Re: Superposed observers (was No new Einstein)

rof@xxxxxxxxxxxx ha scritto:

...

> ... essentially it comes down to your claim in quant-ph/0007117
> that it is possible to distinguish between cases (a) and (b) above.
>
> As I said, I believe there's a calculational error in that paper and
> that in both cases the detector clicks 1/8 of the time, so it
> can't be used to detect superpositions.

I am confident that the calculations are correct. In order to
facilitate verification however, I will provide a rationale for the
result.
The key point is very simple. The shutter on the upper photon induces a
random phase change, which destroys any interference pattern between
the upper and the lower photon. This is quite intutive and is actually
proven at the bottom of page 5 of [1]. At the upper beam-splitter you
have an incoming upper photon with probability 1/4 and and incoming
lower photon with probability 1/2. Since they do not interfere, after
being halved passing through the beam splitter , they will yield a
1/4+1/8=3/8 probabilty at the detector. In the non-superposed shutters
situation the phases stay put and you get cancellation at the upper
beam-splitter, so the result is 1/8.

Your calculation is based on an interpretation of entanglement that I
consider erroneous, although it may provide a handy rule of thumb in
trivial situations.
I have tackled the issue of entanglement in my post [2] (cf. [3],[4]).
Indeed the whole procedure I am proposing depends on the fact that
entanglement is an observer-dependent property of the measurement
process, not of the wave function, i.e. as long as evolution is
unitary, entanglement has no object.
In the setting we are discussing that implies that prior to measurement
all photons can be summed and subtracted, as long as their phases can
be tracked.

>
> Here's another reason why it's impossible to detect superpositions
> that way. Your claim is that you can distinguish between a series
> of states (a_n|SA>+b_n|SB>)/sqrt(2) where the a_n and b_n are
> random complex numbers of modulus 1, and a series of states
> each of which is |SA> or |SB>, with 50% probability of each.
>
> The reason that I have subscript n on a and b is because, as you
> say, the relative phases are uncontrollable, and hence must
> be different each time you prepare the system and make one of
> your weak measurements.
>
> Now, I claim that if you can do the above, namely distinguish between
> the cases (a) and (b) which you described above through any series
> of measurements, then you can signal faster than light.
>
> Proof: Let the superposition (a_n|SA>+b_n|SB>)/sqrt(2) arise
> in the following way. Alice and Bob, situated far away from one another,
> receive spin-half particles in the singlet state
> (|up>|down>-|down>|up>)/sqrt(2). Alice can measure the spin of
> her particles along the z-axis or along the x-axis.
>
> Bob allows his incoming particle to interact with the beam absorber
> so that if the x-spin of the particle is up, the beam absorber does
> not block the beam, while if the x-spin is down, the absorber does
> block the beam.
>
> That is, Bob allows the incoming particle to interact with the
> absorber with an interaction which implements the following:
>
> |up_x>|SA> --> |up_x>|SA>
>
> |down_x>|SA> --> |up_x>|SB>
>
> (It is an exercise in quantum mechanics to show that the above can
> be implemented with a unitary time-evolution operator.)

As I said, in this context I regard this notation as meaningless. Since
entanglement is a property of the measurement process, introducing it
within a unitary process leads to results that, if I am right, are
testably erroneous.

>
> Now, from Alice's point of view, if she measures along the x-axis,
> then, since she adopts the Copenhagen interpretation (which gives
> the same experimental predictions as anybody's favourite interpretation,
> and which she is entitled to use), she considers that Bob is receiving
> particles which are _either_ |up_x> or |down_x> and not a superposition
> of the two. Hence, if she measures along the x-axis, then she can
> say with certainty: "Bob's results, whatever they are, will be
> consistent with the results that he would get if he was receiving
> a beam of particles, each of which is in either |up_x> or |down_x>.
> That is, Bob's beam absorber is _either_ in the state |SA> or
> in the state |SB>."
>
> On the other hand, if Alice measures along the z-axis, then
> Bob (as far as Alice is concerned) is receiving particles
> with a well-defined value of z-spin. That is, Bob will receive
> particles whose states are either |up_x>+|down_x> or
> |up_x>-|down_x>. Consequently, Bob's absorber will end up
> in the states |SA>+|SB> or |SA>-|SB>.
>
> Notice the crucial point - Alice can *control* whether Bob's
> absorber is in a superposition or not.

This is meaningless.
In your setting Alice cannot control whether the system is in a
superposition or not in Bob's perspective. The information she
transmits must in any case be unknown to Bob, otherwise there can be no
question of superluminal signalling. Actually Alice is in the situation
of Nick inside the box, who, if we have not agreed otherwise
beforehand, can either use the Geiger counter or use his
free will to arrange the shutters.

The key point here is that using a Geiger counter, tossing a coin or
resorting to one's free will have exactly the same effect on the
procedure, i.e. they will create a superposition in the perspective of
an external observer. From the point of view of an external observer
impredictability and free will are indistinguishable/identical. When
Alice decides which bit to send next, she is acting as a Geiger
counter, creating uncertainty and therefore a superposition of herself
in Bob's perspective.

In a nutshell, Nick or Alice using their free will are not different
from a Geiger counter.

For an the external observer (Bob in this case) , the shutters will
simply stay in a superposition, as long as he does not know in which
position they are, either checking directly or because the procedure
has been agreed beforehand with him.

Alice's logs may not coincide with Bob's when they finally exchange
information . Bob's logs will appear fudged by input from shadow
instances of Alice, preventing superluminal communication, but that's
the whole point of detecting superposed observers.

>
> Consequently, if Bob can detect whether or not his absorber
> is in a superposition, by any method at all, then he can
> receive signals from Alice, faster than light. This concludes
> the proof.
>
> Now, this does not mean that it is impossible to detect
> superpositions at all. It only shows that it is impossible
> to detect superpositions when the relative phase of the
> superposed states is unknown. In the case above, Alice
> can control whether Bob's absorber is in a superposition,
> but she can't control the phase - it might be |SA>+|SB>
> and it might be |SA>-|SB>, and Alice has no control over
> that. If she could control that, she could signal faster
> than light.
>
> Similarly, with your interferometry experiment, because
> the relative phase b/a is unknown, it is impossible to
> detect the presence of superpositions. As I said in my
> previous post, if you repeat the calculation taking into
> account the entanglement of the photon with the beam
> absorber, the result is that the detector clicks one
> eighth of the time, and hence the superposition can't
> be detected.

We'll see. I certainly do not expect my argument to carry the day based
on its theoretical merits, but the effect I am describing should nag
"interaction-free" measurements as soon as they are used to actually
extract information that the observer does not hold yet. Experiments of
that kind have been in the pipeline for a while (cf. [5],[6]).

Cheers and thanks for the stimulating feedback.


IV


[1] http://arxiv.org/abs/quant-ph/0007117
[2]
http://groups.google.com/group/sci.physics.research/msg/6bcaaef954a7bb83
[3] I. Vecchi "Is entanglement observer-dependent?" at
http://xxx.lanl.gov/abs/quant-ph/0106003
[4] G. Ortiz, R. Somma, H. Barnum, E. Knill, L. Viola "Entanglement as
an Observer-Dependent Concept ..." at
http://lanl.arxiv.org/abs/quant-ph/0403043
[5]
http://bulldog2.redlands.edu/facultyfolder/deweerd/research/IFM-AJP.pdf
[6]
http://www.physics.uiuc.edu/People/Faculty/profiles/Kwiat/Interaction-Free-Measurements.htm#Now

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