Graduate Consistent Histories and Locality

[Morbert]

Yes, that's my intention. The idea is that the outcomes are specific and identical, let's say both a +1 outcome at 19 degrees (I just picked this out of thin air). Or both -1 at 56 degrees (also out of thin air). Variations of that can be tested with the cited Hensen experiment (although they use different angle settings, the expectation values match QM; and CH must too).

So by what standard can Griffiths say "appropriate measurements can reveal quantum properties possessed by the measured system before the measurement took place" and it NOT mean the same thing as Bell or myself? He is apparently drawing some distinction between: a) "hidden variables" (those don't exist); and b) "true" realism (which because it is different than Bell realism, means that Bell inequalities don't apply).

Well, my example does not rely on Bell or CHSH. It is back to EPR-type reasoning (elements of reality). So either the measurement outcome of A is independent of the setting at distant B; or it isn't. Which? Because if they match, the question immediately becomes: how do 2 independently oriented electrons suddenly match spins without violating the very Einsteinian locality that is central to CH? Given they had those "quantum properties" prior to measurement, also according to CH.

Then let's try to discuss the ideal experiment (so no loopholes) where a two-particle system is prepared in a correlated state such that no matter the aspect chosen, the outcomes will be identical with 100% certainty. Below is my attempt to formalise this without recourse to CH. I use unitary evolution of the entire system, but I don't ascribe meaning to it beyond a tool for computing probabilities.

We have a two-particle system prepared in a Bell state $|\phi^+\rangle = \frac{1}{\sqrt{2}}(|\uparrow_\omega\uparrow_\omega\rangle + |\downarrow_\omega\downarrow_\omega\rangle)$, a quantum random number generator (QRNG) that chooses an aspect $\omega$ and distant labs $A$ and $B$ that each measure one of the photons in the $\omega$ basis, with each with outcomes 1 or 0. Quantum mechanics predicts identical outcomes regardless of aspect.

The three components above are prepared in the initial state $$|\Psi_0\rangle = |\Omega\rangle_{QRNG}|\Omega\rangle_{AB}|\phi^+\rangle_{12}$$Evolving to the moment the measurements conclude, we have the state $$|\Psi_1\rangle = \sum_{\omega}\frac{c_\omega}{\sqrt{2}}|\omega\rangle_\mathrm{QRNG}(|11\rangle_{AB12} + |00\rangle_{AB12})$$We want the probability that the outcomes will be correlated, given any aspect. I.e. $\mathrm{Pr}(11 \lor 00 | \omega)$. Dropping the subscripts where possible:
\begin{eqnarray*}
\mathrm{Pr}(\omega) &=& \mathrm{tr} \left[\Psi_1\right]\otimes\left[\omega\right]\otimes I_{AB12} &=& |c_\omega|^2\\
\mathrm{Pr}(\left[11 \lor 00\right]\land\omega) &=& \mathrm{tr}\left[\Psi_1\right]\otimes\left[\omega\right]\otimes(\left[11\right] + \left[00\right])&=& |c_\omega|^2\\
\mathrm{Pr}(11 \lor 00 | \omega) &=& \frac{\mathrm{Pr}(\left[11 \lor 00\right]\land\omega)}{\mathrm{Pr}(\omega)} &=& 1
\end{eqnarray*}Before discussing how CH would introduce realism to this scenario, maybe it is good to first see if we agree with this simple model of the experiment.

[edit] - Clarified some things
[edit 2] - Simplified Bell state expression and removed potentially misleading coefficients

 

[DrChinese]

Then let's try to discuss the ideal experiment (so no loopholes) where a two-particle system is prepared in a correlated state such that no matter the aspect chosen, the outcomes will be identical with 100% certainty. Below is my attempt to formalise this without recourse to CH.

We have a two-particle system prepared in a Bell state $|\phi^+\rangle = a_\omega|\uparrow_\omega\uparrow_\omega\rangle + b_\omega|\downarrow_\omega\downarrow_\omega\rangle$, a quantum random number generator (QRNG) that chooses an aspect $\omega$ and distant labs $A$ and $B$ that each measure one of the photons in the $\omega$ basis, with each with outcomes 1 or 0. Quantum mechanics predicts identical outcomes regardless of aspect.

The three components above are prepared in the initial state $$|\Psi_0\rangle = |\Omega\rangle_{QRNG}|\Omega\rangle_{AB}|\phi^+\rangle_{12}$$Evolving to the moment before the measurements are carried out, the systems are in the state $$|\Psi_1\rangle = \sum_{\omega}c_\omega|\omega\rangle_\mathrm{QRNG}(a_\omega|11\rangle_{AB12} + b_\omega|00\rangle_{AB12})$$We want the probability that the outcomes will be correlated, given any aspect. I.e. $\mathrm{Pr}(11 \lor 00 | \omega)$. Dropping the subscripts where possible:
\begin{eqnarray*}
\mathrm{Pr}(\omega) &=& \mathrm{tr} \left[\Psi_1\right]\otimes\left[\omega\right]\otimes I_{AB12} &=& |c_\omega|^2\\
\mathrm{Pr}(\left[11 \lor 00\right]\land\omega) &=& \mathrm{tr}\left[\Psi_1\right]\otimes\left[\omega\right]\otimes(\left[11\right] + \left[00\right])&=& |c_\omega|^2\\
\mathrm{Pr}(11 \lor 00 | \omega) &=& \frac{\mathrm{Pr}(\left[11 \lor 00\right]\land\omega)}{\mathrm{Pr}(\omega)} &=& 1
\end{eqnarray*}Before discussing how CH would introduce realism to this scenario, maybe it is good to first see if we agree with this simple model of the experiment.

Sure, but I can already see where we will deviate.

Let’s say I concur/concede that CH can explain realism and locality in this case.

My issue is that we can have the exact same situation as you have well-defined above when there is a swap. But the swap can occur anywhere in spacetime without regard to locality (in principle). And the two particles in the Bell state you describe need never have interacted or been present in any common backward light cone.

In other words: let’s start exactly where you want and assume I will go with that. I do want to see how you explain the realism issue. But it is the next step after that I want to understand. It attacks the locality assumption at every turn.

 

[Morbert]

@DrChinese So let's now compute this probability using the CH formalism. First, we construct a set of consistent histories $\{C_\omega\}$, where$$C_\omega = \left[\Psi_0\right]\odot\left[\omega\right]\odot\left[00\lor11\right]$$Computing the same probability $\mathrm{Pr}(00 \lor 11 | \omega)$
\begin{eqnarray*}\mathrm{Pr}(\omega) &=& \mathrm{tr} C_\omega \left[\Psi_0\right]C_\omega^\dagger&=& |c_\omega|^2\\\mathrm{Pr}(\left[11\lor 00\right]\land\omega) &=& \mathrm{tr} C_\omega \left[\Psi_0\right]C_\omega^\dagger&=& |c_\omega|^2\\
\mathrm{Pr}(00 \lor 11 | \omega)&=&\frac{\mathrm{Pr}(\left[11\lor 00\right]\land\omega) }{\mathrm{Pr}(\omega) }&=& 1\end{eqnarray*}Alternatively, if we want to describe a realistic measurement, we can construct histories $\{C_{\omega,\uparrow_\omega},C_{\omega,\downarrow_\omega}\}$ where \begin{eqnarray*}C_{\omega,\uparrow_\omega}&=& \left[\Psi_0\right]\odot\left[\uparrow_\omega\uparrow_\omega\right]\odot\left[\omega\right]\odot\left[11\right]\\C_{\omega,\downarrow_\omega}&=& \left[\Psi_0\right]\odot\left[\downarrow_\omega\downarrow_\omega\right]\odot\left[\omega\right]\odot\left[00\right]\end{eqnarray*}We can use the coarse-graining $C_\omega = C_{\omega,\uparrow}+C_{\omega,\downarrow}$ to compute the probabilities above, but now we can also determine if a realistic measurement occurs. Griffiths et al identify a measurement with the conditions \begin{eqnarray*}\mathrm{Pr}(\omega\land\uparrow\uparrow) &=& \mathrm{Pr}(\omega\land11) &=& \mathrm{Pr}(\omega\land\uparrow\uparrow\land11)\\\mathrm{Pr}(\omega\land\downarrow\downarrow) &=& \mathrm{Pr}(\omega\land00) &=& \mathrm{Pr}(\omega\land\downarrow\downarrow\land11)\end{eqnarray*}These conditions hold in this case, so we can use this set of histories to describe the realistic scenario where a measurement reveals a pre-existing property.

 

[Morbert]

PS the above is a more succinct version of what Griffiths discusses explicitly in chapters 23 and 24 of his book

https://quantum.phys.cmu.edu/CQT/chaps/cqt23.pdf
https://quantum.phys.cmu.edu/CQT/chaps/cqt24.pdf

 

[Morbert]

My issue is that we can have the exact same situation as you have well-defined above when there is a swap. But the swap can occur anywhere in spacetime without regard to locality (in principle). And the two particles in the Bell state you describe need never have interacted or been present in any common backward light cone.

In other words: let’s start exactly where you want and assume I will go with that. I do want to see how you explain the realism issue. But it is the next step after that I want to understand. It attacks the locality assumption at every turn.

I'll see about applying the CH formalism to the Ma paper you linked above. Hopefully it will not become too involved.

 

[DrChinese]

@DrChinese So let's now compute this probability using the CH formalism. First, we construct a set of consistent histories $\{C_\omega\}$, where$$C_\omega = \left[\Psi_0\right]\odot\left[\omega\right]\odot\left[00\lor11\right]$$Computing the same probability $\mathrm{Pr}(00 \lor 11 | \omega)$
\begin{eqnarray*}\mathrm{Pr}(\omega) &=& \mathrm{tr} C_\omega \left[\Psi_0\right]C_\omega^\dagger&=& |c_\omega|^2\\\mathrm{Pr}(\left[11\lor 00\right]\land\omega) &=& \mathrm{tr} C_\omega \left[\Psi_0\right]C_\omega^\dagger&=& |c_\omega|^2\\
\mathrm{Pr}(00 \lor 11 | \omega)&=&\frac{\mathrm{Pr}(\left[11\lor 00\right]\land\omega) }{\mathrm{Pr}(\omega) }&=& 1\end{eqnarray*}Alternatively, if we want to describe a realistic measurement, we can construct histories $\{C_{\omega,\uparrow_\omega},C_{\omega,\downarrow_\omega}\}$ where \begin{eqnarray*}C_{\omega,\uparrow_\omega}&=& \left[\Psi_0\right]\odot\left[\uparrow_\omega\uparrow_\omega\right]\odot\left[\omega\right]\odot\left[11\right]\\C_{\omega,\downarrow_\omega}&=& \left[\Psi_0\right]\odot\left[\downarrow_\omega\downarrow_\omega\right]\odot\left[\omega\right]\odot\left[00\right]\end{eqnarray*}We can use the coarse-graining $C_\omega = C_{\omega,\uparrow}+C_{\omega,\downarrow}$ to compute the probabilities above, but now we can also determine if a realistic measurement occurs. Griffiths et al identify a measurement with the conditions \begin{eqnarray*}\mathrm{Pr}(\omega\land\uparrow\uparrow) &=& \mathrm{Pr}(\omega\land11) &=& \mathrm{Pr}(\omega\land\uparrow\uparrow\land11)\\\mathrm{Pr}(\omega\land\downarrow\downarrow) &=& \mathrm{Pr}(\omega\land00) &=& \mathrm{Pr}(\omega\land\downarrow\downarrow\land11)\end{eqnarray*}These conditions hold in this case, so we can use this set of histories to describe the realistic scenario where a measurement reveals a pre-existing property.

OK, I'm going to accept this as I promised. There's no issue with the math. I would refer to this as the first half of the EPR logic (modified for spin/polarization). Basically, we make the eminently reasonable deduction that there must be an "element of reality" associated with a 100% certain outcome when an undisturbed system has interacted with another system in the past. This deduction rests on the assumption that the existence of the element of reality preceded the measurement. Of course you use the same assumption as a definition: "...a realistic measurement, we can construct histories...". So that construction defines the realism we seek to discuss. That's fair, in fact that's one of the points of CH. But that assumption is of course fair game for experimental disproof.

So let's start with the EPR elements of reality being equivalent to the CH realistic measurement. This applies very nicely to the initial thoughts we might have regarding normal PDC entangled pair production. I won't bring Bell into the discussion, so we can skip any debate about whether Bell's Theorem/Inequality applies.

So your next step should be to apply your well-presented explanation of PDC pair production to pair production by swapping. Ideally, you would initially tackle one of the following scenarios (both require explanation, of course):

a) The Ma paper, where the entanglement is created unambiguously (all reference frames) in the future of the entangled 1&4 pair. Obviously, the challenge here is to explain how your equations {Pr(ω∧↑↑)=Pr(ω∧11), Pr(ω∧↓↓)=Pr(ω∧00)} - which assume the ↑↑/↓↓ occurs before the 11/00 - works when the measurements occur before there exists any entangled relationship whatsoever between the 2 photons (1 & 4).

After all, your math in #33 assumed State(ω∧↑↑) -> Pr(ω∧11), State(ω∧↓↓) -> Pr(ω∧00). That relationship is by delayed choice in this case, so the states leading to the certain outcome don't yet exist.


b) The Sun paper, where the measurements occur distant to each other, and the 1 & 4 photons never coexist in any common backward light cone. Presumably, 2 such photons could have never been placed in an entangled state in the past (that being ruled out experimentally) if Einstein locality holds.

After all, how could we ever get to State(ω∧↑↑) or State(ω∧↓↓) if they didn't first interact? Which they are not allowed to do in the experiment, having been created by distant (25 km apart) sources.

BTW: Thanks for your time working on this with me.-DrC

 

[Morbert]

@DrChinese

I have started with the Ma paper, and in anticipation of a discussion about coexisting photons, I have modified it so that photons 1 and 4 never coexist, where "never coexist" means in the rest frame of the labs that measure photons 1 and 4, there is no time interval that contains both 1 and 4 a la Megidish. I also use the time labels of Fig. 1 in Megidish: I, II, III, IV, V. I have taken the liberty of some simplifications that make the maths easier but (hopefully you agree) do not detract from any relevant aspect. The simplifications are:

i) I use a BSM that has a 50/50 chance of occuring, but if it occurs, is perfect, i.e. is a measurement in the basis $\{\psi^+,\psi^-,\phi^+,\phi^-\}$ as opposed to the more accurate $\{\phi^+,\phi^-,\mathrm{fail}\}$.
ii) I also assume the measurements that occur at I and V by devices A and B are pre-aligned to the aspect $\omega$, such that a successful BSM will select perfect correlation/anti-correlation between 1 and 4.

Attempting to model the unitary evolution of the experiment: In a conventional swapping experiment described by Ma, the initial state of the 4-photon system is $$|\Psi\rangle = |\psi^-\rangle_{12}|\psi^-\rangle_{34}$$In the Megidish experiment, it is $$|\Psi\rangle = |\psi^-\rangle_{12}|\emptyset\rangle$$where $|\emptyset\rangle$ are the relevant degrees of freedom so that $U(t_{III},t_I)|\emptyset\rangle = |\psi^-\rangle_{34}$. The initial state including all relevant measurement systems is $$|\psi^-\rangle_{12}|\emptyset\rangle|\Omega\rangle_A|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM}$$The destructive measurement of photon 1 by $A$ is modeled like so:
\begin{eqnarray*}U(t_{II},t_I)|h_\omega\rangle_1|\Omega\rangle_A&=&|\omega,1\rangle_{1,A}\\
U(t_{II},t_I)|v_\omega\rangle_1|\Omega\rangle_A&=&|\omega,0\rangle_{1,A}\\
U(t_{II},t_I)|\psi^-\rangle_{12}|\Omega_A\rangle &=& \frac{1}{\sqrt{2}}(|\omega,1\rangle_{1,A}|v_\omega\rangle_2 - |\omega,0\rangle_{1,A}|h_\omega\rangle_2) = |\chi^{\psi^-}\rangle_{12,A}\end{eqnarray*}and so the full evolution to II is$$U(t_{II},t_I)|\psi^-\rangle_{12}|\emptyset\rangle|\Omega\rangle_A|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM} = |\emptyset\rangle|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM}|\chi^{\psi^-}\rangle_{12,A}$$Next, we evolve to III, the creation of the 34 pair$$U(t_{III},t_{II})|\emptyset\rangle|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM}|\chi^{\psi^-}\rangle_{12,A} = |\psi^-\rangle_{34}|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM}|\chi^{\psi^-}\rangle_{12,A}$$In evolving to IV, let's say there is a 50/50 chance the BSM occurs. A BSM is modelled like so\begin{eqnarray*}U(t_{IV},t_{III})|\Omega\rangle_{BSM}|\chi^{\psi^-}\rangle_{12,A}|\psi^-\rangle_{34}&=&\frac{1}{2}(|\chi^{\psi^+}\rangle_{14,A}|\Psi^+\rangle_\mathrm{23,BSM}\\&&-|\chi^{\psi^-}\rangle_{14,A}|\Psi^-\rangle_\mathrm{23,BSM}\\&&-|\chi^{\phi^+}\rangle_{14,A}|\Phi^+\rangle_\mathrm{23,BSM}\\&&+|\chi^{\phi^-}\rangle_{14,A}|\Phi^-\rangle_\mathrm{23,BSM})\\&=&|\chi^+\rangle_{1234,A,\mathrm{BSM}}\end{eqnarray*}where the $\chi$ terms are of the form, for example $$|\chi^{\phi^+}\rangle_{14,A} = \frac{1}{\sqrt{2}}(|\omega, 1\rangle_{1,A}|h_\omega\rangle_4 + |\omega, 0\rangle_{1,A}|v_\omega\rangle_4)$$If a BSM doesn't occur, the evolution is \begin{eqnarray*}U(t_{IV},t_{III})|\Omega\rangle_{BSM}|\chi^{\psi^-}\rangle_{12,A}|\psi^-\rangle_{34}&=&\frac{1}{2}|\Omega\rangle_\mathrm{BSM}(|\chi^{\psi^+}\rangle_{14,A}|\psi^+\rangle_\mathrm{23}\\&&-|\chi^{\psi^-}\rangle_{14,A}|\psi^-\rangle_\mathrm{23}\\&&-|\chi^{\phi^+}\rangle_{14,A}|\phi^+\rangle_\mathrm{23}\\&&+|\chi^{\phi^-}\rangle_{14,A}|\phi^-\rangle_\mathrm{23})\\&=&|\chi^-\rangle_{1234,A,\mathrm{BSM}}\end{eqnarray*}So the evolution to IV where the BSM may or may not occur is $$U(t_{IV},t_{III})|\Omega\rangle_B|\Omega\rangle_{BSM}|\chi^{\psi^-}\rangle_{12,A}|\psi^-\rangle_{34} = \frac{1}{\sqrt{2}}|\Omega\rangle_B(|\chi^+\rangle_{1234,A,\mathrm{BSM}}+|\chi^-\rangle_{1234,A,\mathrm{BSM}})$$Finally, to model the measurement at V, we use the evolution that acts on states like$$U(t_V,t_{IV})|\Omega\rangle_B|\chi^{\phi^+}\rangle_{14,A} = \frac{1}{\sqrt{2}}(|\omega,0\rangle_{1,A}|\omega,0\rangle_{2,B}+ |\omega,1\rangle_{1,A}|\omega,1\rangle_{2,B}) =|\zeta^{\phi^+}\rangle_{14,AB}$$Like before, if a BSM previously occurred, we have\begin{eqnarray*}U(t_V,t_{IV})|\Omega\rangle_B|\xi^+\rangle_{1234,A,\mathrm{BSM}}&=&\frac{1}{2}(|\zeta^{\psi^+}\rangle_{14,A}|\Psi^+\rangle_\mathrm{23,BSM}\\&&-|\zeta^{\psi^-}\rangle_{14,A}|\Psi^-\rangle_\mathrm{23,BSM}\\&&-|\zeta^{\phi^+}\rangle_{14,A}|\Phi^+\rangle_\mathrm{23,BSM}\\&&+|\zeta^{\phi^-}\rangle_{14,A}|\Phi^-\rangle_\mathrm{23,BSM})\\&=&|\zeta^+\rangle_{1234,AB,\mathrm{BSM}}\end{eqnarray*}If no BSM occurred, we have
\begin{eqnarray*}U(t_V,t_{IV})|\Omega\rangle_B|\xi^-\rangle_{1234,A,\mathrm{BSM}}&=&\frac{1}{2}|\Omega\rangle_\mathrm{BSM}(|\zeta^{\psi^+}\rangle_{14,A}|\psi^+\rangle_\mathrm{23}\\&&-|\zeta^{\psi^-}\rangle_{14,A}|\psi^-\rangle_\mathrm{23}\\&&-|\zeta^{\phi^+}\rangle_{14,A}|\phi^+\rangle_\mathrm{23}\\&&+|\zeta^{\phi^-}\rangle_{14,A}|\phi^-\rangle_\mathrm{23})\\&=&|\zeta^-\rangle_{1234,AB,\mathrm{BSM}}\end{eqnarray*}Putting all of this together, we have the full evolution$$U(t_V,t_I)|\psi^-\rangle_{12}|\emptyset\rangle|\Omega\rangle_A|\Omega\rangle_B|\Omega\rangle_\mathrm{BSM} = \frac{1}{\sqrt{2}}(|\zeta^+\rangle_{1234,AB,\mathrm{BSM}} + |\zeta^-\rangle_{1234,AB,\mathrm{BSM}})$$Before proceeding, maybe it is good to see if you object to this model of the time-evolution of a (simplified) entanglement swapping experiment where photons 1 and 4 never coexist, or want clarification at any stage outlined above.

 

[Demystifier]

"Measurement reveals pre-existing properties" or "appropriate measurements can reveal quantum properties possessed by the measured system before the measurement took place" (quoted from the abstract here) sounds realistic in the same sense I mean.

It's much easier to understand on a concrete example. When you say that there is some pre-existing property, you may have some specific property in mind. For example, the spin in z-direction. Or the spin in x-direction. But not both. You will not say that both spin in x-direction and spin in z-direction are pre-existing, am I right? On the other hand CH will say that both are simultaneously pre-existing, but in different frameworks. What does that mean? Honestly, I have no idea how to explain it in a way that I believe it would make sense to you or Bell. Simultaneous pre-existence of incompatible variables (like spin in x- and z-direction) in different frameworks is a notion that is very different from the notion of pre-existence that Bell had in mind. Does that help?

 

[Morbert]

It's much easier to understand on a concrete example. When you say that there is some pre-existing property, you may have some specific property in mind. For example, the spin in z-direction. Or the spin in x-direction. But not both. You will not say that both spin in x-direction and spin in z-direction are pre-existing, am I right? On the other hand CH will say that both are simultaneously pre-existing, but in different frameworks. What does that mean? Honestly, I have no idea how to explain it in a way that I believe it would make sense to you or Bell. Simultaneous pre-existence of incompatible variables (like spin in x- and z-direction) in different frameworks is a notion that is very different from the notion of pre-existence that Bell had in mind. Does that help?

This is Griffiths attempt to analogize frameworks.

Similarly, choosing a framework is something like choosing an inertial reference frame in special relativity. The choice is up to the physicist, and there is no law of nature, at least no law belonging to relativity theory, that singles out one rather than another. Sometimes one choice is more convenient than another when discussing a particular problem; e.g., the reference frame in which the center of mass is at rest. The choice obviously does not have any influence upon the real world. But again there is a disanalogy: any argument worked out using one inertial frame can be worked out in another; the two descriptions can be mapped onto each other. This is not true for quantum frameworks: one must employ a framework (there may be several possibilities) in which the properties of interest can be described; they must lie in the event algebra of the corresponding [projective decomposition].

For a more picturesque positive analogy consider a mountain, say Mount Rainier, which can be viewed from different sides. An observer can choose to look at it from the north or from the south; there is no “law of nature” that singles out one perspective as the correct one. One can learn different things from different viewpoints, so there might be some Utility in adopting one perspective rather than the other. But once again the analogy fails in that the north and south views can, at least in principle, be combined into a single unified description of Mount Rainier from which both views can be derived as partial descriptions. Let us call this the principle of unicity. It no longer holds in the quantum world once one assumes the Hilbert space represents properties in the manner discussed above.

https://arxiv.org/pdf/1105.3932

Ultimately it is to accommodate the association of physical properties with subspaces in Hilbert space, there is a subspace for "spin-z = up" and for "spin-x = up" but not "spin-z = up and spin-x = up". The last property, not associated with any subspace, cannot be said to exist according the CH.

 

[javisot20]

Can we distinguish a pair of particles that show quantum correlation after having interacted in some sense from the same pair of particles that show the same quantum correlation without having interacted in any sense?

 

[pines-demon]

Can we distinguish a pair of particles that show quantum correlation after having interacted in some sense from the same pair of particles that show the same quantum correlation without having interacted in any sense?

For single pairs of particles no. If you have a source of pairs, then you can ask if the pairs emitted by the source are entangled by doing various measurements. If a pair is entangled because a previous local interaction or because some sophisticated entanglement swapping protocol, you cannot tell from the measurements alone.

 

[Demystifier]

This is Griffiths attempt to analogize frameworks.

Excellent dis-analogies!

 

[DrChinese]

@DrChinese

I have started with the Ma paper, and in anticipation of a discussion about coexisting photons, I have modified it so that photons 1 and 4 never coexist, where "never coexist" means in the rest frame of the labs that measure photons 1 and 4, there is no time interval that contains both 1 and 4 a la Megidish. I also use the time labels of Fig. 1 in Megidish: I, II, III, IV, V. I have taken the liberty of some simplifications that make the maths easier but (hopefully you agree) do not detract from any relevant aspect. The simplifications are:

i) I use a BSM that has a 50/50 chance of occuring, but if it occurs, is perfect, i.e. is a measurement in the basis $\{\psi^+,\psi^-,\phi^+,\phi^-\}$ as opposed to the more accurate $\{\phi^+,\phi^-,\mathrm{fail}\}$.
ii) I also assume the measurements that occur at I and V by devices A and B are pre-aligned to the aspect $\omega$, such that a successful BSM will select perfect correlation/anti-correlation between 1 and 4.

Attempting to model the unitary evolution of the experiment: In a conventional swapping experiment described by Ma, the initial state of the 4-photon system is $$|\Psi\rangle = |\psi^-\rangle_{12}|\psi^-\rangle_{34}$$In the Megidish experiment, it is $$|\Psi\rangle = |\psi^-\rangle_{12}|\emptyset\rangle$$where $|\emptyset\rangle$ are the relevant degrees of freedom so that $U(t_{III},t_I)|\emptyset\rangle = |\psi^-\rangle_{34}$. The initial state including all relevant measurement systems is $$|\psi^-

OK, a few points to work out. I'm good with using whatever mix of the Megadish/Ma experiments you like, I can follow your presentation just fine as you have it.

1. The chances of a BSM (2 & 3 arriving within the time window) is not 50/50, it is 100%. The issue is that only 2 of the 4 possible Bell states (as you specify) are identifiable with current optical technology. That really is not important in any way, we could simply agree that all of the BSMs are one particular Bell state, and model that one.

2. I don't agree with your characterization of the state as of either time I or II. There is only a single entangled Φ- pair at those points, and there is no particular connection to any other pair in existence then. And certainly not with the one the swap is being made with. So I would opt for a simpler description at I.

Also, at II: the V/H polarization of photon 1 is now determinate and known (I guess you would say in the V/H framework?). Is the polarization of photon 2 now determinate and known as well? Or would you say it is in a superposition still of all possible polarization states?

3. Either way, if we are respecting Einsteinian locality/causality: there is only one photon in existence (after II). If possible, I think we should model that alone until we get to III.

Thanks,

-DrC

 

[PeterDonis]

This is Griffiths attempt to analogize frameworks.

All that does is raise my Bayesian estimate for "useless sophistry". He's just waving his hands and saying it's perfectly okay for QM not to obey the "principle of unicity"--without ever addressing the fact that that principle is part of the bedrock of our mental model of reality. It's not even clear what it would mean to violate it, and saying "well, Hilbert space just violates it, that's all there is to it" is no help at all.

 

[gentzen]

This is Griffiths attempt to analogize frameworks.

Excellent dis-analogies!

All that does is raise my Bayesian estimate for "useless sophistry". He's just waving his hands and saying it's perfectly okay for QM not to obey the "principle of unicity"--without ever addressing the fact that that principle is part of the bedrock of our mental model of reality. It's not even clear what it would mean to violate it, and saying "well, Hilbert space just violates it, that's all there is to it" is no help at all.

I agree that this was not one of Griffiths’ strong moments:

I haven‘t seen him „hand wave“ them away. Griffiths has occasionally weak spots, but nothing serious, especially compared to Goldstein who simply is wrong about CH.

Heisenberg got much closer to clarify why they are needed with his „boundary condition“ analogy. In the context of that analogy, their non-uniqueness also becomes more interesting. Of course, there cannot be a finest framework. But also the opposite direction is interesting: Often there is a coarsest framework for a given approach to model some given physical situation. But each such approach has a limited accuracy. However, it is also unclear whether CH can really reach arbitrary high accuracy, or whether it is actually limited as soon as QFT effects get important.

 

[Morbert]

1. The chances of a BSM (2 & 3 arriving within the time window) is not 50/50, it is 100%. The issue is that only 2 of the 4 possible Bell states (as you specify) are identifiable with current optical technology. That really is not important in any way, we could simply agree that all of the BSMs are one particular Bell state, and model that one.

Yes, and so below I've constructed histories for the actual $\Phi^+, \Phi^-, \mathrm{fail}$ scenarios. Hopefully it will not be too confusing and we can revisit the quantum state of the experiment if need be.

2. I don't agree with your characterization of the state as of either time I or II. There is only a single entangled Φ- pair at those points, and there is no particular connection to any other pair in existence then. And certainly not with the one the swap is being made with. So I would opt for a simpler description at I.

I am including degrees of freedom of the labs so that the experiment can be approximated by a pure state and unitary time evolution. This simplification is common but in the end it will only be relevant if you object to my claim that the histories I construct below are in fact decoherent.

Also, at II: the V/H polarization of photon 1 is now determinate and known (I guess you would say in the V/H framework?). Is the polarization of photon 2 now determinate and known as well? Or would you say it is in a superposition still of all possible polarization states?

The quantum state is not interpreted as real, and instead only yields probabilities for real alternatives. So yes in reality at II the polarization would be known to the experimenter.

Let's now construct a set of histories relevant to describing this experiment. For expediency I will only describe histories for which the probability is > 0. We can place the projector $\left[\psi^-_{12}\right]$ at time $I$ as the first element of our histories. To describe the measurement at $II$, we will add a branch just before, giving us$$\left[\psi^-_{12}\right]\odot\begin{cases}\left[h_\omega\right]_1&\odot&\left[1\right]_{A}\\\left[v_\omega\right]_1&\odot&\left[0\right]_{A}&\end{cases}$$Along the upper branch, the measurement result 1 reveals a horizontal polarization (relative to the chosen aspect), along the lower branch, the measurement result 0 reveals a vertical polarization. We will follow the upper branch, with the understanding that the lower branch has a similar structure. The next relevant event is at $III$, the creation of the 34 pair
$$\left[\psi^-_{12}\right]\odot\left[h_\omega\right]_1\odot\left[1\right]_{A}\odot\left[\psi^-\right]_{34}$$Before the BSM at $IV$ we can describe the polarization of photon 4 with the branch$$\left[\psi^-_{12}\right]\odot\left[h_\omega\right]_1\odot\left[1\right]_{A}\odot\left[\psi^-\right]_{34}\odot\begin{cases}
\left[h_\omega\right]_4\\
\left[v_\omega\right]_4
\end{cases}$$Then, at $IV$, we have the BSM. The $v_\omega$ polarization of 4 will be correlated with a failed BSM, so we have a branching of the form
$$\left[\psi^-_{12}\right]\odot\left[h_\omega\right]_1\odot\left[1\right]_{A}\odot\left[\psi^-\right]_{34}\odot\begin{cases}
\left[h_\omega\right]_4&\odot&\begin{cases}\left[
\Phi^+\lor\Phi^-\right]_\mathrm{BSM}\\
\left[\mathrm{fail}\right]_\mathrm{BSM}\end{cases}\\
\left[v_\omega\right]_4&\odot&\left[\mathrm{fail}\right]_\mathrm{BSM}
\end{cases}$$Following the upper branch again and extending it to the measurement of 4, we have a completed history$$\left[\psi^-_{12}\right]\odot\left[h_\omega\right]_1\odot\left[1\right]_{A}\odot
\left[\psi^-\right]_{34}\odot\left[h_\omega\right]_4\odot\left[\Phi^+\lor\Phi^-\right]_\mathrm{BSM}\odot\left[1\right]_B
$$Each of these histories $C$ has the probability of occurring $p(C) = \mathrm{tr}C^\dagger\left[\Psi_0\right] C$ and we can user them to compute the relevant conditional probabilities that correlate a successful BSM with certain correlation between the polarizations of 1 and 4.

Unlike a scenario where the BSM instantly influences photon 4, and retroactively influences photon 1 to entangle them, these histories describe a scenario where a successful BSM reveals a correlation between the polarization of 1 and 4.

 

[Morbert]

(I guess you would say in the V/H framework?)

As an aside, although we are considering a set of histories with a single aspect adopted by both A and B, we could generalize and model arbitrary choices of aspects by A and B (similar to how Gell-Mann describes the EPRB here https://www.webofstories.com/play/murray.gell-mann/165 ) Though this perhaps would just lead us further down a side track.

 

[Morbert]

Excellent dis-analogies!

All that does is raise my Bayesian estimate for "useless sophistry". He's just waving his hands and saying it's perfectly okay for QM not to obey the "principle of unicity"--without ever addressing the fact that that principle is part of the bedrock of our mental model of reality. It's not even clear what it would mean to violate it, and saying "well, Hilbert space just violates it, that's all there is to it" is no help at all.

These analogies are intended to emphasize that the choice of framework is a choice of description, and hence different frameworks do not assert different realities. The SFR itself is unambiguous, and the deeper consequences of the SFR (that these analogies do not address) are explored in plenty of literature, by people both for an against it.

It should also be noted that this ontology is not a necessary feature of CH. Roland Omnes gives an alternative account of consistent histories where the only commitments to what is real is what is in the lab. Both accounts preserve locality.

 

[PeterDonis]

different frameworks do not assert different realities.

But they do! That's the whole point of Griffith's tossed off comment that the "principle of uniticity" is violated in QM.

 

[Morbert]

different frameworks do not assert different realities.

But they do! That's the whole point of Griffith's tossed off comment that the "principle of uniticity" is violated in QM.

Griffiths's point is the opposite. From the paper

However, [a phase space analogy] is still helpful in illustrating some aspects of the quantum situation, and in avoiding the misleading idea that the relationship between different quantum frameworks is one of mutual exclusivity.

Different frameworks offer complementary descriptions of the same reality.

 

[PeterDonis]

Different frameworks offer complementary descriptions of the same reality.

That seems to contradict what Griffiths says about the "principle of uniticity" being violated by QM, because "complementary descriptions of the same reality" seems like the principle of uniticity is not violated.

 

[gentzen]

Different frameworks offer complementary descriptions of the same reality.

What do you mean by „descriptions of the same reality“? Those are your words. Some frameworks are unrelated to physical reality. Computations using those frameworks won‘t produce contradictions, but they are useless nevertheless. As Roland Omnes quipped: They belong in the waste-paper basket.

 

[javisot20]

"descriptions of the same reality" = descriptions/correspondences capable of explaining with the same precision the set of experimental data that make up QM

https://en.wikipedia.org/wiki/Penrose_interpretation, this is an example of incorrect interpretation because it does not achieve the objective

 

[Morbert]

What do you mean by „descriptions of the same reality“? Those are your words. Some frameworks are unrelated to physical reality. Computations using those frameworks won‘t produce contradictions, but they are useless nevertheless. As Roland Omnes quipped: They belong in the waste-paper basket.

I mean very plainly that they are descriptions of the same reality. From Griffith's Consistent Quantum Theory textbook

In order to avoid the mistake of supposing that incompatible descriptions are mutually exclusive, it is helpful to think of them as referring to different aspects of a quantum system

From https://arxiv.org/abs/1704.08725

In classical physics it is usually the case that only a single sample space need be considered when discussing a particular physical problem, and so its choice needs no emphasis, and it may not even be mentioned. In quantum physics this is no longer the case: many mistakes and numerous paradoxes, e.g., the Kochen-Specker Paradox (see Sec. V E), are based on not paying sufficient attention to the sample space in circumstances in which several distinct and incompatible sample spaces may seem like reasonable choices. For this reason it is convenient to use a special term, framework, to indicate the sample space or the corresponding event algebra which is under discussion.

Any system, quantum or classical, can be described by many different sample spaces and event algebras. For a classical system, we can identify a maximally refined sample space for which all other sample spaces are coarse grainings. We cannot do this for a quantum system, but the different samples spaces describing a quantum system are still describing that system.

Re/ Omnes: I suspect I know the context of his statement but to be sure, can you give me the source?

 

[gentzen]

I mean very plainly that they are descriptions of the same reality.

Then your statement is simply wrong, and no amount of quotes from Griffiths which don‘t use the word „reality“ can fix this.
Edit: Feel free to defend your statement in your own words as your own reasoning. I am simply opposed to use it to describe Griffiths‘ position.

Re/ Omnes: I suspect I know the context of his statement but to be sure, can you give me the source?

It is from his book „quantum philosophy“, where he addresses the objection that the existence of incompatible frameworks would undermine CH.

 

[Morbert]

Then your statement is simply wrong, and no amount of quotes from Griffiths which don‘t use the word „reality“ can fix this.

Why do you think it is wrong? How, for example, do you interpret the following from Griffiths?

https://arxiv.org/abs/quant-ph/9708028

There are many posable consistent families or frameworks which can be used to describe the same quantum system, and the choice of which of these to employ is a choice made by the physicist based upon pragmatic considerations, namely the type of physical problem he is trying to study or the question he wants to answer.

It is from his book „quantum philosophy“, where he addresses the objection that the existence of incompatible frameworks would undermine CH.

i) Remember that Omnes's position is different from Griffiths. Omnes understands CH as offering "trustworthy" logic for quantum systems. Griffiths more ambitiously presents CH as a realistic interpretation.
ii) His waste-paper basket comment is about frameworks not relevant to what a physicist is specifically interested in. Different frameworks will be useful/useless depending on what the physicist is investigating. [edit to add] There may be some frameworks that are useless for all experiments (e.g. ones with macroscopic quantum superpositions) but they still pertain to the same system.

 

[gentzen]

Why do you think it is wrong? How, for example, do you interpret the following from Griffiths?

The question is how to interpret your words. I have no trouble interpreting Griffiths words in various defensible ways. I don‘t see how to do that with your words, and PeterDonis seems to have a similar problem.

You don’t seem to see the difference between the concrete (physical) reality and an abstract quantum system.

 

[Morbert]

The question is how to interpret your words. I have no trouble interpreting Griffiths words in various defensible ways. I don‘t see how to do that with your words, and PeterDonis seems to have a similar problem.

You don’t seem to see the difference between the concrete (physical) reality and an abstract quantum system.

I'm asking you to point out where my reading of Griffiths is wrong.

 

[gentzen]

I'm asking you to point out where my reading of Griffiths is wrong.

When Griffiths is talking about different aspects of a quantum system, I can interpret quantum system just like in other discussions I had here, for example with vanhees71 when we talked about the polarization of silver atoms leaving an „oven“. Here a silver atom „prepared“ in this way (and especially its polarization) is interpreted as a quantum system.

This sort of interpretation would feel very forced for your words „same reality“.

 

[Morbert]

When Griffiths is talking about different aspects of a quantum system, I can interpret quantum system just like in other discussions I had here, for example with vanhees71 when we talked about the polarization of silver atoms leaving an „oven“. Here a silver atom „prepared“ in this way (and especially its polarization) is interpreted as a quantum system.

This sort of interpretation would feel very forced for your words „same reality“.

Assuming you are referring to the Stern-Gerlach experiment: Griffiths, in the same paper I referenced re/ his ontology

Choosing a description is choosing what to talk about, and what physicists choose to talk about has very little (direct) influence on what actually goes on in the world. Shall we discuss the location of Jupiter’s center of mass, or its rate of rotation? Either is possible, and neither has the slightest influence on the behavior of Jupiter. What about a silver atom approaching a Stern-Gerlach apparatus? Shall we discuss its (approximate) location or the value of Sx or the value of Sz ? Any one of these is possible, and the single framework rule allows location to be combined with Sx, or with Sz . But not with both, since it is impossible to put both Sx and Sz , at least when referring to a single particle at a particular time, in the same framework. In this case, no less than for Jupiter, the physicist’s choice of framework has not the slightest influence on the silver atom

It's clear here that Griffith believes CH can offer a realistic description of the silver atom. It's also clear he believes the multiple frameworks possible when describing the silver atom are different descriptions of the same reality (the silver atom) and not different, mutually exclusive versions of reality.