A Consistent Histories and Locality

[Morbert]

TL;DR Summary

Continuation of a discussion in a previous thread

Griffiths only talks about statistical properties. This is not what most people (including me) mean by realistic. In fact, I claim that CH itself can only talk about statistical properties. This is a nontrivial claim, and it could be wrong. But not in the way Griffiths argues against it, by simply ignoring the issue.

I don't understand this accusation. Given a single system prepared in some state $\psi = \sum c_i|i\rangle$, the probability $\mathrm{tr} |\psi\rangle\langle\psi|i\rangle\langle i|$, according to Griffiths, is the probability that the system has the property $i$. This is in contrast to statistical interpretations that present QM as a theory about infinite ensembles rather than single systems.

 

[Morbert]

My head is spinning.

I didn’t see the non-hidden variable mechanism that would then need to exist in CH. (We see that in MWI. We see that in retrocausal type explanations.) On the other hand, in your post #69: you say CH is realistic, but denies hidden variables. I am not sure how it can be realistic, which implies a pre-existing and determinate outcome for measurements at all angles independent of a setting elsewhere.

And he does accept a form of “proper” nonlocality. But I am very open to better understanding what is being presented, because it doesn’t seem to fit together as I read it.

These papers might be useful https://arxiv.org/abs/1105.3932 , https://arxiv.org/abs/1704.08725

While, according to Griffiths, measurements reveal pre-existing properties of systems, this doesn't mean we can write down some comprehensive state $\lambda$ describing all properties a system has because, unlike in classical physics, there is no maximally refined sample space covering all properties of the system.

 

[gentzen]

I don't understand this accusation. Given a single system prepared in some state $\psi = \sum c_i|i\rangle$, the probability $\mathrm{tr} |\psi\rangle\langle\psi|i\rangle\langle i|$, according to Griffiths, is the probability that the system has the property $i$.

Let us first agree that CH is consistent, and that Goldstein was wrong when he claimed that CH us inconsistent, and accused the framework rule to simply forbid to talk about it, without removing the inconsistency.

Therefore, Griffiths is not free to simply declare that probabilities predicted by CH are the probability of a single system to have the property $i$. Only the entire framework is allowed to be interpreted, not a single isolated part like the preparation without the remaining context.

My claim is that the interpretation of the probabilities for the framework are only statistical, i.e. not for single systems.

This is in contrast to statistical interpretations that present QM as a theory about infinite ensembles rather than single systems.

In the end, A. Neumaier and me had the same disagreement with vanhees71. The problem is the nature of predictions which apply to single systems. Just because you say „my probabilities talk about a single system“ doesn‘t make this true.

 

[Morbert]

Let us first agree that CH is consistent, and that Goldstein was wrong when he claimed that CH us inconsistent, and accused the framework rule to simply forbid to talk about it, without removing the inconsistency.

Therefore, Griffiths is not free to simply declare that probabilities predicted by CH are the probability of a single system to have the property $i$. Only the entire framework is allowed to be interpreted, not a single isolated part like the preparation without the remaining context.

My claim is that the interpretation of the probabilities for the framework are only statistical, i.e. not for single systems.

A framework is just a sample space, which exists in both classical and quantum theories. In quantum theories there isn't a unique sample space for which all other sample spaces are coarse grainings, but that doesn't prevent statements like "There is a probability p that system A has a property X"

In the end, A. Neumaier and me had the same disagreement with vanhees71. The problem is the nature of predictions which apply to single systems. Just because you say „my probabilities talk about a single system“ doesn‘t make this true.

It's not about being true or false. It's about consistency and sufficiency of interpretation. To insist probabilities yielded by quantum theories must be fundamentally about samples or ensembles, you would presumably have to argue that bayesian or propensity interpretations of probabilities are inconsistent or deficient.

 

[gentzen]

A framework is just a sample space, which exists in both classical and quantum theories.

All statements within CH have to be inside of some framework. Especially, taking about the preparation and properties $i$ of that preparation as if it were independent of a framework is not allowed.

This goes both ways, for Goldstein who cannot use this to prove CH inconsistent, but also for Griffiths who cannot use this to claim that CH is realistic.

In quantum theories there isn't a unique sample space for which all other sample spaces are coarse grainings, but that doesn't prevent statements like "There is a probability p that system A has a property X"

Careful, Copenhagen has its own ways to make such statements sometimes valid. But CH is more strict about which statements are allowed and forbidden, therefore it is not enough that such statements are not always strictly invalid.

It's not about being true or false.

As long as people like vanhees71 believe that the minimal statistical interpretation can make statements about single systems, I prefer the clarity of saying that this is simply not true.

It's about consistency and sufficiency of interpretation.

CH is consistent. Whether Bohmian mechanics is sufficient for all scenarios where QM or QFT apply is disputed. But it is not important for our current discussion whether CH is much better than Bohmian mechanics in this respect. I hope we can both agree that there are many scenarios where CH is sufficient.

Where we disagree is whether CH is realistic. I claim that CH is not realistic in the sense that Bell, DrChinese, and many other people understand that concept.

To insist probabilities yielded by quantum theories must be fundamentally about samples or ensembles, you would presumably have to argue that bayesian or propensity interpretations of probabilities are inconsistent or deficient.

The Bayesian interpretation of probabilities doesn‘t help either to turn some non-realist interpretation of QM into a realist one.

 

[DrChinese]

... according to Griffiths, measurements reveal pre-existing properties of systems, this doesn't mean we can write down some comprehensive state $\lambda$ describing all properties a system has because, unlike in classical physics, there is no maximally refined sample space covering all properties of the system.

If I can predict the precise outcome of any polarization measurement on a photon that has not been locally disturbed, altered or otherwise examined during its existence: How does the above make sense?

(Admittedly we can't make a statement about all properties simultaneously.)

My claim is that the interpretation of the probabilities for the framework are only statistical, i.e. not for single systems.

If I can predict the precise outcome of any polarization measurement on a photon that has not been locally disturbed, altered or otherwise examined during its existence: How is this "only statistical"?

I can do this for each and every identifiable Bell state resulting from a swap. That doesn't seem statistical to me.

 

[DrChinese]

These papers might be useful https://arxiv.org/abs/1105.3932 , https://arxiv.org/abs/1704.08725

While, according to Griffiths, measurements reveal pre-existing properties of systems, this doesn't mean we can write down some comprehensive state $\lambda$ describing all properties a system has because, unlike in classical physics, there is no maximally refined sample space covering all properties of the system.

Thanks for the references. I will work through them a bit closer to see if I can understand how "pre-existing" properties could possibly be made to yield the usual correlations for entangled photons: cos^2(A-B) where A=Alice's future angle setting and B=Bob's future angle setting.

I can't see how that is possible in a realistic interpretation. And not surprisingly, there is not a single specific example of how that could work. I would sure like to see one that reproduces both the quantum expectation for A<>B and A=B!

 

[gentzen]

If I can predict the precise outcome of any polarization measurement on a photon that has not been locally disturbed, altered or otherwise examined during its existence: How is this "only statistical"?

As I wrote in the other thread, those cases are not „only statistical“. It is the other cases where CH can only talk about statistical properties.

I can do this for each and every identifiable Bell state resulting from a swap. That doesn't seem statistical to me.

You can only do this for certain frameworks. And typical Bell inequality violation experiments cannot be described in those frameworks. But because you are not allowed to mix incompatible frameworks in CH, you cannot conclude anything from the possibility of those exact predictions.

 

[Demystifier]

I claim that CH is not realistic in the sense that Bell, DrChinese, and many other people understand that concept.

Yes, that's definitely true. CH is an attempt to give a realist interpretation of quantum complementarity. While many other interpretations say that complementarity (i.e. dependence on the framework) is really a dependence on the measurement setting, CH insists that complementarity has intrinsic ontological meaning independent on measurement. Some CH proponents say that CH is the Bohr's interpretation done right. To people with a kind of thinking similar to Bell's, that's a way too weird notion of ontology.

On top of that, CH in the Griffiths version adds a non-classical logic as a correct way of thinking about different frameworks, which is why Goldstein et al call it inconsistent, from the perspective of classical logic.

 

[Demystifier]

I can't see how that is possible in a realistic interpretation.

It's not, if the word "realistic" is interpreted in your way. But CH interprets the word "realistic" in a very different way.

 

[Demystifier]

Let us first agree that CH is consistent, and that Goldstein was wrong when he claimed that CH us inconsistent, and accused the framework rule to simply forbid to talk about it, without removing the inconsistency.

CH claims that reality depends on the framework. If we stretch this principle a bit, there is a framework, the framework of classical logic, in which Goldstein is right that CH in the Griffiths's version is inconsistent.

 

[Morbert]

If I can predict the precise outcome of any polarization measurement on a photon that has not been locally disturbed, altered or otherwise examined during its existence: How does the above make sense?

Consider a measurement on the photon with a random choice of aspect $\omega$ and measurement outcome $\{1_\omega,0_\omega\}$. The photon is in the initial state $|\psi\rangle = a_\omega|+_\omega\rangle + b_\omega|-_\omega\rangle$ and the measurement apparatus is in the initial state $|\phi\rangle = \sum_\omega c_\omega |\omega\rangle$. We can model this scenario with the time-evolution $\mathcal{U}$$$\begin{eqnarray*}\mathcal{U} &=& \sum_\omega U^{(\omega)}|\omega\rangle\langle\omega|\\U^{(\omega)}(t_0,t_1)|\psi\rangle|\omega\rangle &=& |\psi\rangle|\omega\rangle\\U^{(\omega)}(t_1,t_2)|\psi\rangle|\omega\rangle &=& a_\omega|+_\omega\rangle|1_\omega\rangle+b_\omega|-_\omega\rangle|0_\omega\rangle \end{eqnarray*}$$First, we can construct histories $\{C_\alpha\}$ reflecting the ordinary quantum description of this experiment. $$C_\alpha = \left[\psi,\phi\right]\odot\left[\omega_\alpha\right]\odot\left[W_\alpha\right]$$where $\left[W_\alpha\right]$ is either $\left[1_{\omega_\alpha}\right]$ or $\left[0_{\omega_\alpha}\right]$, depending on $\alpha$. These histories describe the preparation at $t_0$, the choice of aspect at $t_1$, and the outcome at $t_2$. The probability of a history occurring is $\mathrm{Pr}(C_\alpha) = \mathrm{tr}C_\alpha^\dagger\left[\psi,\phi\right]C_\alpha$. If, instead, we are concerned with a "realistic" description, where the measurement reveals a pre-existing property $\{+_\omega,-_\omega\}$, we can construct the histories $$C_\alpha = \left[\psi,\phi\right]\odot\left[\omega_\alpha\right]\odot\left[\lambda_\alpha\right]\odot\left[W_\alpha\right]$$ where a measurement outcome $W_\alpha$ reveals the property $\lambda_\alpha$. We'll get the same probabilities as before. Note that unlike Bell's hidden variables, the microscopic properties $\{\lambda_\alpha\}$ don't permit a joint probability distribution like $\mathrm{Pr}(\lambda, \omega)$. So "realistic" in the sense of measurements revealing pre-existing properties, but not hidden variables, and hence not running afoul of Bell's theorem.

 

[Morbert]

CH claims that reality depends on the framework. If we stretch this principle a bit, there is a framework, the framework of classical logic, in which Goldstein is right that CH in the Griffiths's version is inconsistent.

It's not that reality depends on the choice of framework. It's that a description of reality requires multiple frameworks, with the logic of propositions about reality being specific to frameworks. This framework dependence is true in classical theories too. The difference being in classical theories we can always identify a unique framework for which all other frameworks are coarse-grainings.

This is quite opposite to the common misunderstanding of CH where we can decree whatever we like to be true by choosing the right framework.

 

[Morbert]

Where we disagree is whether CH is realistic. I claim that CH is not realistic in the sense that Bell, DrChinese, and many other people understand that concept.

This might be true. It is certainly true for Bell. "Measurements reveal pre-existing properties" is a weaker condition than what Bell addresses.

Your other objections seems to amount to pointing out that CH does not render alternative interpretations incorrect. This is also true. All that can ultimately be shown is that CH is a coherent, unambiguous, and robust interpretation of any quantum theory.

 

[gentzen]

CH claims that reality depends on the framework. If we stretch this principle a bit, there is a framework, the framework of classical logic, in which Goldstein is right that CH in the Griffiths's version is inconsistent.

What do you by „the framework of classical logic“? The technical meaning of „framework“ in CH does not talk about stuff like intuitionistic logic. Morbert tried to make sense of your remark, and came up with:

The difference being in classical theories we can always identify a unique framework for which all other frameworks are coarse-grainings.

Is this what you mean? Or was it just a play of words never intended to make technical sense?

 

[Demystifier]

Or was it just a play of words never intended to make technical sense?

Yes.

 

[Demystifier]

It's not that reality depends on the choice of framework. It's that a description of reality requires multiple frameworks, with the logic of propositions about reality being specific to frameworks. This framework dependence is true in classical theories too. The difference being in classical theories we can always identify a unique framework for which all other frameworks are coarse-grainings.

In other words, used e.g. by some Bohmian-type realists, in classical theories there is primitive ontology, while the CH interpretation lacks primitive ontology.

 

[gentzen]

This might be true. It is certainly true for Bell. "Measurements reveal pre-existing properties" is a weaker condition than what Bell addresses.

Good. I guess/hope this is enough to not confuse DrChinese unnecessarily.

Your other objections seems to amount to pointing out that CH does not render alternative interpretations incorrect. This is also true.

I am not sure what to make of this. I certainly tried to prevent an unproductive discussion with DrChinese, caused by a „niche“ interpretation of „realistic“ and a missing sensibility of non-statistical properties, or maybe better „non-ensemble“ single system properties.

But I also brought up Bohmian mechanics and Copenhagen, so maybe you are referencing to that. Maybe more importantly, when I found out that Barandes only talks about statistical properties in his latest paper

And his "Causally Local Formulation" turns out to be just the well known "no signaling" property of QM

I got quite disappointed and worried, why he „lost his track“. But it is worse for Barandes (who is explicitly concerned with primitive ontology) than for CH (where the formulation itself „needs no ontology“).

All that can ultimately be shown is that CH is a coherent, unambiguous, and robust interpretation of any quantum theory.

We definitively agree in this point, except that it may not be „all that can ultimately be shown“.

 

[Morbert]

Good. I guess/hope this is enough to not confuse DrChinese unnecessarily.

I am not sure what to make of this. I certainly tried to prevent an unproductive discussion with DrChinese, caused by a „niche“ interpretation of „realistic“

See this post where the meaning was clarified.

and a missing sensibility of non-statistical properties, or maybe better „non-ensemble“ single system properties.

I still don't understand this. This sensibility isn't missing in CH. One of the primary motivations for CH was its applicability to single systems like the universe.
https://www.webofstories.com/play/murray.gell-mann/163

 

[DrChinese]

These papers might be useful : https://arxiv.org/abs/1704.08725

1. "... let us now turn to the source of the mistaken notion that the quantum world is somehow pervaded by nonlocal influences, which even their proponents admit cannot be used to transmit information, and are hence experimentally undetectable."

Whoops! In the Ma experiment, the nonlocal influence is clearly detectable. See figure 3a versus 3b. But yes, we all otherwise agree that these influences cannot be used to transmit information.


2. "Einstein locality: Objective properties of isolated individual systems do not change when something is done to another non-interacting system."

In the Ma experiment and many others of a similar vein, this is precisely what is violated. Objective properties of distant systems change, or at least apparently so. "Apparently" here meaning that the most obvious conclusion is that the experimenter can freely choose to change an observable statistical resultset. But there are of course interpretations that can address that (Bohmian Mechanics for example). Most are nonlocal or retrocausal.

 

[DrChinese]

It's not, if the word "realistic" is interpreted in your [DrChinese] way. But CH interprets the word "realistic" in a very different way.

Point taken, Demystifier, thanks. But...

"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.

And yet the only elements used to make a 100% certain quantum prediction for distant observations is the relative (nonlocal) settings of both measurements. So do the observed quantum properties exist and have those values prior to the measurement settings? Or do the values change as the distant relative settings change?

In other words: I can make up a definition of "realistic" that differs from how Bell uses that concept. Then I claim QM is realistic in my sense but not in Bell's. But aren't I obligated to address how my definition applies to various experiments directly probing the matter? Or do I get off scot-free because I merely claim my definition works?

So I am asking for a simple application of any CH definition of realism as it applies to this loophole free test where settings are changed midflight, and the systems (1.3 km apart) do not have time to communicate after the settings* are determined. What, exactly, pre-exists before the measurement takes place? Keeping in mind that the A and B electrons being spin-measured have absolutely no relationship to each other at the beginning of each run. (They are not part of a single quantum system after reset, they become entangled later on a another distant spot C.)

Can anyone please explain? Because I don't think there is a meaningful difference in "DrChinese (or Bell) realistic" vs. "CH realistic". Certainly nothing that justifies claiming that QM is "CH" realistic and Einstein local.


*For sake of this discussion, let's pretend the settings are always the same for A and B when measuring the electron spins, even though they vary from run to run. That way we always get perfect correlation.

 

[Morbert]

And yet the only elements used to make a 100% certain quantum prediction for distant observations is the relative (nonlocal) settings of both measurements. So do the observed quantum properties exist and have those values prior to the measurement settings? Or do the values change as the distant relative settings change?

In other words: I can make up a definition of "realistic" that differs from how Bell uses that concept. Then I claim QM is realistic in my sense but not in Bell's. But aren't I obligated to address how my definition applies to various experiments directly probing the matter? Or do I get off scot-free because I merely claim my definition works?

Maybe my post #17 was not relevant to what you are asking about. What is meant here by 100% certain quantum predictions? Some preparation and measurement where the probability of a specific outcome is 1?

 

[DrChinese]

Maybe my post #17 was not relevant to what you are asking about. What is meant here by 100% certain quantum predictions? Some preparation and measurement where the probability of a specific outcome is 1?

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.

 

[gentzen]

possessed by the measured system

The trick is what is meant by measured system. In Copenhagen, this would be the state before measurement (i.e. the analog of an initial state in a deterministic theory) and the system dynamics, typically given by a Hamiltonian. But in CH, the framework is also part of the measured system.

That is the context of that statement. However, for Einstein locality, Griffiths tried to show more, and was more liberal with respect to the framework. But his calculations are just for statistical properties, i.e. basically the no signaling property, or at least not significantly more.

 

[DrChinese]

And just to drop another point:

"Reichenbach's Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a -so called (Reichenbachian) common cause, which brings about the correlation."-Miklos Redei

Since Redei (see here, 2003) holds locality (what he calls "local primitive causality") true by assumption (which he claims is justified, i.e. "this condition has been verified in many concrete models"). Using that assumption, he is able to "prove" cause precedes effect in basic entanglement setups (such as PDC).

On the other hand: If you believe the above Common Cause Principle is true (as he does), then Delayed Choice Entanglement Swapping experiments prove that cause need not precede the effect. That's because the causal agent unambiguously occurs in the future, and not in the past.

---

Note yet again: There is no consideration of Delayed Choice Entanglement Swapping experiments in his 2003 paper, because those had not been widely published and discussed until about 2008. And also not surprisingly: none of Redei's later papers in the arxiv ever acknowledge the existence of swapping experiments. Therefore he merrily proceeds to prove mathematically something that is disproven by experiment. My point is... don't be like Redei!

 

[DrChinese]

DrChinese said:
"possessed by the measured system" [quoting Griffiths]

The trick is what is meant by measured system. In Copenhagen, this would be the state before measurement (i.e. the analog of an initial state in a deterministic theory) and the system dynamics, typically given by a Hamiltonian. But in CH, the framework is also part of the measured system.

Griffiths specifically added "before the measurement took place", which you left off of the quote. That's realism as he defines it himself, and close enough to how I define it not to see a difference. If there is such realism, there cannot be locality. Such locality would NOT be excluded because of Bell's Theorem; it would be excluded by experiment.

 

[gentzen]

That's realism as he defines it himself, and close enough to how I define it not to see a difference.

This is irrelevant, whether you believe you know Griffiths and CH better than me. There is a context in which those statements are correct, and this context doesn‘t change just because you highlight some word in bold.

 

[DrChinese]

This is irrelevant, whether you believe you know Griffiths and CH better than me. There is a context in which those statements are correct, and this context doesn‘t change just because you highlight some word in bold.

I don’t think I know it better than you, I am the one asking.

Is there a preexisting value for A and B (or relationship between them) prior to their measurement? I’m trying to get specific details of what Griffith is asserting. I don’t understand why you truncated my Griffiths quote where you did, because the “prior” nature is fundamental to my question. If you agree/disagree a future action can affect (or appear to affect) a past outcome, tell me either way yes or no.

I am presenting specific experimental evidence that appears diametrically opposite to CH. I am trying to walk through using standard terminology. Words/phrases like “frameworks” (not present virtually anywhere else) and “ill defined concepts of measurement” (which seems pretty well described in actual experiments) make the discussion more difficult.

I am simply trying to map CH ideas to entanglement swapping, which I have yet to see mapped. All I have seen is general dismissals without addressing substance.

 

[gentzen]

Words/phrases like “frameworks” (not present virtually anywhere else) … make the discussion more difficult.

It makes no sense to talk about CH without its „framework“ concept:
https://plato.stanford.edu/entries/qm-consistent-histories/#PDIsFramSingFramRule

A central principle of the histories approach is that quantum propositional reasoning must always employ a single PDI, referred to as a framework. While the choice may be implicit, it is often helpful to make it explicit when carrying out a logical argument. The single framework rule states that any logical argument must use a particular framework; combining results from two incompatible frameworks is illegitimate.

If you agree/disagree a future action can affect (or appear to affect) a past outcome, tell me either way yes or no.

It is unclear whether CH can even talk about this, in the way you „want“ to talk about those things. When it comes to Einstein locality, Griffiths tries hard to do this, but I am not convinced that he succeeded.

When it comes to pre-existing properties revealed by measurements, there is little point in trying to go beyond the single framework rule in CH, because otherwise it would be simple to show that CH is inconsistent, if that were allowed.

 

[DrChinese]

It makes no sense to talk about CH without its „framework“ concept:
https://plato.stanford.edu/entries/qm-consistent-histories/#PDIsFramSingFramRule


When it comes to pre-existing properties revealed by measurements, there is little point in trying to go beyond the single framework rule in CH, because otherwise it would be simple to show that CH is inconsistent, if that were allowed.

Thanks for the reference.

Griffiths specifically talks about spin x not being a compatible framework with spin z. Ok, accepted. But that makes discussion of 4 fold measurements on photons 1/2/3/4 a compatible framework, because no counterfactual statements are discussed. Only physically realizable measurements. We are looking at polarization: 1&4 is L/R measured early, 2&3 is V/H measured late. Distances between measurements can be arbitrarily large.

So the question is: does this consistent framework indicate a failure of locality, as it appears to be in the experiment? No counterfactual or inconsistent reasoning is employed. And Bell is not a factor. Just actual results.

I say CH forbids this type of swap by its stated premises. In contradiction to the experiment.

 

[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.