This is more of a physics question than a philosophy question, but Einstein's argument was just against the idea that the quantum state gives the complete information about the particle's state, since if it were complete the correlated values after measurement would have to be "spooky action at a distance"--he preferred an idea like you suggest where the correlations are explained in terms of the particles already having definite values for multiple non-commuting variables like sz1 and sx1, but this would imply the quantum state is incomplete.

However, later work by John Bell showed that this simple way of avoiding the conclusion of spooky action at a distance (non-locality) doesn't work in terms of explaining the statistics of measurements on entangled particles, see my physics stack exchange post here.

@Hypnosifl I what way is non-locality avoided here? The conclusion could be that the entangled non-local states have determined precise observable values. So s1z and s2z have precise (and correlated) values, at the same time as s1x and s2x have precise (and correlated) values. Which flies in the face of standard QM, saying that non-commuting operators, like sx and sz, can have no precise values at the same time.

If not for the violation of Bell inequalities, what need would there be for non-locality if you explain things in this way? Both entangled particles emanated from a common source, so if you believe they had fixed spins on both x and z axes prior to being measured and the measurements merely revealed these pre-existing values, in that case you could say that source created them with identical spins on both axes and then they were sent off in opposite directions--a purely local explanation for the fact that the experimenters always get the same results when they measure the same axis.

@Hypnosifl But how can you measure sx and sz of one particle at the same time? Put two measuring devices around the electron? One for sz and one for sx?

You don't have to be able to measure them at the same time to believe in an ontological sense that they have well-defined values for both sz and sx before you measure one of them (which could have the effect of disturbing the value of the other one, so it would be impossible to actually measure both pre-existing values--this would be a 'local hidden variables' model of QM, which would say a particle's state at any given moment has more variables than could be simultaneously measured, thus the additional variables are 'hidden' from experimenters).

@Hypnosifl But doesn't the fact that you can measure them precisely at the same time imply they have precise values at the same time?

You can't measure both values for the same particle at the same time. You can measure s1z for particle #1 and s2x for particle #2 at the same time, but unless you assume a local hidden variables model from the start (with Bell's theorem showing fundamental problems with trying to explain all measurement statistics using such a model), there's no reason to think particle #1 had a pre-existing value for spin along the x axis that was identical to what you found when you measured s2x for particle #2.

@Hypnosifl But if you measure sx2 at the same time as sz1, you know both sz1 and sx1 at the same time.

Again, you only "know" that if you assume a local hidden variables model from the start, if you don't assume that measurements of each particles' spins merely revealed preexisting values, and that the explanation for identical results on trials where you measure the same axis is that both particles have identical preexisting values prior to measurement, then in that case there is no reason to believe sx2 tells you anything about what sx1 would have been had you measured particle #1's spin on the x-axis.

Also, hopefully you understand that if you actually measure sx1 for particle #1 an arbitrarily short time after you measured sx2 and sz1, you only have a 50% chance of getting the same spin that you got when you measured sx2 for particle #2.

@Hypnosifl But what if you measure sz1 and sx2 at the same time? If you know sx2 you know sx1 also. So you know both sz1 and sx1 at the same time.

That's exactly what I addressed! Measuring sx2 doesn't tell you anything about sx1 unless you make the philosophical assumption that particle #1 has a definite spin on the x-axis even though you didn't measure it, and that it must be the same as particle #2's spin on the x-axis. Why would you make this assumption, unless you are assuming a local hidden variables model from the start? In a non-local interpretation, it could be that each particle only "decides" the spin they will have on a measured axis at the moment of measurement, and they coordinate their "choices" instantaneously.

@Hypnosifl But entanglement is a fact. If you measure s1z you know s2z and if you measure s2x you know s1x. So you need two measurements only to know all four spins.

If you measure s1z you only know s2z if in fact you measure particle #2 on the z-axis. Absent additional philosophical assumptions, this gives no reason to believe that in a different experiment where you measure s1z but do not measure s2z, particle #2 nevertheless had an unseen pre-existing value on the z-axis that was identical to what you saw when you measured s1z for particle #1. The quantum theory of entanglement certainly doesn't make such a claim.

Identical pre-existing values may seem like an intuitive explanation for the fact that you always get the same value when you measure both on the same axis, but did you read my lotto card analogy in the other post? Do you understand the argument that if we assume each card had identical pre-existing fruit in each box, that'd imply that when we scratch different boxes, we should get the same fruit at least 1/3 of the time? If we find that we only get the same fruit 1/4 of the time, that shows a problem for the model of pre-existing values.

@Hypnosifl Boxes with fruit are classical systems. The content of one box isn't correlated with the others, as in quantum boxes. If the s1x fruit is up the sx2 is down. If the sz2 fruit is down, the sz1 fruit is up. Measure sx1 and sz2 simultaneously and you know sx1 and sz1 simultaneously.

Did you read the post? It wasn't about classical boxes with physical fruits, it was a thought-experiment about pairs of scratch lotto cards where you could only pick one of three "boxes" (i.e. covered square regions on the card) to scratch, and when you did an image of a fruit was revealed. And it was a premise of the thought experiment that on any trial where both experimenters chose the same box on their respective card to scratch (say, box #2), they would both find the same fruit--if one found a cherry the other would too, if one found a lemon the other would too.

But the whole idea was that the statistics of what happened on trials where the experimenters chose different boxes to scratch showed some seemingly magical properties that would be impossible to explain with a "classical" model where each card in a pair is just manufactured with the same pre-existing combination of fruit in each box (say, both cards having a cherry in box #1, a lemon in box #2, and a lemon in box #3). So you're forced to imagine weirder explanations, like the cards communicating faster than light and only "deciding" what fruit image to create when the boxes are scratched.

@Hypnosifl The magical influence that connects the boxes is the influence of the manufacturer. He made the boxes with the fruit beneath it. There is no correlation because of entanglement. The boxes have nothing to do with the spins of the electron. If you measure sx1 and sz2 simultaneously you know sx1 and sz1 simultaneously, implying a violation of uncertainty, according to which you can't know them.

Again, did you read the post? The statistics seen on trials where the experimenters pick different boxes to scratch specifically rule out the theory that the 100% match rate in trials where they pick the same box can be explained just in terms of the manufacturer printing pairs of cards that have matching pre-existing fruit images behind each box. If you refuse to read the post to understand the reasoning for ruling out the theory "they were just manufactured to have matching fruits", there's no point in continuing (but if you do read and have questions, I'm happy to address them).

@Hypnosifl The boxes have nothing to do with QM because they are classical. There is no motion of fruit beneath the boxes. I don't speak of boxes on classical cards but measurements of QM spins. Fact is that you can measure sz1 and sx1 (via sx2) at the same time. So they can have precise values simultaneously. Squares with fruit beneath them have nothing to do with this. The fruit beneath each square has no entanglement with the other fruits beneath the squares (unless you make the connection beforehand).

The boxes are part of a thought-experiment, an imaginary scenario, so you can't just assume they behave as real classical cards would behave. The thought-experiment is specifically laid out so that the statistics are identical to those in quantum spin experiments, so they cannot be explained classically. The usefulness of the thought-experiment is just to make some of the abstract terms like "hidden variables" used in discussions of QM a little more concrete--it's easier to think about the idea that there are actually drawings of fruits under each square before you scratch off the covering.

"Fact is that you can measure sz1 and sx1 (via sx2) at the same time. So they can have precise values simultaneously." Yes, and if we label the three boxes on each card x, y, and z, then you can scratch the z box on card #1 and see a cherry, and the x box on card #2 and see a lemon, but because of the statistics in the thought-experiment (statistics exactly identical to those in QM), this cannot lead you to conclude that the x box on card #1 also had a lemon, i.e. the theory that the correlations can be explained by both cards being manufactured with the same hidden fruits.

To see why you can't just explain the results of the thought-experiment in terms of all pairs of cards being manufactured with identical combos of fruits, you have to follow the reasoning about how this hypothesis would imply the experimenters would get matching fruits on at least 1/3 of the trials where they chose different boxes to reveal, whereas the actual match rate in this thought-experiment is 1/4 (which can also be arranged in some quantum spin experiments). Do you disagree that the classical explanation where they were manufactured identically leads to the "at least 1/3" prediction?

@Hypnosifl Do you know in the box schratching that if you scratch a cherry behind the first aquare over here, there is a cherry behind the other first square over there too, and that if you scratch a lemon behind the second square over there, there is a lemon over here too? So if you scratch two different squares, you know four fruits?

In the thought experiment, the two experimenters in each trial are only able to choose one of the three boxes on their cards to reveal. If we look at the subset of trials where they both happened to choose the same box on their respective cards, they did both see the same fruit, 100% of the time. But in the subset of trials where they chose different boxes, they only saw the same fruit 1/4 of the time, which is less than the 1/3 that would be predicted if you assume the two cards in each trial had the identical pre-existing drawings of fruits behind each box.

If it helps to think about it, instead of a card we could imagine each experimenter is given a digital tablet with three boxes on the screen, & if they tap one of the three boxes the image changes to a fruit, but after that they can't tap any of the other boxes. So then the question is whether the two tablets already had stored values for the fruit that would be revealed if any of the three boxes were tapped, or whether they "chose" which fruit to display only at the moment of the tapping, perhaps coordinating their choices using something like a wi-fi signal (but possibly faster than light).

@Hypnosifl i'm sure it's an interesting card experiment but the cards are not what's the spin experiment here is about. I'm describing two measurements of spins of entangled electrons. If you measure the sz of one of both it gives the value of the other sz. The possible combinations are + - and - +. Likewise for sx, which you can measure at the same time by means of measuring the sx of the other. So you know both sx and sz at the same time. So there are non-local hidden variables.

It's a thought-experiment where the statistics exactly match those of the quantum experiment with entangled particles--you could even replace the fruit images with "+" or "-" signs revealed behind the chosen box to make the match more obvious. Again, on every trial where both choose the same box, they get the same symbol--do you think this proves that both cards (or tablets) must have pre-set symbols behind each of the three boxes? If not, then you can't have any rational reasoning for believing that the the particles have pre-set spins along each of the three axes.

@Hypnosifl Are there two possible outcomes beneath each square they scratch? Can you scratch both a + and a -? I don't think the spins are preset. There is a correlation though between the spins.

Yes, two possible outcomes, though on any single trial you only see either one or the other (just like on any given spin measurement you only get one outcome or another). As I said in my original description, each card "has three boxes that, when scratched, can reveal either a cherry or a lemon".

@Hypnosifl I don't see the link though with the simultaneous precise values of sx1 and sz1 (or sx2 and sz2) which can't have simultaneous precise values according to the uncertainty principle, since sx and sz don't commute. Doesn't this prove that God don't play dice?

@Hypnosifl "Yes, two possible outcomes, though on any single trial you only see either one or the other" Is there a superposition?

"Is there a superposition?" A "superposition" is an aspect of how we model quantum systems mathematically, you don't ever actually see one in measurements, you only see definite results like "spin-up along x-axis" or "spin-down along z-axis". Similarly in the card thought-experiment you only get definite results like "I revealed the x-box and saw the symbol +" or "I revealed the z-box and saw the symbol -". You would be free to model this in terms of there being a superposition of symbols under the boxes until you revealed one, of course.

"I don't see the link though with the simultaneous precise values of sx1 and sz1" There is no evidence that sx1 and sz1 have simultaneous precise values at the time you measure only one, just like there is no evidence that the x-box and the z-box on a single card have simultaneous precise values at the time you reveal just one. As always, all the observable results in the card thought-experiment are identical to those involving entangled particles, so there can be no rational basis for concluding preexisting simultaneous values in one case and not the other.

@Hypnosifl There is no evidence that sx1 and sz1 have simultaneous precise values at the time you measure only one, But you measure both. sz1 on electron 1 and sx2 on electron 2. If you know sx2 then you know sx1. So you know both sz1 and sx1 at the same time. Which shouldn't be possible. Only sz and the norm of s can be measured simultaneously, so the story goes.

But you aren't giving any reason why it might be rational to conclude the existence of simultaneous values in the case of the electrons but not the cards when all empirical measurements are identical in the two cases, by construction. Again, in the thought-experiment it's true that on any trial where they choose the same box, they get the same result. Do you think this means that when experimenter #1 chooses the x-box on card #1 and sees a +, and experimenter #2 chooses the z-box on card #2 and sees a -, that we can conclude there must have been a - behind the z-box on card #1 too?

@Hypnosifl If the cards are entangled and if #1 sees under the x-box of card1 a +, then he knows that the x-box of card2 has a -. Likwise, if #2 sees, at the same time, under the z-box of card2 a +, then he knows that the z-box of card1 contains a -. So both the x-box and z-box of #1 are known. As are those of #2.

"Entangled" just refers to a mathematical rule for predicting what we will observe empirically when we measure particles prepared in a certain way, it's not as if there is some truth about whether a given system "really is" entangled beyond those empirical measurements. Are you imagining there could be two different scenarios, one where the cards "are entangled" and another where they aren't, even if all empirical observations about the symbols seen when boxes are scratched are identical in both scenarios?

@Hypnosifl Entangled" just refers to a mathematical rule for predicting what we will observe empirically when we measure particles prepared in a certain way, it's not as if there is some truth about whether a given system "really is" entangled beyond those empirical measurements. I think that here the realism question can be considered. I think entanglement is a real state. The spin of electron 1 is really entangled with that of electron 2. Non-locally. So if you measure the spin of one of them the superimposed state collapses into + - or - +.

@Hypnosifl Are you imagining there could be two different scenarios, one where the cards "are entangled" and another where they aren't, even if all empirical observations about the symbols seen when boxes are scratched are identical in both scenarios? Exactly.

But then do you also think there could be two possible worlds where the measurements of electrons in these experiments are identical (and physicists in both worlds use the same mathematical formalism of 'entanglement' to predict experimental results), but in one world the electrons are "really entangled" and in the other they aren't? If so, could you explain more what conceptual content you assign to "really entangled" if it goes beyond the mathematical formalism? Does it have to do with the idea that a superposition of spin states in the formalism means they "really" have both spins?

@Hypnosifl But then do you also think there could be two possible worlds The two possible worlds exist in the case of the cards. There is only one possible world for the electrons. Namely the world in which the spins are entangled, and non-local hidden variables make that happen. Which means that sx1 and sz2 can have simultaneously precise values. I think it's this what Einstein argued against God playing dice.

If you think there is a possible world where the observable statistics of the cards can be explained without them being "really entangled" and without preexisting hidden variables, how can you say there isn't a possible world where the statistics of the electrons can also be explained without entanglement/preexisting hidden variables, given that the observable statistics are identical in both cases? Are you literally saying it's logically impossible, or some other idea like it being physically impossible given your assumptions about how physics works?

@Hypnosifl I can say that because the electrons are different from the cards. Entanglement is a fact. You can use it to give two non-commuting operators, sx and sz, precise values at the same time. It could be that I misunderstand your cards and what's beneath the boxes. So the fruit of the cards is entangled, like the spin? Before scratching the cards have a superposition lemon-lemon and cherry-cherry? So if the first box shows lemmon or cherry, so does the the first box of the second card? But then what about the second box?

"Entanglement is a fact" The empirical results seen in experiments with electrons are a fact, and it is a fact that a certain mathematical formalism which physicists call "entanglement" accurately predicts those results (with the cards we would see the same empirical results, and the same type of mathematical formalism could be used for predictions about the cards). But if you are using entanglement to refer to claims about "reality" that go beyond the empirical results and the predictive power of the formalism, that is not a scientific fact, it's a philosophical interpretation.

@Hypnosifl Not sure I follow you here. If we see an empirical correlation between the direction of two spins, then the entanglement is not real? I still can't see a relation with the cards. Imagine the first experimenter finds a + behind the first box. Then he can find a plus behind the second too. Two lemons. Or two cherries. Or a lemon and a cherry.

"If we see an empirical correlation between the direction of two spins, then the entanglement is not real?" You didn't answer my questions earlier about what you meant when you said entanglement is "real" so it's hard to answer. Does "entanglement is real" means anything more than "the mathematical formalism of entanglement makes accurate predictions about empirical measurements"? If "entanglement is real" has any additional content for you beyond that, then I would say that what you mean by "entanglement is real" must be including philosophical claims that are not empirically testable.

And note that if "entanglement is real" does just mean "the mathematical formalism of entanglement makes accurate predictions about empirical measurements" and nothing more, then you would have to say "entanglement is real" in the case of the cards too, since the empirical measurements are the same so one could use the same mathematical formalism to predict these measurements.

@Hypnosifl Entanglement being real just means there is a real connection between the spins. By, say, hidden variables.

Does "real connection" mean something beyond the observed statistical correlations between measurements of spins, and the accuracy of the mathematical formalism used to predict those correlations? If it goes beyond that, then you're talking about something non-empirical, it's a metaphysical claim rather than a scientific one.

@Hypnosifl Then the electron is metaphysical too. Still, I don't think the electron as metaphysical. It's rather physical. So why hidden variables not too?

Mathematical models involving elements called "electrons" can be used to accurately predict the result of empirical experiments, that's not a metaphysical claim. But if two people agreed on that, yet one said electrons "really exist" and the other said they didn't, that would be a metaphysical disagreement which science can't settle, at least not without an agreed-upon definition of what it means for something to "really exist" phrased in terms of measurable data and mathematical models used to predict that data. Even saying that "human beings exist" is metaphysical, absent such a defintion.

@Felicia - "But do mathematical models really exist. I can say they don't really exist." Whether mathematical models exist or not is also a metaphysical question, science can only deal with testable questions. You can write down a mathematical model and use it to make predictions about empirical results and compare with reality, but that doesn't tell you if the model, or any element of the model, "exists".