Re: A Look at Quantum "Spookiness"


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"Erops" <erops@xxxxxxx> wrote in message
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A Look at Quantum "Spookiness"

The results of quantum theory were described as "spooky" by
Drs.
Einstein, Podalsky, and Rosen because quantum theory seemed to
reject
"objective reality". They believed that all observed effects
must be
produced by "local" causality. Their conclusion resulted from
their
firm
belief that information could not travel faster than the
velocity of
light. Indeed, if this were the case, quantum theory would
indeed be
"spooky". Quantum theory required, for example, that "paired
photons"
maintain polarizations which were opposite in direction . If the
polarization angle of one of the "paired photons" were changed,
the
polarization angle of the other photon of the pair must
instantaneously
change to match.

Actually this is not a very accurate description of the situation.
It
is
about determining the polarization, not changing it. The key thing
is
that
one can measure the polarization in two directions which are
incompatible
observables in quantum mechanics, so the results cannot be
explained
merely
by the states the particles started off in. Which direction to
measure
can
be chosen by a central controller communicating simultaneously
with
each
end
of the experiment.

If you read the papers about Aspect's experiment (which verified
Bell's
theorem empirically), as I have, you will see Bell's result is
only
visible
in the statistics, not in any individual measurement. No-one has
found
any
way to use this effect to communicate faster than light, and very
few
people
expect this to ever happen.

Although, they have proven conclusivly that the aformentioned
effect is
real. Not by statistics, but by concrete mathimatics. Instead of
coupling two photons, they used three photons, and the math became
"Always" (Quantum Spookiness) or "Never" (Einstein Hidden
Variables).
Turns out its "Always"

It is possible my intended meaning was not clear. The effect is in
the
statistics of the measurements, and can be seen when one collates the
experimental data from both (or all three) afterwards, but at the
time,
there is no way for any of the participants to use the effect to
communicate
with another at faster than the speed of light.

One article I read on this said that when a participant makes a
measurement
it "determines an aspect of reality". This aspect of reality is
immediately
known at another point connected by quantum entanglement. The reason
this
cannot be used to communicate is that the person who made the
measurement
did not specify what the aspect of reality was (for example, whether
the
photon was vertically polarised), he merely measured it. From
measurements
made at the other end, the other participant cannot determine
anything
about
what the first person did. We can only be sure that if the other
person
measures the vertical polarisation, his measurement will be
determined by
the measurement that the first person made.

Is this clear now?

Yes... But, what if entangled particles are actually different facets
of the same underlying particle? Maybe there is no such thing as "one
photon" entangled with another "one photon". Maybe once they are
entangled, the actually become a single entity that exists at more
than
one physical point.


Yes, it's absolutely right to think of the two photons as two parts of a
single quantum mechanical object.(called a pair of coupled photons)
until a
measurement of the polarization of one of them is made. This
"immediately"
fixes the polarization of the other photon, but also breaks the
coupling, in
the sense that nothing you do to one of the photons afterwards will have
any
further implications for the other photon (i.e. if anyone made any
measurements on one of the photons, the new information would not be
useful
to predict what a measurement on the other photon would be.

Fortunately this is made clear by the "bra-ket" formulation due to
Dirac.
When you write down the quantum states, you can see the way two photons
can
be "entangled" and it also shows how when one makes a measurement they
become "untangled".

Let a single photon 1 in a vertically polarized state be:

| V1 >

and a single photon 1 in a horizontally polarized state:

| H1 >

If photon 1 is unpolarized it is a state which is 50% of each:

1/2 | V1 > + 1/2 | H1 >

{ a photon could also be polararized at any other chosen angle, and
these
states can be mixed together in various ways, but that's just confusing
the
issue unnecessarily }

Suppose we start with paired photons of random polarisation. We write
the
state where they are both vertically polarised

| V1 > (x) | V2 >

and the state where they are both horizontally polarised

| H1 > (x) | H2 >

Each of these is what is called a tensor product of the individual
photon
states. A state that is a tensor product of states of the individual
photons
is not entangled, because the behaviour of each of the photons does not
depend at all on anything you do to the other one.

The initial state in the experiment is given as:

1/2 | V1 > (x) | V2 > + 1/2 | H1> (x) | H2 >

Which says there's 50% chance they are both vertically polarized and 50%
chance they are both horizontally polarized. This is an "entangled", or
"coupled" or or "correllated" state. This is because it is impossible
to
write this as a single tensor product of the states of two different
photons. It is entangled because any measurement of polarization of one
photon acts on this state in a way which gives you information about the
other photon.

If we make a measurement of the polarization of one of the photons in
the
vertical direction, mathematically we apply an operator to that state
above
and the rules say that we get either:

| V1 > (x) | V2 >

or else

| H1 > (x) | H2 >

{with a real polarizing filter, the latter would be the case where it
reflected}

Either of these is no longer an entangled state, as it is a tensor
product
of the states of two separate photons, so it behaves like two separate
photons. No measurement on one photon will tell you any more about the
other
one.

At the risk of being repetitive, if the state of two particles cannot be
written as a tensor product of a state of one particle and a state of
the
other particle, they are entangled.

That explains some to me, thanks.

So, what happens if I somehow alter (without mesuring) the polarization
of one of the photons? Does the other photon change? In other words,
given A is entangled with B, if you change A by 10 degrees, does B
change by 10 degrees as well?

What about alterations that are unobservable?

Sorry if these are newb questions.


Well the answer in the case we were looking at is this.

starting with the state:

1/2 | 0_1 > (x) | 0_2 > + 1/2 | 90_1> (x) | 90_2 >

i.e. 50% chance that both are polarized at 0 degrees and 50% chance that
both are polarized at 90 degrees.

If you make a measurement of the polarization of one photon in _any_
direction, you immediately know what the polarization of the other photon
is. For example, if you measure the polarization of the 1st photon in the
direction 45 degrees, you either find it is polarized at 45 degrees or it
is polarized at 135 degrees (just like horizontal = not vertical, only
twisted around a bit).

When you have done this, you immediately know the second photon has the
same polarization, either 45 degrees in one case, or 135 degrees in the
other. This gives us a state like:

| 45_1 > (x) | 45_2 >

i.e. both photons are polarized at 45 degrees.

But once you have done this once, the photons are unentangled, and you
can't do anything to one photon that tells you anything more about the
other one. For instance, once you have found the polarization of both
photons is at 45 degrees, if you then measure the polarization of the
first photon at 90 degrees, there's a 50% chance you find it is and a 50%
chance that it isn't (which means it's polarized at 0 degrees).

But the polarization of the second photon stays at 45 degrees, since it is
no longer coupled. So we just get a state like:

| 0_1 > (x) | 45_2 >

or

| 90_1 > (x) | 45_2 >

There are very simple mathematical rules for exactly what happens when you
apply any measurement to any sum of tensor products of states that gives
exactly the results above.


Unfortunately, I overstated the case above. Doing the calculation, it is
clear that one does not get a certain polarization for the second photon for
all possible choices of the angle of the filters, just a increased
probability in general. My argument for this is as follows:

The rules for measurement on 1 photon are easy. If it is polarized at an
angle a and you measure the polarization at angle b , the probabilty is
(cos(a-b))^2 that it gives "true" and 1-(cos(a-b))^2 that it gives "false".

i.e. < M(b) | a_1 > = cos(a-b)^2

If you tensor this photon state with the state of another photon,
measurements applied to the first photon don't affect the state of the
second:

< M(b) | a_1> (x) | c_2 > = cos(a-b)^2 | c_2>

and the rule can be extended to a applying a measurement to one of a pair of
coupled photons, by applying it to each component separately:

<M(b)| applied to 1/2 | 0_1 > (x) | 0_2 > + 1/2 | 90_1 >
(x) | 90_2 >

= (1/2)(cos(b))^2 | 0_2 > + (1/2)(cos(90-b))^2 | 90_2 >

so in this case, our knowledge about the polarization of the second photon
is only perfect when a measurement was made at either 0 degrees or 90
degrees (180 or 270 degrees are the same, of course).

Unfortunately, the worst case is where we make a measurement at 45 degrees
(or 135, 225, 315 ), when this is:

(1/2)(cos(45))^2| 0_2 > + (1/2)(cos(45))^2 | 90_2 > = (1/4) | 0_2 > + (1/4)
| 90_2>

so in this case we know nothing about the polarization of the other photon
(any measurement of the polarization of the second photon gives a positive
result 50% of the time).

However, it is possible to have situations where _any_ measurement of
polarization of the first photon will give some (but not perfect)
imformation about the second photon. For example, if we know that the two
photons have identical polarization, but no idea what that is, the state is
an integral from theta=0 to theta=2*pi of

1/(2*pi) | theta_1 > (x) | theta_2 >

{a state where both photons are polarized at the same random angle theta
between 0 and 2*pi}

Measuring the polarization of the first photon at some direction phi gives
an integral over theta of 1/(2*pi) (cos(phi-theta))^2 | theta_2 >

Making a measurement of the polarization of the second photon at angle phi
gives an integral over theta of 1/(2*pi) * (cos(phi-theta))^4

The ratio of the probabilities (which is the conditional probability that
the polarization of the second photon agrees with the measurement of the
first) is the ratio of the integral of cos^4(theta) to the integral of
cos^2(theta).

This is 0.75.

So, in this case there is a 75% chance that the polarization of the second
photon agrees with that of the first, for measurements of polarization in
any direction.

Any questions/comments/improvements?


.

Relevant Pages

  • Re: A Look at Quantum "Spookiness"
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  • Re: A Look at Quantum "Spookiness"
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