Using the QM Formalism To Make Predictions

[Morbert]

[Moderator's note: Thread spun off from previous discussion as it is more technical.]

Yes. The math by itself does not tell you when to apply Rule 7. A human always has to put that in by hand.

Adding to this: the formalism of QM is quite flexible when it comes to applying the projection postulate. We can even apply it before the time of measurement.

Consider a preparation (at time $t_0$) of a quantum system $\psi$ and detector $D$ that performs a nondestructive, projective measurement of observable $A = \sum_i a_i |a_i\rangle\langle a_i|$ at time $t_1$. The outcome is perceived by the experimenter at $t_2$. This scenario implies the unitary evolution $$U(t_1,t_0)|\psi,D\rangle = \sum_i c_i |a_i,D_i\rangle$$ We might consider the collapse to eigenvalue $a_i$ occur at time $t_2 > t > t_1$ E.g. The non-unitary evolution $$C^\dagger_{i}|\psi,D\rangle = U(t_2,t)|a_i\rangle\langle a_i|U(t,t_1)U(t_1,t_0)|\psi,D\rangle$$ with probability $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ or we can consider the collapse to occur before the measurement at $t_1 > t > t_0$ E.g. $$C'^\dagger_{i}|\psi,D\rangle = U(t_2,t_1)U(t_1,t)|a_i\rangle\langle a_i|U(t,t_0)|\psi,D\rangle$$ and the probability will be $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C'_i\} = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ We can liberally construct these "collapsing" non-unitary evolutions so long as it is always the case that $$\mathrm{Re}\left[\mathrm{Tr}\{C^\dagger_i|\psi,D\rangle\langle\psi,D|C_j\}\right] \approx 0, i\neq j$$ (see equation 4.5 and 4.6 in [1])

Of course, if we commit ourselves to a particular interpretation that says the projection postulate is an actual physical process, then additional constraints will come into play. But so long as the above condition is met, the formalism will return consistent and reliable probabilities for whatever and whenever we project.

[1] https://arxiv.org/pdf/gr-qc/9210010.pdf

 

[PeterDonis]

We can liberally construct these "collapsing" nonunitary evolutions so long as it is always the case that

This condition basically means the alternatives don't interfere. So, for example, we could not construct such a "collapsing nonunitary evolution" between the slits and the detector screen for the standard double slit experiment (no which-way detectors), because the alternatives do interfere.

 

[Morbert]

This condition basically means the alternatives don't interfere. So, for example, we could not construct such a "collapsing nonunitary evolution" between the slits and the detector screen for the standard double slit experiment (no which-way detectors), because the alternatives do interfere.

Yes, so I think one possible lesson is so long as the requisite non-interference between alternatives is secured by correlating the possible eigenvalues with the possible detector states, the formalism does not insist on a specific moment of application of the projection, before or after measurement.

[edit] - Added "before or after measurement" to the end.

 

[Morbert]

I'm not sure what you mean by this. The only way to "secure" non-interference between alternatives is to design the experiment to prevent it. But doing that often would remove the very thing you want the experiment to test.

I'm just making it clear that the flexibility in applying wavefunction collapse is a robust feature of the formalism, rather than a hole in the formalism to be solved. Specific interpretations might demand additional work. But the predictions of QM don't change depending on where collapse is applied.

Take the double-slit experiment + which-way detectors experiment, characterised by the wavefunction $$|\Psi\rangle = \frac{1}{\sqrt{2}}(|\phi_\mathrm{left},D_\mathrm{left}\rangle + |\phi_\mathrm{right},D_\mathrm{right}\rangle)$$ If we want to predict the distribution of dots on the screen, we could collapse the wavefunctions at the which-way detector, and sum the resultant additive probabilities $$p(x) = \mathrm{Tr}\{0.5|\phi_\mathrm{left},D_\mathrm{left}\rangle\langle\phi_\mathrm{left},D_\mathrm{left}|x\rangle\langle x|\} + \mathrm{Tr}\{0.5|\phi_\mathrm{right},D_\mathrm{right}\rangle\langle\phi_\mathrm{right},D_\mathrm{right}|x\rangle\langle x|\}$$ Or we could not collapse the wavefunction at the which-way detectors and compute the distribution as $$p(x) = \mathrm{Tr}\{|\Psi\rangle\langle\Psi|x\rangle\langle x|\}$$ both computations will produce the same (non-interference) pattern.

 

[PeterDonis]

Take the double-slit experiment + which-way detectors experiment, characterised by the wavefunction $$|\Psi\rangle = \frac{1}{\sqrt{2}}(|\phi_\mathrm{left},D_\mathrm{left}\rangle + |\phi_\mathrm{right},D_\mathrm{right}\rangle)$$

No. This wave function only applies after the electron has passed one or the other slit but we have not yet looked to see which which-way detector registered. And really it doesn't even apply then (unless you're using the MWI), since we'll throw away one of the terms and re-normalize as soon as we look to see which which-way detector registered.

Before the electron reaches the slits, the correct wave function is

$$
|\Psi\rangle = \frac{1}{\sqrt{2}} \left(|\phi_\mathrm{left} \rangle + |\phi_\mathrm{right} \right) |D_\mathrm{ready}\rangle
$$

where $|D_\mathrm{ready}\rangle$ means neither which-way detector has fired so we have no which-slit information (yet). And if it turned out that neither which-way detector fired at all (maybe there was a malfunction or they weren't set up right), then that would be the wave function we used to predict what would be seen at the detector (leaving out the $D$ factor altogether since it would be immaterial), and it would show interference.

both computations will produce the same (non-interference) pattern.

Because they're the same computation, just arranged differently. To say that the second of these does not "collapse the wavefunction", while the first does, is misleading.

What you are describing is not the computation of the probability of where a single dot, from a single electron, will appear on the detector screen. What you are describing is a computation of the distribution of dots over a large number of runs of the experiment. But that large number of runs will involve a large number of firings of which-way detectors and individual dots on the screen, and if we want to correctly predict the result for a single run, we can't use your computation at all.

To correctly predict the result of a single run, we have to do it in two stages. First, we predict the probability for which which-way detector will register. That probability is based on nothing more than the wave function for the electron by itself, prior to the slits, which is

$$
|\psi_\mathrm{electron} \rangle = \frac{1}{\sqrt{2}} \left(|\phi_\mathrm{left} \rangle + |\phi_\mathrm{right} \rangle \right)
$$

This predicts a 50-50 probability for each which-way detector, since we know that $|\phi_\mathrm{left} \rangle$ will make the left which-way detector register and $|\phi_\mathrm{right} \rangle$ will make the right which-way detector register.

Then, once we know which which-way detector has registered, we collapse the wave function for the electron to the appropriate one, either $|\phi_\mathrm{left} \rangle$ or $|\phi_\mathrm{right} \rangle$, and use that to predict the probabilities for where a dot will appear on the detector screen.

 

[Morbert]

No. This wave function only applies after the electron has passed one or the other slit but we have not yet looked to see which which-way detector registered. And really it doesn't even apply then (unless you're using the MWI), since we'll throw away one of the terms and re-normalize as soon as we look to see which which-way detector registered.

Yes it is time-evolved to a time after the electron correlates with the detector, and I can write $U|\Psi\rangle$ if you like, but the wavefunction I wrote down absolutely characterises the experiment in question, and the formalism does not insist we must collapse the wavefunction after this correlation is established. Where we collapse the wavefunction is only informed by what is most convenient in generating the boolean logic that contains the predictions we are interested in computing.

Before the electron reaches the slits, the correct wave function is
$$
|\Psi\rangle = \frac{1}{\sqrt{2}} \left(|\phi_\mathrm{left} \rangle + |\phi_\mathrm{right} \right) |D_\mathrm{ready}\rangle
$$
where $|D_\mathrm{ready}\rangle$ means neither which-way detector has fired so we have no which-slit information (yet). And if it turned out that neither which-way detector fired at all (maybe there was a malfunction or they weren't set up right), then that would be the wave function we used to predict what would be seen at the detector (leaving out the $D$ factor altogether since it would be immaterial), and it would show interference.

I assumed the detector was working perfectly. We can of course accommodate an imperfect detector by modifying the time evolution to produce e.g. $$U|\Psi\rangle = c_1|\phi_\mathrm{left}, D_\mathrm{left}\rangle + c_2|\phi_\mathrm{right},D_\mathrm{right}\rangle + c_3(|\phi_\mathrm{left}\rangle + |\phi_\mathrm{right}\rangle)|D_\mathrm{failed}\rangle$$

Because they're the same computation, just arranged differently. To say that the second of these does not "collapse the wavefunction", while the first does, is misleading.

What you are describing is not the computation of the probability of where a single dot, from a single electron, will appear on the detector screen. What you are describing is a computation of the distribution of dots over a large number of runs of the experiment. But that large number of runs will involve a large number of firings of which-way detectors and individual dots on the screen, and if we want to correctly predict the result for a single run, we can't use your computation at all.

You can interpret $p(x)$ in terms of relative frequencies pertaining to multiple runs or in terms of Bayesian probability e.g. informing betting odds for a single run. In the rest of your post, you describe a perfectly fine procedure for computing probabilities for a single run. But you don't say why we can't use my computation. The same quantity is computed.

Now, if we are interested in computing probabilities for both the detections on the screen far from the slits *and* the clicks of the which-way detectors, then we would have to use the appropriate which-way projectors.

 

[PeterDonis]

the formalism does not insist we must collapse the wavefunction after this correlation is established

How you apply the formalism depends on what you are computing. Computing a probability distribution over many runs is different from computing probabilities for a single run.

I assumed the detector was working perfectly.

I know. That's the usual assumption. But the OP of this thread is trying to learn the basics of QM, and I am posting as much for his benefit as for yours.

you don't say why we can't use my computation. The same quantity is computed.

No, it's not. Your computation does not take into account which which-way detector actually registered for a given run. Mine does. So if you and I both make bets on individual runs, I will win more often than you, because I am making use of information that you are not.

 

[Morbert]

How you apply the formalism depends on what you are computing. Computing a probability distribution over many runs is different from computing probabilities for a single run.

No, it's not. Your computation does not take into account which which-way detector actually registered for a given run. Mine does. So if you and I both make bets on individual runs, I will win more often than you, because I am making use of information that you are not.

You are using von Neumann's rule to condition probabilities on outcomes recorded by the which-way detectors, which is fine, but in order to make use of this information we would have to place bets mid-run. If bets are placed at the start of the single run, you'd have to weigh these conditional probabilities appropriately.

The disinction I am making is between computing $p_\Psi(x)$ directly (no VN rule applied), and computing $p_\Psi(x)$ by considering the which-way measurement outcomes $$p_\Psi(x) = p_\Psi(\mathrm{left})p_\mathrm{left}(x) + p_\Psi(\mathrm{right})p_\mathrm{right}(x)$$
where e.g. $$p_\mathrm{left}(x) = \mathrm{Tr}\{\rho_\mathrm{left}\Pi_x\}, \rho_\mathrm{left} = \frac{\Pi_\mathrm{left}\rho_\Psi\Pi_\mathrm{left}}{\mathrm{Tr}\{\Pi_\mathrm{left}\rho_\Psi\Pi_\mathrm{left}\}}$$ (VN rule applied to obtain $\rho_{\mathrm{left}},\rho_{\mathrm{right}}$)

[edit] - Technically I have shown Lueder's rule rather than VN but I don't think the difference is important here)

 

[PeterDonis]

You are using von Neumann's rule to condition probabilities on outcomes recorded by the which-way detectors, which is fine, but in order to make use of this information we would have to place bets mid-run. If bets are placed at the start of the single run, you'd have to weigh these conditional probabilities appropriately.

Yes, agreed. As I said, it depends on what you are computing. If you are limiting yourself to computing the probability distribution at the detector for a single run without knowing which which-way detector registers on that run, then yes, you don't apply Rule 7 (the projection postulate) at the which-way detectors, because you aren't making use of that information.