Superposed observers (was No new Einstein)
- From: "I.Vecchi" <vecchi@xxxxxxxxxxxxx>
- Date: Sun, 14 Aug 2005 17:28:36 +0000 (UTC)
rof@xxxxxxxxxxxx ha scritto:
> "I.Vecchi" <vecchi@xxxxxxxxxxxxx> writes:
> >Any state (including |a1>+|a2>) has "a well-defined value of some
> >observable", the challenge is to implement the measurement
> >corresponding to that observable in practice.
>
> Agreed, and it will involve constructing some machine which
> interacts with Nick in a very invasive way (sorry, Nick).
Never. In my experiments Nick will be treated like the gentleman he
is.
He may just be asked to signal the cat's good health by putting his
hand across one arm of a Mach-Zehnder interferometer.
...
> The hypothetical experiment involves Nick looking at Schrodinger's
> cat, so the system starts off in the state:
> |Nick with his eyes closed>(|deat cat>+|live cat>)
>
> After this, Nick opens his eyes and sees the cat, so the system
> evolves to the state:
> |Nick seeing a live cat>|live cat> + |Nick seeing a dead cat>|dead cat>
>
> I call a state of a two-component system entangled when the state
> cannot be factorised into:
> |state of one subsystem>|state of the other subsystem>.
> The state above, of the Nick-cat system is such a state, and so
> I say that Nick is entangled with the cat. I believe my usage
> of the term entangled is standard.
That's where the problem lies. You regard entanglement as a property of
the system's wave function. My point is that is not so. Entanglement is
a property of the measurement process. Roughly speaking, entanglement
is the observer's blueprint for state vector reduction ([1], cf. [2]
and [3]). The direct product notation you are using is widespread and
handy, but it is based on a conceptual mistake which comes home to
roost in your argument and in large swaths of current QM research. In a
nutshell, the |N1>|C1> representation encodes the erroneous assumption
that the system already knows in which basis it is going to be
measured, whereas "entanglement is an observer-dependent concept, whose
properties are determined by the expectations of a distinguished set of
observables without reference to a preferred subsystem decomposition,
i.e., it depends on the physically relevant point of view" ([2]). It is
not a property that gets encoded in the wave function.
> I'll ignore the notation |a1> and so on and rewrite the above state
> as |N1>|C1>+|N2>|C2>, so that |N1> means |Nick seeing a live cat>
> and so on. Now, if I want to observe a superposition of |N1> with
> |N2>, I cannot do this by doing any kind of measurement on Nick
> alone. In order to detect Nick in a state like |N1>+|N2>, for
> example, the system would have to actually be in a state which
> looked like (|N1>+|N2>)|C3>, where |C3> is some state of the cat.
> However, the system isn't factorizable into a state of Nick times
> a state of the cat.
>
> In order to get the system into a configuration where I can observe
> the superposition of Nick alone, |N1>+|N2>, I would have to make the
> cat undergo some procedure whereby both |C1> and |C2> evolved to |C3>,
> so that the system could be in a state (|N1>+|N2>)|C3>, after which
> point I could ignore the cat and focus my attention on Nick, who
> is now in a genuine superposition.
> Now, this is impossible already (because of unitarity, |C1>
> and |C2> can't both evolve to a single state |C3> (proof: if
> that were the case, |C1>-|C2> would time-evolve to zero)). Instead,
> you'd need to have the cat "re-interact" with Nick with some
> interaction which makes the system evolve like:
> |N1>|C1> --> |N3>|C5>
> |N2>|C2> --> |N4>|C5>
> so that the system becomes factorizable into (|N3>+|N4>)|C5>, where
> |C5> is some state of the cat and |N3> and |N4> are the states
> "Nick, who saw a live cat turn into a cat in state C5" and
> "Nick, who saw a dead cat turn into a cat in state C5", respectively.
> After this procedure, the information about whether the cat was
> alive or dead has been erased (from the state of the cat, if not from
> Nick's memory).
>
> This is why it is necessary to erase the information (from the cat)
> about whether the cat was alive or dead in order to observe
> interference of Nick.
The state of the system Nick+cat IN OUR PERSPECTIVE (the state vector
encondes the observer's knowledge, i.e. the information defining his
image of reality) after Nick has looked at the cat is given by a vector
|NC>, which we may or may not express as |NC>=|NC1>+|NC2>, where |NC1>
and |NC2> are respectively "Nick sees that the cat is dead" and "Nick
sees that the cat is alive". Notice that in |NC> the cat is either
dead or alive in Nick's perspective, but not in ours.
The state vector of the system in our perspective may be different from
that of each instance of Nick+cat, since "the quantum state of a system
is always a state of that system with respect to a certain other
system. More precisely, when we say that a physical quantity takes the
value v, we should always (explicitly or implicitly) qualify this
statement as: the physical quantity takes the value v with respect to
the so and so observer"([4]).
Now, what we want to do is to show that a system that subsists in state
|NC> (that is a state where we DO NOT measure Nick and the cat so as to
extract information on whether the cat is dead or alive) yields
measurement outcomes that are different from those which are obtained
under the assumption that the state is either |NC1> or |NC2>. That's
what detecting superpositions (of Nick, cats, SQUID currents,
electrons or poker hands) is all about.
> This can be compared to a two-slit interferometry experiment
> in which filters are put over the slits, so that particles
> going through one slit have a given polarization while
> particles going through the other slit have a different
> polarization. After passing through the screen with the
> slits and filters, a particle is in the state
> |x1>|up> + |x2>|down> (in entanglement speak, its position
> is entangled with its spin). Here, |x1> refers to the
> position of one of the slits and so on. The particle subsequently
> evolves towards a detecting screen where its position is measured.
>
> Suppose an interference pattern looks bright at x3 and
> dark at x4, while no interference pattern looks half-illuminated
> at both x3 and x4. This is the case if |x1> -> |x3>+|x4>,
> while |x2> -> |x3>-|x4>. So if there had been no filters
> over the slits, the particle would be in the state (|x1>+|x2>)|S>
> after passing the screen, where |S> is some spin state which
> won't affect the interference pattern, and the particle would
> evolve to the state |x3>|S> at the detecting screen, at which
> point there would be a flash at x3 but not at x4. That is, without
> the filters, there would be an interference pattern.
>
> On the other hand, if the filters are there, the final state
> of the particle at the detecting screen is
> (|x3>+|x4>)|up> + (|x3>-|x4>)|down>. If the detecting screen
> measures only position, then a beam with this state is
> indistinguishable from a beam which consists of 50% particles
> in the state |x3>+|x4> and 50% in the state |x3>-|x4>. That
> is, x3 and x4 will be equally illuminated and there will be
> no interference.
>
> Similarly in the case of Nick and the cat. If Nick is entangled
> with the cat, you have to disentangle them before you can do
> an interference experiment on Nick alone. This process erases
> any information the cat has about what its original state was,
> or, more precisely, in order to observe interference of Nick,
> you have to remove any information that the cat might be able to
> reveal about which state Nick was in.
That's simply not what I plan to do, see above. By the way, the
experiment's structure should be exactly the same if I wanted to detect
only the cat's superpositions or the superpositions of shutters.
...
> >In your overall argument you are confusing the result of the final
> >measurement B, which we carry out in the standard basis (|Nick seeing
> >cat dead>, |Nick seeing cat alive> with the result of measurement A we
> >carry out to detect superpositions, where we do not extract
> >information in that basis, so that we don't collapse ourselves into one
> >of Nicks standard branches.
>
> I don't think I am. I'm claiming that the first measurement (in
> which interference is observed) cannot be made without interfering
> with the result of the second measurement (live cat or dead cat).
> That is, the information which the system contains about the result
> that the second measurement (live or dead) would give if done at
> the beginning, has to be destroyed in order to observe the
> superposition.
>
> You may, after the first measurement, make the second measurement
> and you will find a result, but that result is not the same result
> that would would have gotten if you had left out the interference
> experiment and simply asked Nick what he had seen.
> This is a classic case of what Niels Bohr would have called
> complementarity. Measure A, measure B, measure A again. Do you get
> the same result for the first and the third measurements? No. A and
> B are complementary. In this case, observation of the superposition
> of an observer is complementary to the observation of the memory
> of the observer about which branch of the superposition he was in.
>
As I said in my previous post, the measurement to detect superpositions
is ideally a non-demolition measurement, where the state vector stays
what it is. In practice that may be hard to achieve perfectly, but it
should be possible to conduct the experiment in a way that operatively
approximates the ideal situation.
First however we have to get rid of the conceptual mess created by
decoherence theory, where entanglement is muddled into the
wave-function instead of being recognised as a constraint on
measurement outcomes.
After realising the mistake one may harness decoherence (which in this
context means only random perturbation of relative phases and is a
LOCAL process) to detect residual superpositions in composite systems
where only one component (the one containing Nick and the cat)
decoheres.
Decoherence being local, the actual interaction between a
non-decohering subsystem and a decohering one may yield different
results than the hypothetical overall decoherence of the composite
system. Such discrepancy may reveal the persistence of superpositions.
IV
[1] I. Vecchi "Is entanglement observer-dependent?" at
http://xxx.lanl.gov/abs/quant-ph/0106003
[2] 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
[3] G. Mahler "The partitioned quantum universe: entanglement and the
emergence of functionality" at www.mindmatter.de/mmpdf/mahler.pdf
[4] C. Rovelli "Relational Quantum Mechanics"
http://arxiv.org/abs/quant-ph/9609002
.
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