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LHVs vs qm some newer ideas

  1. Apr 25, 2005 #1

    vzn

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    hi all in another recent thread there was an epic battle between
    a local realist & a qm formalist. good theatre. but a stalemate in
    the end.

    I would like to post just a few key recent refs I was reminded of by that
    thread, which might be useful for anyone interested in the area. from
    your pt of view, either the LHV position is increasingly untenable for
    anyone other than fanatics. or, another idea-- there is only a very
    narrow slice of LHV theories remaining that have not been ruled out so far,
    which have to have a higher degree of sophistication than some of the
    "toy models" considered by bell.

    more discussion on the "qm2" mailing list

    http://groups.yahoo.com/group/qm2/

    a. phd & nobel prize winner t'hooft is working on LHVs somewhat
    recently which can be found in this paper. the idea is to use a set
    of simple harmonic oscillators as the hidden systems. this would tend
    to refute the idea that there is no point in working on LHVs, that the
    whole matter is closed, that there is no possibility, etc. some other
    researchers are already building on it

    How Does God Play Dice? (Pre-)Determinism at the Planck Scale
    http://www.arxiv.org/abs/hep-th/0104219

    b. I think bell came up with a wonderful analysis of this problem but
    in my opinion his form of the LHV is too strict. (or equivalently, you
    can see it as ruling out all but a very restricted class of LHVs, which
    have not been explored too much). lets look at the probability
    distribution of the single hidden variable as we rotate analyzers in the
    bell experiment. bells proof assumes this distribution does not change
    as their difference angle changes. reasonable for a toy LHV model.

    however there do exist LHV models such that the lambda probability
    distribution can change as the analyzers change. the details are subtle
    but mostly unexplored. therefore it makes a lot of sense to look
    at LHV theories that are maximally compatible with the predictions of QM.

    c. a very neat experiment purports to do a "efficient detection" of
    a bell experiment for the 1st time. published in nature.
    "experimental violation of a bells inequality with efficient detection"
    by rowe et al, 2001. link to the paper & my analysis of this here.

    http://groups.yahoo.com/group/qm2/message/9730

    basically I agree its a
    beautiful tour-de-force qm experiment which measures what it
    purports to measure, but from the pt of view of bell rigor, the experiment
    is very much lacking in rigor. the authors trapped two ions in an
    atom trap. but the bell experiment consisted of a single laser and single
    detector for BOTH ARMs of the experiment, which are traditionally at
    least spatially separated!!

    d. it seems to me the ultimate hidden variable in QM is simply
    "phi", the phase angle in the wavefn. it appears EVERYWHERE in
    qm derivations yet the theory denies that it can be measured. suppose
    there were some new theory that proposed that phi could somehow
    be measured via some clever advancements (theoretical/experimental).
    phi is clearly a hidden variable in QM--therefore "HVs" cannot
    be so controversial. the major problem is figuring out how
    to make a theory in which you have "locality", LHV.
     
  2. jcsd
  3. Apr 25, 2005 #2

    vzn

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    oops, one other tidbit I forgot to mention. something remarkable
    I ran across recently.

    e. eric reiter claims on his site to be doing a classic
    grangier,roger,aspect beamsplitter experiment with
    gamma rays, using a slab of aluminum as a beamsplitter (or other variations).
    the possible problem for this experiment is that I dont know of any
    literature which allows for the possibility of using a beamsplitter
    for gamma rays. so anyone can criticize his experiment on that
    element.

    I tend to think, barring contrary evidence, he may be
    doing what he says he is doing. this is the most sophisticated
    qm experiment by a non-university affiliated researcher Ive
    ever seen. very impressive on that level alone.
    he told me the whole effect can be demostrated
    with a setup for less than $500. I would very much like to see
    "professionals" analyze this experiment.

    http://unquantum.com

    conceptually, he looks at the histogram of timing differences between an
    "idler" and a "signal" photon and finds a nonrandom peak at t=0.
    in other words, the gamma ray photon goes "both ways" in the beamsplitter,
    and energy is conserved on average but not in individual events.

    to interpret his results reiter goes back to a "loading theory" of
    plank in which quantization is a property of matter, not of light.
    this is the basic thrust of semiclassical theories and
    "stochastic electrodynamics".

    now suppose this effect cannot be
    seen with low energy photons, where visible light does not
    have enough energy. that might explain why it has never
    been observed so far. maybe the effect is proportional to photon
    energy & is way too weak with visible light. although he does not
    have a quantitative theory yet, reiter seems to be
    proposing something along those lines.
     
  4. Apr 25, 2005 #3

    DrChinese

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    First, no one is saying that research should not be done in the area of EPR/Bell. However, I and others object to dismissing the existing experimental results, which are quite convincing.

    Second, the existing experiments are not toys and they do not test toy models. They go straight to the heart of what EPR discussed. And this is in fact the heart and soul of all LHV theories: that the spin components are determined at the time the particles are created - and NOT at the time of the measurement.

    Third, t'hooft says the following about LHV theories in the cited reference:

    "The Einstein-Rosen-Podolsky paradox and the violation of the Bell inequalities. This is surely the most dicult aspect to be addressed, and a completely satisfactory response has not yet been given. ..."

    So it is not like this isn't taken seriously by him - he recognizes that Bell is a significant obstacle in LHV development.
     
  5. Apr 25, 2005 #4

    DrChinese

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    This is not a comparable test to the Grangier experiments by a long shot. There is nothing to independently measure that a single photon is entering the apparatus.

    By the way: Peer review is as open to Reiter as anyone. And you can see from the picture that he has a lot more that $500 worth of gear.
     
  6. Apr 25, 2005 #5

    DrChinese

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    This criticism of Rowe is misplaced. Aspect and others showed through time-varying analyzers with spatial separation that the Bell inequality was still violated. Therefore, one concludes that separation is not an issue.

    Once a variable is ruled out as affecting application of a theory, it does not need to be addressed again in every permutation of that experiment. The simplest example of this concept is whether the results of experiments would be different during the day versus during the night, on Mondays versus Tuesday, in the lower hemisphere versus the upper hemisphere, etc. Failing to account for these permutations is not a scientific flaw, because the laws of physics have already been accepted as invariant to these. Ditto with the EPR "loopholes" which are largely closed at this time.
     
    Last edited: Apr 25, 2005
  7. Apr 25, 2005 #6

    vanesch

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    If you make allusions to my discussion with nightlight, you might have misunderstood the debate (or I might !).
    I was not defending QM for QM sake. I'm pretty open to theoretical challenges of QM. But the point was that nightlight was making an (in my honest opinion ; this was not some "point of view" matter) elementary mistake in what he claimed was *A QUANTUM THEORY PREDICTION*, in that he claimed that QED did NOT predict anticorrelation of photons in a setup a la Thorn.
    Now, or I have a serious misunderstanding of quantum theory myself, or this prediction (contradicted by all people working in the field) is not a QED prediction.

    I think I somehow proved that QED, in this particular case, did predict anti-correlation. Nightlight claimed that I could not use the same operator for the two detectors + electronics when there is and when there is not a beamsplitter in place, which I think is absurd because the beamsplitter didn't have anything to do with the "measurement apparatus", but just determined what state was presented to the measurement apparatus.

    Now, if someone thinks that *I* made a serious mistake there, I'd like to know!

    But the discussion was not about LHV versus QM. It was: "what is a correct prediction of QM".

    cheers,
    Patrick.
     
  8. Apr 25, 2005 #7

    vzn

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    vanesch.. a question for you .. I think you have referred
    to the "many worlds" interpretation of qm at different
    points. are you familiar at all with the "Ghirardi-Rimini-Weber"
    model? at first I thought it was only a philosophical interpretation,
    but I realized slightly to my shock
    recently after skimming a pretty good ref on the
    subject of bell thms (baggott, _the meaning of quantum theory_)
    that GRW actually proposed a slight physical twist to
    the collapse of the wavefn, with physically measurable parameters.

    also acc to baggott, as I recall at this moment, he says they have
    a LOCAL theory given only slight modifications to quantum mechanics.

    from what I can tell
    GRW seems to have highly influenced modern talk about
    "decoherence" which is quite mainstream and
    not at all as disreputable and stigmatized
    as "LHV" investigation. in fact afaik, there are even now experiments
    that "measure" "decoherence"... ?

    if anyone understands the GRW model or knows of a nice intro
    on it, please let me know
     
  9. Apr 25, 2005 #8
    I don't know GRW, but consider this : take original Bell's Ansatz for EPRB :

    local hidden variable (whatever they are physically) in A : the result is

    [tex] A(n_A,\lambda) [/tex] n_a is the direction of measurement in A, hence representing the measuring apparatus.

    Interpreations :

    1) Bell : [tex]\lambda[/tex] is a hidden variable (what's this ?) causing the result [tex] A(n_A,\lambda) [/tex] in A.. NB : the result can depend on the configuration of the measurement apparatus

    2) MW : [tex] A(n_a,\lambda) [/tex] is interpreted as : the result of measurement given by the apparatus in confiuration n_a, IN THE UNIVERSE indexed by the extra coordinate [tex]\lambda[/tex]

    So that in fact hidden variables and many-worlds are equivalent, they just are two different interpretations of the same physics (mathetmatical formula)
     
  10. Apr 25, 2005 #9
  11. Apr 26, 2005 #10

    vanesch

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    Well, I may have misunderstood what you say here, but if you mean that "the choice of the branch" by Alice is the "hidden variable", then this is in a certain way true. However, this "hidden variable" can stay local, thanks to an original "trick" in MWI: the "measurement" at Bob's doesn't have to be decided at the moment when Alice "chooses her branch" !
    Bob's "measurement" (from the point of view of Alice) only makes sense for Alice when she LEARNS ABOUT IT (through a "classical" communication channel). It is only at THAT MOMENT that the "hidden variable" (the chosen branch by Alice) determines Bob's outcome (from Alice's point of view).
    But at that moment, there is no space-like separation anymore between the two "decision points". So this can happen "locally".
    What Bell's theorem implies is that there cannot be a local hidden variable in the system which decides what will be Bob's outcome AT THE MOMENT WHEN ALICE FINDS HER OUTCOME. And MWI solves the riddle by *postponing* Bob's outcome (he's in a superposition from Alice's point of view) until she learns about Bob's "result", locally, at Alice's place.

    cheers,
    Patrick.
     
  12. Apr 26, 2005 #11

    vzn

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    fyi, just to correct myself.
    I was just rereading baggott. he said something about GRW
    reproducing classical predictions which I misinterpreted. later he says
    that GRW does not give a local theory for eg the bell experiments.
    I must say however that there is a lot of literature on what is
    called "dynamical collapse" by semimainstream physicists
    which is a kind of hidden variable theory--
    but generally not a local one.
     
  13. Apr 26, 2005 #12
    It's funny, you always speak about time, when the formla has absoutely nothing to do with it.

    In fact time is one possible way out of Bell's intelligent trap (not admitted by everyone like everytime)

    assume the result is time dependent : [tex] A(n_a,\lambda,t) [/tex]

    then the correlation is : [tex] C(A,B,t)=\int A(n_a,\lambda,t)B(n_b,\lambda,t)d\lambda [/tex] assuming the result are simultanous (in the lab frame for example).

    Then if you plug into CHSH, the correlation is measured in 4 different times :

    [tex] CHSH=|C(A,B,t_1)-C(A,B',t_2)+C(A',B,t_3)+C(A',B',t_4)| [/tex]

    since the correlation depends on t_i, and that those are all different, then you cannot factorize a la Bell, and you get CHSH<4.

    Note that every "visible" variable allow this, hence every variable still present in the correlation, as an extra-parameter more than the angles of measurement.
     
  14. Apr 27, 2005 #13

    vzn

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    fyi, one fairly straightfwd LHV theory apparently not covered
    by the bell impossibility/"no-go" thm:

    let the hidden variable determine whether the particle
    is detected or not, in conjunction with local polarizer angle.

    from what I can tell (working from memory)
    it is true that bell thms are fairly general and would tend
    to rule out a LHV such that whether the particle is detected
    at a remote arm is fixed by a hidden variable at the time of
    generation.

    however, if one examines a hidden variable that interacts
    with the polarizer angle, the bell thm cannot rule it out. I noticed
    that one can get different distributions of hidden variable
    lambdas at each end, depending on polarizer orientation, a
    situation not covered by his thms. (bell has a later thm that
    shows that if the lambda distribution is the same at each end,
    independent of polarizer angle, his thm still holds.)

    hence, a perfect LHV may be possible. gisin has a paper with a very
    similar idea.

    A local hidden variable model of quantum correlation exploiting the detection loophole
    http://arxiv.org/abs/quant-ph/9905018

    I tend to disagree with them that the above model necessarily involves
    a detector efficiency issue. you can model the dynamics in two ways:
    a) lack of detection is due to detector inefficiency
    b) lack of detection is not due to detector inefficiency. ie model holds
    even with 100% efficient detector

    the point is that (a) and (b) have IDENTICAL mathematics yet (b)
    is NOT MEASURABLE. ie the difference between (a) and (b) cannot
    be discriminated by experiment.

    it is counterintuitive in the sense that it is not
    deterministic in this way: not every "particle" or "wave"
    launched at the source can be detected, no matter how the
    measuring system is altered. yet it is deterministic in the sense
    of having cause & effect, but just not being able to associate an
    effect with every cause (not a 1-1 correspondence). it proposes
    "unavoidable information loss" so to speak.

    note this loophole is emphatically NOT the same as the efficiency
    of detection loophole, although conceptually they seem almost
    identical.

    this "lambda and polarizer determine detection"
    loophole involves the idea that even with a 100% efficient
    detector, hidden variables control whether the particle is
    detected.

    it actually fits into the copenhagen interpretation in
    a remarkable sense: it argues there
    is a physically real parameter that influences
    detection, but that there is no way to measure it.

    it also has a strong resemblance to the heisenberg uncertainty
    principle, which says we cannot sharpen complementary
    measurements past a certain point.
     
  15. Apr 27, 2005 #14

    DrChinese

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    This variation has no effect on Bell, it still applies. The question is whether Malus' law [tex]i_{source}cos^2\theta[/tex] applies as QM predicts and has been amply verified. If that law applies, then Bell applies and the detection HV is simply mixed in with the ensemble that becomes [tex]\lambda[/tex]. This then leads to the contradiction.

    Even in the scenarios in which the detectors are alleged to see an "unfair" sample - and perhaps you are referring to this element in your detection variation - there are serious problems: the "bias" (presumably a result of a local HV) must sometimes be positive and sometimes be negative, and must vary according to a very exacting formula intended to yield the observed results. (Experiments in which fair sampling is not an issue do not show any different results anyway.)

    In addition, this hypothetical local HV would need to apply ONLY to photon pairs in the singlet state! Because nothing like this has ever been otherwise observed in classical optics, which agrees with the predictions of QM.
     
    Last edited: Apr 27, 2005
  16. Apr 28, 2005 #15

    vanesch

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    The formula only has a meaning when you think that the process that "made the measurement happen", happened about simultaneously at Bob's and at Alice's, because the factorization (the locality condition) only applies at that point.
    This means that somehow, Bob DID a measurement, and Alice DID a measurement, and they HAVE a result.
    MWI "weasels out" by having Bob, and his measurement, in a "suspended state" (entanglement) until Alice learns about it, which is a local interaction at Alice's (I'm giving here purely the Alice point of view). So there's (from Alice's point of view) no "probability" to be associated with Bob's "measurement" at all. The only two real measurements are done by Alice:
    She first does her measurement on the photon, and then she does her second measurement on the messenger from Bob.
    Bob never measured anything, he's just part of the outer world with respect to Alice.
    And as the only probabilities generated are now locally, at Alice's, indeed, Bell's theorem doesn't apply here because everything happens locally at Alice's.
    The only thing she could eventually deduce is that Bob got some *information* from her first decision, by calculating if her second measurement contains information from her first CHOICE (not measurement). She will then find out that no such information transfer took place.

    cheers,
    Patrick.
     
  17. Apr 28, 2005 #16
    Patrick,

    The truth is that I understand almost nothing from the text. I rather believe in the principle : a formula is worth 1000 words.

    If you compute from basic quantum axioms the probabilities of a local measurement made by Alice only, you get that a parameter local at Bob's place from the definition of the probability in QM, gets then local in A...however there is no indication if there is a continuous displacement of that parameter. This is summarized by the equation of the prob. of let say + in A by : ([tex] \Psi[/tex] is the singlet state)

    [tex] p(+_A)=|<+_A\phi|\Psi>|^2 [/tex]

    here [tex]\phi[/tex] is located in B

    [tex] \Rightarrow p(+_A)=\frac{1}{4}(1-\cos(\theta_A-2\phi) )[/tex]

    here [tex]\phi[/tex] is located in A

    If you now have an interpretation of this quantum formalism then I think you're lucky...personally I think the formulas are self-explaining, and that phrasal intepretations are just making the clearness of the mathematical language opaque.
     
  18. Apr 29, 2005 #17

    vanesch

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    Ok, so let's do it formula-wise:



    Let us take as initial state a singlet state of 2
    1/2 spin particles:
    |psi_start> = 1/sqrt(2) (|z+>|z-> - |z->|z+>)

    We know that if the axis n makes an angle b with the
    z-axis, then we can write:
    |z+> = cos(b/2) |n+> + sin(b/2)|n->
    |z-> = -sin(b/2)|n+> + cos(b/2)|n->

    Rewriting psi_start with the n-axis for the second
    particle, we have:

    |psi_start> = 1/sqrt(2) {
    -sin(b/2)|z+>|n+> + cos(b/2)|z+>|n->
    - cos(b/2)|z->|n+> - sin(b/2)|z->|n-> }

    Note that the association of the first and second
    ket of each tensor product could just as well be obtained
    by an association of indices, with each ket living individually
    in its own Hilbert space.

    Let us also introduce "initial states" for Alice and Bob:
    |A0> and |B0>.

    So our starting state is |psi_start>|A0>|B0>

    Now, Alice "measures" the z-component and Bob "measures" the
    n-component (where Bob decides about the b-angle).
    This "measurement" is just a LOCAL interaction, where, for
    Alice, this corresponds to the following time evolution
    operator:

    U_A: |z+>|A0> -> |noz> |A+>
    |z->|A0> -> |noz> |A->

    (|noz> corresponds to an absorbed particle)

    This U_A operator acts only on the Alice space and the first particle
    space.

    In the same way, there is an operator corresponding to Bob:

    U_B(b): |n+> |B0> -> |noz> |B+>
    |n-> |B0> -> |noz> |B->

    After Alice and Bob's respective LOCAL interactions, (after
    applying the time evolution operator U_A x U_B(b) ), we have:

    |state1> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A+>|B+> + cos(b/2)|A+>|B->
    - cos(b/2)|A->|B+> - sin(b/2)|A->|B-> }

    Again, the association in tensor products can be replaced by
    indices which have been inherited ; the evolutions take place
    individually, in each factor space.

    Now let us assume that we take the viewpoint of Alice as an
    observer. This means that at this point, Alice will only observe
    one of her two branches, with probability given by the Born rule.
    She can be in the branch A+ with a probability given by the sum of the
    absolute square of all amplitudes of A+:
    ((-sin(b/2))^2 + (cos(b/2))^2)/2 = 1/2
    or she can be in the branch A- with a probability given by:
    ((-cos(b/2))^2+(-sin(b/2))^2)/2 = 1/2

    Let us, for sake of example, suppose that our Alice
    as an observer is in the A+ branch; which we will indicate by *:

    |state1> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A+*>|B+> + cos(b/2)|A+*>|B->
    - cos(b/2)|A->|B+> - sin(b/2)|A->|B-> }

    Next, there is a transmission of information from Bob to Alice,
    because Bob travelled to Alice's:
    this is implemented by the following time evolution operator:
    (which can now act LOCALLY upon the Bob and Alice, because both
    are now in the same spot at the same time:

    U_T : |A+>|B+> -> |A++>|B++>
    |A+>|B-> -> |A+->|B-+>
    |A->|B+> -> |A-+>|B+->
    |A->|B-> -> |A-->|B-->

    (note that A-+ means that Alice first had locally - and then saw
    Bob with a +, while B+- means that Bob first had locally a + and
    then saw Alice with a -)

    So now we have:

    |state2> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A++*>|B++> + cos(b/2)|A+-*>|B-+>
    - cos(b/2)|A-+>|B+-> - sin(b/2)|A-->|B--> }

    But this is again something that is an observation from Alice's point
    of view (because it affects her body state).
    So we have to apply again Born's rule, which has now to choose
    between the A++ and the A+- state, with relative probabilities:

    sin(b/2)^2 and cos(b/2)^2

    So let us suppose that the first case wins:

    We now have:

    |state2> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A++*>|B++> + cos(b/2)|A+->|B-+>
    - cos(b/2)|A-+>|B+-> - sin(b/2)|A-->|B--> }

    Alice observed first a z+ state, with probability of 1/2.

    Next she observed Bob, when he came to see her,
    to be in a state B+ with (relative)
    probability (sin(b/2))^2

    Knowing what Bob did, she deduces that this means that Bob
    "measured" an n+ state. But that's a deduction, which does not
    need to imply that Bob somehow "was" in an n+ state and so that
    something happened at Bob's.

    We can also tell the story from Bob's point of view. It is quite
    different, but the states are the same.

    After the "measurement" interactions, we have the state:

    |state1> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A+>|B+> + cos(b/2)|A+>|B->
    - cos(b/2)|A->|B+> - sin(b/2)|A->|B-> }

    Because this changed Bob's body state, and Bob is now our observer,
    we have that Bob is in the B+ state with probability 1/2 and
    in the B- state with probability 1/2.
    Let's assume that our observer Bob is associated with the B- state,
    which we indicate with #:

    |state1> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A+>|B+> + cos(b/2)|A+>|B-#>
    - cos(b/2)|A->|B+> - sin(b/2)|A->|B-#> }

    Now, Bob goes and sees Alice ; the same state2 as before is there:

    |state2> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A++>|B++> + cos(b/2)|A+->|B-+#>
    - cos(b/2)|A-+>|B+-> - sin(b/2)|A-->|B--#> }

    But because we see things from our Bob observer point of view,
    Bob will observe only one of his body states B-+ or B--, according
    to the relative Born rule: cos(b/2)^2 versus sin(b/2)^2.

    Let us suppose that, with a probability cos(b/2)^2, our Bob observer
    is associated with B-+:

    |state2> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A++>|B++> + cos(b/2)|A+->|B-+#>
    - cos(b/2)|A-+>|B+-> - sin(b/2)|A-->|B--> }

    We could eventually consider both adventures together:

    |state2> = 1/sqrt(2) |noz>|noz> {
    -sin(b/2)|A++*>|B++> + cos(b/2)|A+->|B-+#>
    - cos(b/2)|A-+>|B+-> - sin(b/2)|A-->|B--> }

    Alice "observer" ended up in the A++ body state, and in that branch, observes a Bob body state which is in a B++ state, and whose body will
    act (like talking) accordingly ; so for Alice observer, it is clear that
    "Bob" saw an n+ state.

    For Bob "observer", he ended up in the B-+ body state. In that branch,
    he observes Alice in a body state +-, so it is clear for him that
    Alice observed + and that he observed - (the Alice body in his branch
    will agree with that).

    cheers,
    Patrick.
     
  19. Apr 30, 2005 #18
    You always remain local by an a priori position :

    1) you speak about the measurement operator : [tex] (\vec{\sigma}\cdot\vec{n}_A)\otimes(\vec{\sigma}\cdot\vec{n}_B) [/tex] which is not local, because it disturbs both systems A and B.

    The local measurement operator for A is [tex] (\vec{\sigma}\cdot\vec{n}_A)\otimes(I_2) [/tex]. This measurement is LOCAL in A in the sense : it does not disturb B.

    The correlation between A and B is then given by a superposition of both type of measurements.

    2) Then there is no indication of how many time the collapse in a measurement takes, but it is a unitary operation of course, so you abusively speak about time evolution.

    Following Copenhagen QM :

    a) there is no indication if the collapse is composed by steps (continuous or discrete) between the initial state and the final wavefunctions.

    Let's put the hypothesis the process is composed by steps :

    b) These steps can have another physical dimension than time

    b) If this process is continuous for example, then it can be non unitary between the initial and the final points. Expressed mathematically, the collapse of the WF could be put as the Ansatz :

    [tex] \psi(\epsilon=0)=|\Psi\rangle [/tex] singlet state
    [tex] \psi(\epsilon=1)=|+_A\rangle [/tex]

    We can naturally have [tex] ||\psi(0<\epsilon<1)||^2\neq 1 [/tex].


    3)

    a) if I'm not mistaken what you think is : p(+_A)=p(-_A)=p(+_B)=p(-_B)=1/2.
    if A and B measure along the same direction.

    b) What quantum mecahnics gives you locally is, in common with my previous message :

    [tex] p(-_A)=\frac{1}{4}(1+\cos(\theta_A-2\phi)) [/tex]
    [tex] p(+_B)=\frac{1}{4}(1+\cos(\theta_B-2\phi)) [/tex]
    [tex] p(-_B)=\frac{1}{4}(1-\cos(\theta_B-2\phi)) [/tex]

    hence :

    [tex] p(+_A)+p(-_A)+p(+_B)+p(-_B)=1 [/tex]

    with [tex] \max_{\phi} p(+_A)=1/2 [/tex]

    so that the sum of all the single event is one, not only one side of the system...because QM always considers the wholeness of the system , since it is described by a single wf.

    This gives a simpler way than yours, since for example let say : [tex]\theta_A=\theta_B=0 [/tex]. (A and B measure along the same direction)

    Let's assume : hypothesis : [tex] p(-_A)=0 [/tex].

    From this we are locally sure that the result is +_A.

    If you take the above probabilities, then you see this implies : [tex] p(+_B)=0 [/tex].

    Hence the result is -_B in B.

    As summary : taking the basic axioms of Copenhagen QM allows you to explain only with a single variable [tex] \phi [/tex], how the measurement on 1 side affects the other. NB: [tex]\phi[/tex] is just some eigenvector of the identity operator.
     
    Last edited: Apr 30, 2005
  20. Apr 30, 2005 #19

    vanesch

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    Ah, well, then apply first the A "measurement" operator (U_A x 1), and later apply the B "measurement" operator. That comes down to the same thing.
    Note that these operators are just the time evolution operator corresponding to the interaction of Alice (Bob) with some apparatus and the system under study, which must have a Hamiltonian associated with it. It is an abuse of language to call it a "measurement" operator ; it is just a time evolution like any other time evolution.

    Exactly. However, it is not [tex]
    \vec{\sigma}[/tex] ; it is much more complicated and describes the typical action of a measurement device (but expressed as a physical system, with its interaction hamiltonian), and Bob's (Alice's) body state with all its atoms and so on.

    Well, there is a certain interaction time needed for Bob's body and his measurement apparatus to interact with the particle ; it is this complicated interaction that is symbolised by the time evolution operator U_B, in a very complex Hilbert space (containing all of Bob's body states).
    But somehow we can assume that after a reasonable amount of time (say, a few seconds) Bob's body is in a certain state which corresponds to "Bob saw the red LED of his apparatus flash".
    As I cannot write down of course explicitly the hamiltonian (with all interactions in his retina, his nerve cells and so on on the molecular level: go figure!) I just write it down symbolically with U_A(t, t+a few seconds).


    But I'm giving you an MWI QM picture, which DOES suppose that there is no such "collapse" but only unitary interactions!

    Well, the MWI viewpoint is that it is just a time evolution like any other (of which the time derivative is the hamiltonian).

    Yes, but that's explicitly non-unitary (you cannot have it unitary). And now you will have to tell me what physical processes happen according to this evolution equation, and what physical processes happen according to the unitary time evolution. This is the dilemma of the Copenhagen interpretation.



    No, in all cases ! Even if they don't measure according to the same direction.
    That's true, no matter what "approach" (MWI, Copenhagen...) you take, no matter who measures first etc...
    It is essential, in that entanglement does not allow for information transfer.
    Alice will see half of her electrons spin up, and the other half: spin down ; and for Bob, the same ; no matter what angles they happen to chose.

    Well, that's wrong then. At least if you mean by p(-_A) the probability that Alice will measure spin down (unconditionally). This is 1/2. No matter what Bob does, and no matter which axis Alice chooses.
    We should first clear out this.


    Well, it is clear that p(-_A) + p(+_A) = 1, because Alice can only obtain 2 possible answers: spin up or spin down, right ?
    So an event will have given spin down at Alice's or it will have given spin up at Alice's. The sum of these probabilities must be equal to 1!

    Let's clear out this problem first...

    cheers,
    Patrick.
     
  21. Apr 30, 2005 #20
    That does NOT comes to the same thing....But to explain you the BASICS of QM I think it really becomes a hard work: Let start with the singlet state [tex] \Psi [/tex] :

    Apply AxI, suppose you get + in A, hence the final state is [tex] \psi_1=|+_A>|\phi> [/tex].

    [tex] |\phi> [/tex] is any unitary vector of R^2.

    Apply IxB on the state obtained..suppose B got -. Then the final state is

    [tex]\psi_2=|\phi>|-_B> [/tex].

    However applying AxB gives you the final state [tex] \psi_3=|+_A>|-_B> [/tex]

    if you suppose you get the same results of measurement (+ in A and - in B)

    You immediatly sees this are not the same final states...

    This is exactly the difference between local operators AxI, IxB and the non-local one AxB.

    I thought the time evolution operator should contain time, but I learn everyday like everyone else.



    It's known that calculable theoretical physics will never be able to reproduce the whole reality, by chance to give work to some physicist....However if you want to do numerical large simulation it's up to you, but those will still remain just in the box and not real.



    Well there are two processes you mixed up : the time evolution of the qbit system in a magnetic field for example, because you need the energy (Schroedinger equation)...and the wave function collapse (which is not an evolution) when you try to find out in which state your qbit is..

    You mixed up because in the case of qubit the energy is a multiple of the spin operator (with the gyromegnetic factor aso..)

    You cannot distinguish between a unitary evolution and a non-unitary one in the sense : you only have access to the initial and the final points, which have the same norm...but you don't know what happens inbetween.




    I told you : in the case you measure along the same direction, not in all cases.

    Are you always using a prioris like this ?


    If you want this to be 1/2 then you can forget QM. Just use the basic axioms....you always use your intuition instead of computing...that's misleading.

    For example the exercise is : compute the probability of getting +_A with the local operator AxI on the singlet state. You will see it's not 1/2 but :

    [tex] p(+_A)=|<+_A|<\phi|\Psi>|^2 [/tex] you know this ? This is the definition of the prob. of an outcome of a measurement in A in the z_direction. Put [tex] |+_A>=\left(\begin{array}{c} 1\\0\end{array}\right)[/tex] eigenvector of [tex]\sigma_z[/tex] and [tex] |\phi>=\left(\begin{array}{c}\cos(\phi)\\\sin(\phi)\end{array}\right)[/tex] an eigenvector of I.

    then :

    [tex] p(+_A)=\frac{1}{2}|\left(\begin{array}{c} 1\\0\end{array}\right)\otimes\left(\begin{array}{c}\cos(\phi)\\\sin(\phi)\end{array}\right)|\left(\begin{array}{c} 1\\0\end{array}\right)\otimes\left(\begin{array}{c} 0\\1\end{array}\right)-\left(\begin{array}{c} 0\\1\end{array}\right)\otimes\left(\begin{array}{c} 1\\0\end{array}\right)|^2[/tex]

    [tex] p(+_A)=\frac{1}{2}\sin(\phi)^2[/tex].

    NB :[tex]\phi[/tex] IS NOT THE DIRECTION OF MEASUREMENT IN A !!

    So you see : the prob. is not always 1/2...it CAN BE, but on average over the variable [tex]\phi[/tex] it's 1/4....If you allow the direction in A to change then you get what I wrote in the previous message.

    No, globally there are 4 events : +_A, -_A,+_B,-_B.....why do you split the world in 2 ?

    It's always the same problem : you follow your intuition and hence don't see the globality of the system. What do you do with B ? You again split the world in 2, so that you REDUCE THE UNIVERSE OF PROBABILITIES...but you don't explain why you reduce...and what's the point of reducing it..and why you're allowed to do it.

    Another way to explain the problem is : you have a GLOBAL system at A and B....Let's say you choose randomly one side (they are equivalent if you measure along the same directions) with prob. p(A)=1/2.

    Then you have partial probabailites p(+)=1/2 for + p(-)=1/2 for -..hence the total probability is :

    p(+_A)=p(A)*p(+)=1/2*1/2=1/4.

    This is the same for all single events, hence the events in A and B are equiprobable...but the sum of ALL of them should be 1...

    cheers,
    Patrick.[/QUOTE]
     
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