictions
regarding the future behavior of the system.
Since these conditions constitute an inherent
element of the descriptions of any phenomenon to
which the term physical reality can be properly
attached, we see that the argumentation of the
mentioned authors does not justify their
conculsion that quantum-mechanical description is
essentially incomplete. Niels Bohr (1935)
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Einstein on EPR
It is characteristic of physical objects that
they are thought of as arranged in a space-time
continuum. An essential aspect of the
arrangement is that they lay claim, at a certain
time, to an existence independent of one another,
provided those objects are situated in different
parts of space. The following idea
characterizes the relative independence of
objects (L and R) far apart in space external
influence on L has no direct influence on
R. adapted from letter to M. Born

• The paradox forces us to relinquish one of the
  following two assertions
• the description by means of the ?-function is
  complete
• the real states of spatially separated objects
  are independent of each other.
• Einsten Reply to criticisms

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A shocking surprise
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The Bell game

• Profound argument due to John Bell (1964)

?
?
?
?

• Particles L and R go to separate measuring
  devices.
• L observer can measure A or B
• R observer can measure C or D
• All measurements have possible outcomes ?1
• Local realism requires that
• A, B, C and D all have simultaneous definite
  values
• The choice of measurement on L does not affect
  results of R measurements, and vice versa.

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?AC? -1 agree 0 of time ?AC? 0 agree 50 of
time ?AC? 1 agree 100 of time
?AC? average of the product of the A and C
results
Q A (C D) B (C D)
One of these is zero, and the other is ?2.
Q ?2
?Q? ? 2
Bells inequality
?Q? ?AC? - ?AD? ?BC? ?BD? ? 2
Bells inequality must be true if local realism
is true.
We can measure these in separate experiments.
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Quantum physics of the EPR state

• Measure L and R along two axes A and C

?
?
?
?

• If A and C are 0? apart, the results agree 0 of
  the time.
• If A and C are 45? apart, the results agree 15
  of the time.
• If A and C are 90? apart, the results agree 50
  of the time.
• If A and C are 135? apart, the results agree 85
  of the time.

Likelihood that A and C results agree cos2(?/2)
Mathematical aside
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A and C are 0? apart Results agree 0 of the
time. 100 of the time AC -1 ?AC? -1
A and C are 45? apart Results agree 15 of the
time. 15 of the time AC 1 85 of the time
AC -1 ?AC? -0.7
A and C are 90? apart Results agree 50 of the
time. 50 of the time AC 1 50 of the time
AC -1 ?AC? 0.0
A and C are 135? apart Results agree 85 of the
time. 85 of the time AC 1 15 of the time
AC -1 ?AC? 0.7
Mathematical aside
?AC? - cos ?
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A

• A and C are 135? apart
• ?AC? 0.7
• A and D are 45? apart
• ?AD? - 0.7
• B and C are 135? apart
• ?BC? 0.7
• B and D are 135? apart
• ?BD? 0.7

D
45?
90?
B
90?
C
?Q? ?AC? - ?AD? ?BC? ?BD? 2.8
Two spins in an EPR state can violate Bells
inequality!
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Summary

• Pairs of quantum particles can be in an entangled
  state.
• EPR (1935) argued that the correlations in such a
  state indicate that quantum theory is
  incomplete. There must be things that are
  physically real that are not accounted for in
  quantum theory.
• EPR argue for local realism. which states that
  the world can be divided up into separate pieces
  that act independently.
• John Bell (1964) showed that, if local realism is
  true, then a certain inequality must always hold
  true ?Q? ? 2.
• However, a pair of entangled quantum particles
  can violate Bells inequality ?Q? 2.8.
  Furthermore, many experimental tests have
  verified that this theoretical prediction is
  right.
• Moral Einstein was wrong. Local realism cannot
  be a correct picture of the physical world!

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The horns of the dilemma
Realism The probabilities of quantum mechanics
merely reflect our ignorance of hidden factors.
Locality What happens here has no instantaneous
effect on what happens there.
At least one of these must be wrong. Take your
pick.