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Born vs. ABL

  1. Nov 30, 2008 #1

    Fredrik

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    The Born rule says that if a system is prepared in state |a>, and a measurement of an observable Q is performed, the probability that the result is qi is

    [tex]P(q_i)=|\langle q_i|a\rangle|^2[/tex]

    The ABL rule (Aharonov, Bergmann, Lebowitz) says that if a system is prepared in state |a>, and a measurement of an observable Q is performed, the conditional probability that the result is qi, given that the system is observed in state |b> after the measurement of Q, is

    [tex]P(q_i|b)=\frac{\big|\langle b|q_i\rangle\langle q_i|a\rangle\big|^2}{\sum\limits_j\big|\langle b|q_j\rangle\langle q_j|a\rangle\big|^2}[/tex]

    I'm trying to understand the relationship between these two rules, and what they imply about quantum mechanics and time reversal. It seems to me that QM with the Born rule as a postulate is not invariant under time reversal, and that QM with the ABL rule instead of the Born rule is invariant under time reversal. So these seem to be different theories. On the other hand, it also seems that these rules can be derived from each other.

    What's the correct interpretation here? Are these two different theories or not? Is QM with the Born rule really not invariant under time reversal? Is QM with the ABL rule really invariant?
     
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  3. Nov 30, 2008 #2

    Fredrik

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    I'll post the derivations as well. If they're valid derivations, then these theories are really the same, right?

    The ABL rule can be derived from the Born rule and Bayes' theorem. Consider a physical system prepared in the state |a>. The Born rule says that if we measure Q, the probability that the result will be qi is

    [tex]P(q_i)=|\langle q_i|a\rangle|^2[/tex]

    It also tells us that the probability that the system will be observed in state |b>, given that it has already been observed to be in state |qi> is

    [tex]P(b|q_i)=|\langle b|q_i \rangle|^2[/tex]

    The probability that the system will be observed to be in state |qi> first and |b> second is the product of the two probabilities:

    [tex]P(b,q_i)=P(b|q_i)P(q_i)=\big|\langle b|q_i \rangle\langle q_i|a \rangle\big|^2[/tex]

    The probability that the system will be found to be in state |b>, given that a measurement of Q has been performed (but the result is unknown to us) is

    [tex]P(b)=\sum\limits_j P(b,q_j)=\sum\limits_j \big|\langle b|q_j \rangle\langle q_j|a \rangle\big|^2[/tex]

    What we're looking for is the probability P(qi|b) that the result of the measurement of Q is qi, given that the system was later observed in state |b>. This is where we use Bayes' theorem:

    [tex]P(q_i|b)=\frac{P(b|q_i)P(q_i)}{P(b)}=\frac{\big|\langle b|q_i \rangle\langle q_i|a \rangle\big|^2}{\sum\limits_j \big|\langle b|q_j \rangle\langle q_j|a \rangle\big|^2}[/tex]

    An article I read (a part of) said that we can recover the Born rule by rewriting the right-hand side as

    [tex]\frac{\langle q_i|b\rangle\langle b|q_i \rangle|\langle q_i|a \rangle|^2}{\sum\limits_j \langle q_j|b\rangle\langle b|q_j \rangle|\langle q_j|a \rangle|^2}[/tex]

    and replacing the "property" |b><b| of the final state by the "trivial property" represented by the identity operator. This obviously gives us the desired result, but I don't understand this method.

    If |b> is a vector in an orthonormal basis, this calculation makes more sense to me:

    [tex]P(q_i)=\sum\limits_b P(q_i|b)P(b)=\sum\limits_b\big|\langle b|q_i \rangle\langle q_i|a \rangle\big|^2 =|\langle q_i|a \rangle|^2\sum\limits_b\langle q_i|b\rangle\langle b|q_i \rangle=|\langle q_i|a \rangle|^2[/tex]

    The sum is over all vectors in that basis.

    Edit: No wait, that last calculation doesn't make sense at all. I'm using results obtained from the Born rule to derive the Born rule :rolleyes:.
     
    Last edited: Nov 30, 2008
  4. Dec 1, 2008 #3
    I'm sure you'll answer your own question, and it might be interesting to see how you do it.

    My two cents is that the function describing the wave evolution is invariant under time reversal. The equation of motion is atemporal (ie., time-independent) no matter what rule you might use to produce and evaluate detection probabilities.

    If the Born and ABL rules describe the same sets of irreversible events related to the square of the amplitude of the wave at any point on the function, and if they're related to the same basic equation(s) of motion, then fapp they're the same theory.

    Or is it more complicated than that?
     
  5. Dec 1, 2008 #4

    Fredrik

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    It wouldn't be the first time. :smile:

    I'm still confused though. Right now I don't even understand how the ABL rule is possible at all. A measurement is an interaction that occurs during some time interval. The Born rule simply states that if the system is in state |a> at the start of that time interval, it's in one of several possible states at end of the time interval. But the ABL rule seems to tell us two things about what the state is at the end of the time interval. (Two things that contradict each other).

    One thing that I didn't mention in #1 is that this is the article that's causing most of my confusion. First they describe how quantum mechanics defines an "arrow of time" because of the Born rule. Then they decide to use the ABL rule instead, and claim that it doesn't define an arrow of time, and they keep talking about this as a "generalization" of quantum mechanics.
     
    Last edited: Dec 1, 2008
  6. Dec 2, 2008 #5

    Fredrik

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    Anyone? The part I'm the most curious about right now is this: How does the ABL rule even make sense?

    It seems to say two different things about the state of the system at the end of the time interval during which the measurement is performed. Maybe it isn't supposed to make sense in a formulation of quantum mechanics where states are represented by vectors? Does it make more sense if we use density matrices instead? Is there a way to make sense of the ABL rule when the initial and final states are both mixed states?
     
  7. Dec 2, 2008 #6

    strangerep

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    In the Gell-Mann-Hartle paper you mentioned (gr-qc/9304023v2) they're
    using idealized measurements, expressed as projection operators

    [tex]
    P^k_{\alpha_k}(t_k)
    [/tex]

    A particular history of the system is then described with the help
    of time-ordered products of such projection operators.

    Maybe you should post more of the context of the Gell-Mann-Hartle
    paper in relation to what's bugging you. I read that GMH paper as
    saying that their use of the ABL variant is for a theory of
    quantum histories where both initial and final states are specified
    (as densities operators [itex]\rho_i, \rho_f[/itex]) instead of
    just the initial state. I.e., one theory looks at histories following
    from an initial state (without caring about the final state)
    while the other cares about both.

    You asked earlier about:

    It's not a "method", but rather expresses the difference between one
    theory and another.

    Isn't that exactly what formulae (3.1a, 3.1b) in the GMH paper does?

    PS: Apologies in advance if I take a long time to reply.
     
  8. Dec 2, 2008 #7

    Fredrik

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    D'oh. It didn't even occur to me to think of the (idealized) measurements as instantaneous.

    Hm, if we think of them as instantaneous, then maybe the density matrix version of the ABL rule (3.1 in the G-M & H article) makes sense. But I still don't see how the state vector version of it can make sense. It seems to be saying that at times ta and tb the system is in a pure state, and at some time tQ between those times, it instantly changes to a mixed state and instantly goes back to being a pure state again, but not the same one as before. (I don't mean to imply that the GMH paper suggests that. Just look at the ABL rule as it appears in my #1 and #2).

    The ABL rule itself is what's bugging me. I only have a few smaller concerns about the paper, for example:

    I'm curious why they (and others) refer to to the change from Born to ABL as a generalization of quantum mechanics. If it can be derived from the standard formulation, it isn't really a generalization, is it?

    I also don't really follow the logic of the argument that standard QM defines an arrow of time while "generalized" QM doesn't. It seems to me that we have the freedom to choose to have an initial condition only, a final condition only, or both an initial and a final condition. If that's the case, it's not QM that defines an arrow of time. It's the people who use QM who do that.


    It probably does. I probably just have to see a derivation of that formula to understand it better. I don't have a lot of experience working with density matrices.

    No problem. This isn't the only thing I'm trying to understand better right now, so I can work on the others in the mean time. I'm e.g. making very slow progress with the Geroch notes that George Jones posted in the other thread where you answered some of my questions. (Slow because I tend to get stuck on details and distracted by other things).
     
    Last edited: Dec 2, 2008
  9. Dec 2, 2008 #8

    atyy

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    I think you are right, Aharonov himself claims it's not a generalization.

    New Insights on Time-Symmetry in Quantum Mechanics
    Yakir Aharonov, Jeff Tollaksen
    http://arxiv.org/abs/0706.1232

    "While TSQM is a new conceptual point-of-view that has predicted novel, verified effects which seem impossible according to standard QM, TSQM is in fact a re-formulation of QM. Therefore, experiments cannot prove TSQM over QM (or vice-versa). ................... Nevertheless, the real litmus test of any re-formulation is whether conceptual shifts can teach us something fundamentally new or suggest generalizations of QM, etc. The re-formulation to TSQM suggested a number of new experimentally observable effects, one important example of which are weak-measurements ....."
     
  10. Dec 3, 2008 #9

    strangerep

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    How do you figure that it "changes into a mixed state" at tQ? The expression
    [itex]|q_j\rangle\langle q_j|[/itex] is just a projection operator onto the j'th eigenspace of some
    operator Q (representing an idealized measurement of the property corresponding to Q) which
    gave the result "j" (and implying that the system was therefore in the state [itex]|q_j\rangle[/itex]
    after the measurement -- by the standard collapse axiom).

    There's some interesting ideas in the Aharonov/Tollaksen paper that atyy mentioned,
    but I need to study it more carefully.

    Most of what's needed to understand the ABL stuff seems to be explained
    in the Wiki page on density matrices.
    http://en.wikipedia.org/wiki/Density_matrix
     
  11. Dec 3, 2008 #10

    Fredrik

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    I usually check Wikipedia first when there's something I want to know, but for some reason I didn't read this page until now. It's very well written and it made things a bit clearer for me.

    Yes, we know the system changes into one of the [itex]|q_j\rangle[/itex] states at time tQ, but we don't know which one. That's what it means to say that it changes into a mixed state. In the standard formulation of QM (i.e. when we're using the Born rule), we would describe what's happening like this:

    BEFORE: At time ta, the system is in the state [itex]e^{iH(t_Q-t_a)}|a\rangle[/itex]. It changes with time according to the Schrödinger equation until we perform an idealized (instantaneous) measurement of Q at time tQ>ta.

    DURING: At time tQ the state of the system changes according to the following rule

    [tex]\rho=|a\rangle\langle a|\rightarrow \sum\limits_i |\langle q_i|a\rangle|^2|q_i\rangle\langle q_i| =\sum\limits_i |q_i\rangle\langle q_i|a\rangle\langle a|q_i\rangle\langle q_i| =\sum\limits_i P_i\rho P_i[/tex]

    where I have defined [itex]P_i=|q_i\rangle\langle q_i|[/itex].

    AFTER: At all times t>tQ, the state of the system changes with time according to the Schrödinger equation and the initial condition that the state is [itex]\sum_i P_i\rho P_i[/itex] at time tQ. At time tb>tQ, the state is

    [tex]\sum\limits_i e^{-iH(t_b-t_Q)}P_i\rho P_i e^{iH(t_b-t_Q)}[/tex]

    This answers at least one of my questions from my earlier posts. It's clear that the standard formulation of QM does define an arrow of time. I previously thought that the claim that it does was based on what happens during the measurement, but it's not. It's the assumption that the pure state is a final condition and the mixed state an initial condition that breaks the symmetry.

    My concern about the ABL rule was that I thought it was unacceptable to have a system change instantly from pure to mixed and than immediately and instantly change back to pure again. I still think that's what the ABL rule describes, but I don't find it unacceptable anymore. It wasn't really the discontinuity that bothered me. It was the fact that the "result of the measurement" isn't an initial condition for anything anymore. In order to accept the ABL rule, we have to stop thinking e.g. that we know a particle's position when we have just measured it. Standard QM says that the position is (almost) well-defined right after a measurement. The ABL rule says it isn't.

    Right now I'm OK with the ABL rule. I have some concerns about its derivation though. I feel that since the Born rule says that the state at t>tQ is something that contradicts the ABL rule, it can't be possible to derive the ABL rule from the Born rule. I'm going to try to figure out what's wrong with what I wrote in #2 tomorrow. (Maybe it is possible to derive the ABL expression for the probabilities, even though the boundary conditions come out wrong). It also seems to me that replacing the final condition |b><b| in the ABL rule with a completely unknown state (the identity operator), doesn't really give us the Born rule. It gives us the correct probabilities, but it doesn't give us an initial condition for the state at t>tQ. If we really want to recover the whole Born rule and not just the probabilities, we have to replace [itex]|b\rangle\langle b|[/itex] with [itex]\sum_i P_i\rho P_i[/itex] (instead of with the identity operator) in the ABL rule. This also gives us the correct probabilities, but we still can't claim to have derived the Born rule from the ABL rule, since we assumed that the initial condition for the state at t>tQ is what we want it to be.

    If I'm right about the derivations, then QM with the Born rule replaced by the ABL rule really is a generalization of standard QM. The two rules can't be derived from each other, but we have seen that the Born rule is a special case of the ABL rule. If we allow the initial and final states in the ABL rule to be mixed (and I think we should), then the Born rule is the ABL rule with a pure initial state [itex]\rho[/itex] and a mixed final state [itex]\sum_i P_i\rho P_i[/itex]

    I started reading it today. I agree that it looks interesting.
     
    Last edited: Dec 4, 2008
  12. Dec 4, 2008 #11

    DrChinese

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    This is a fantastic paper and I urge the "regulars" here to take a look at it. It 'solves' (assuming you agree with the argument) an issue that is perplexing to me: what does a 'non-realistic local mechanism' look like if it is not Many Worlds?

    The TSQM (Time Symmetric Quantum Mechanics) solution is that a future context affects the present as does the past. This allows 2 apparently non-local observers (Alice and Bob) to see correlations without violating c as an upper limit. Their future observation contexts help determine entangled particle attributes even when measurement settings are changed mid-flight. This occurs because the context contributions from the future to the present still occurs at the speed of light, though I guess you might call it -c rather than +c. It helps to explain why FTL signaling is not possible, and relativity is otherwise respected as normal. So locality is respected, and all measurements are contextual (i.e. non-realistic).

    A couple of additional points: there is no longer non-local collapse to worry about, and time symmetry is maintained which sits well conceptually.

    They also make some predictions regarding experimental verification using so-called weak measurements.

    A weakness, if you can call it that, is that there is still no obvious reason that we see time moving in one direction other than the basic idea that initial conditions set our universe off in one direction much as we ended up in a matter (as opposed to anti-matter) dominated universe. (Perhaps there was another anti-matter universe which was created by the Big Bang that is headed the other way in time.)

    Frederik, this paper follows up nicely on the earlier one you posted a link to. What are your thoughts?
     
  13. Dec 4, 2008 #12
    The reason for adopting the Born rule in the first place was the relationship, in classical experiments, between the intensity of incident disturbances and the dynamics of detection interactions. It's retained in qm because it works.

    The linear evolution of an incident disturbance is interrupted by interaction with an obstruction of some sort after which a series of amplifying events might or might not produce a detection event. There's no way to know the instant that the incident disturbance is altered (thus 'collapsing' its associated wavefunction) because there's no way to know the instant of its emission or creation.

    In orthodox qm, the description of the total evolution of the 'experimental system', from emission to detection, is an irreversible process. In practice it's also an irreversible process.

    So, I guess I don't understand why a time-symmetric description of a time-asymmetric process is desirable.

    I don't understand exactly what is the instrumental behavior the Gell-Mann and ABL formulations predict that would allow any meaningful differentiation between them and the orthodox formulation.

    If the universe is expanding and evolving, and it seems like a good assumption, then that might be taken as the deep cause of all of our observations of time-asymmetry wrt various behavioral scales. I realize that the reasoning is circular, but it has to be, because the question of why the universe is expanding and evolving is an unanswerable one. Suggestions regarding initial conditions, lower entropy, etc. don't actually explain the observed time-asymmetry, they're just different ways of expressing it.

    So, one has the choice of adopting the view that time-asymmetry is a universal-scale phenomenon because it is a sub-universal scale phenomenon, or that it's a sub-universal scale phenomenon because it's a universal scale phenomenon. I think the latter one makes more sense.

    Using similar reasoning, we can handle the pseudo-problem of nonlocality by positing that the universal-scale rate of expansion sets the limit for any sub-universal scale propagations. Do observations suggest that the universal-scale rate of expansion is approximately c?
     
  14. Dec 5, 2008 #13

    strangerep

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    Aren't these just different ways of saying the same thing? -- An idealized measurement is
    modelled as a projection operator, which in general doesn't have a well-defined inverse.
    I.e., the transformation [itex]\rho \to \rho' = P_i\rho P_i[/itex] is not invertible (where
    [itex]\rho[/itex] is mixed and [itex]\rho'[/itex] is pure). Thus, the best we can do is construct
    an evolution semigroup in such a theory.

    My take on the Born/ABL distinction is this:

    The Born rule answers the question: "if a system is in a state [itex]\rho[/itex] and we
    measure observable B, what is the probability of the result being b ?"

    In contrast, the ABL rule answers the question: "if a system is in a state [itex]\rho[/itex] and we
    measure observable Q but don't record the result, then we measure observable B and find
    a result b, what is the probability that the result of Q was in fact [itex]q_i[/itex] ?"

    So the ABL rule is more applicable to situations where (say) a single photon collision
    with a photomultiplier is amplified to produce a readable result. (We want to use the
    latter to infer something about the microscopic event.)

    (Maybe that's the sort of thing you meant, but I wasn't sure.)

    ??? I think your previous post #2 was ok (deriving ABL from Born and Bayes), except for
    maybe the last two equations. The difference between ABL and Born here is that with
    Born it's assumed we know the result of measuring Q was [itex]q_i[/itex], and the state
    after the measurement is (pure) [itex]|q_i\rangle[/itex].
     
  15. Dec 6, 2008 #14

    Fredrik

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    Yes, the calculation in #2 that derives the mathematical expression for the probability used in the ABL rule from the Born rule and Bayes' theorem is completely correct. (I realized that a few hours after my previous post). But the ABL rule is more than that mathematical expression. The idea of TSQM is to change the postulates of quantum mechanics so that a future boundary condition has the same effect on a measurement as a past boundary condition. The new postulate that replaces the Born rule says that if the state at times t<tQ is

    [tex]e^{-iH(t-t_Q)}|a\rangle[/tex]

    and the state at times t>tQ is

    [tex]e^{-iH(t-t_Q)}|b\rangle[/tex]

    then a measurement of Q at t=tQ will yield the result qi with probability

    [tex]\frac{\big|\langle b|q_i\rangle\langle q_i|a\rangle\big|^2}{\sum\limits_j\big|\langle b|q_j\rangle\langle q_j|a\rangle\big|^2}[/tex]

    The mathematical expression for the probabilities is postulated to be the same as the answer obtained by using standard QM to answer the question "If the state at times t<tQ is [itex]e^{-iH(t-t_Q)}|a\rangle[/itex], and the result of a measurement of B immediately after a a measurement of Q at t=tQ is b, then what's the probability that the result of the measurement of Q was qi?". It's postulated to be the same in order to guarantee that TSQM makes the same predictions as QM about the results of any experiment. The two theories are mathematically inequivalent in the sense that the postulates of TSQM can't be derived from the postulates of QM and vice versa, and they are physically equivalent in the sense that no experiment can distinguish between them. They only disagree about what the system is "doing" when it's not being measured.

    The part of the ABL rule that can't be derived from the Born rule is that the state of the system immediately after the measurement of Q is pure.
     
  16. Dec 6, 2008 #15

    Fredrik

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    I agree that it's a very interesting paper, but unfortunately I still haven't understood what a "weak-measurement" is, so I still have some work to do before I can comment on the rest.
     
  17. Dec 11, 2008 #16

    Fredrik

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    I read a bit more of this paper. Now I think it seems that what they call TSQM (time symmetric reformulation of quantum mechanics) is just what strangerep described in #13 (It's just standard QM applied to a scenario where you do a second measurement before you check the result of the first), and what I called TSQM in #14 is what they call the "destiny generalization" of QM. I find their terminology a bit strange.

    Just wanted to make sure I'm not misleading anyone. :smile:

    I'm probably not going to read the rest of the paper thoroughly, at least not anytime soon. There are too many other things that I'm more interested in right now.

    This discussion was very helpful because I now feel that I understand how standard QM defines an arrow of time, and that it's still physically equivalent to a time-symmetric theory. I also learned a lot about density operators.
     
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