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In Sakurai's book Modern Quantum Mechanics(2nd edition) he states the following in the section on Einstein's Locality Principle and Bell's Inequality (page 241):

"We derive Bell's inequality within the framework of a simple model, conceived by E. P. Wigner, that incorporates the essential features of the various alternative theories. Proponents of this model agree that it is impossible to determine $S_x$ and $S_z$ simultaneously. However, when we have a large number of spin-$\frac{1}{2}$ particles, we assign a certain fraction of them to have the following property:

If $S_z$ is measured, we obtain a plus sign with certainty.

If $S_x$ is measured, we obtain a minus sign with certainty.

A particle satisfying this property is said to belong to type $(\hat{z}+, \hat{x}-)$. Notice that we are not asserting that we can simultaneously measure $S_z$ and $S_x$ to be $+$ and $-$ , respectively. When we measure $S_z$ , we do not measure $S_x$ , and vice versa. We are assigning definite values of spin components in more than one direction with the understanding that only one or the other of the components can actually be measured. Even though this approach is fundamentally different from that of quantum mechanics, the quantum-mechanical predictions for $S_z$ and $S_x$ measurements performed on the spin-up $(S_z +)$ state are reproduced, provided that there are as many particles belonging to type $(\hat{z}+,\hat{x}+)$ as to type $(\hat{z}+,\hat{x}-)$."

Question: I am not quite following how proponents can agree that you cannot determine $S_x$ and $S_z$ simultaneously yet we are able to consider that a particle can belong to a type $(\hat{z} +, \hat{x} - )$? Also if we measure $S_z$ and do not measure $S_x$ as is mentioned, how do we assign it to a type of the form $(\hat{z} +, \hat{x} -)$ given that we are only making single measurement? I don't see how it is allowed to assign definite values of spin components in more than one direction in this sense.

Thanks for any assistance.

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As you read in the section, the concept Einstein and others could not accept was that of non-locality. In their opinion, measurements made by an observer on the system A, could not influence the results of the mesurements made by an other observer on the system B, if the two systems have no way to communicate (light years away for example).

If quantum mechanics is valid, this happens: A and B possess one electron each, the electrons are described by a singlet state. If A measures spin along the axis $\vec{u}$ and gets $|+>$, he projects the state of the two electrons system along that axis. Then he can tell B what he got and B now knows the spin state of his electron: then he can make predictions (giving probabilities) on his measurements along the axis $\vec{v}$. For example, if $\vec{v}=\vec{u}$, he can say to have 100% chance to measure $|->$ (Note that A had to tell B, and this is why we say quantum mechanics doesn't break the speed of light).

Einstein didn't want to deny all the incredible results of quantum mechanincs (of which he was one of the founders), but only this particular aspect. The example you brought from Sakurai is a model made to maintain the probabilistic predictions of quantum mechanics, but respecting the locality principle (A decisions can not influence B's measurements). This is made by introducing the so called "hidden variables": the particle already possess a property that will determine the outcome of the measurement. If we could know that property a priori, we would be certain on the result. But we can't know that property, and that's why we must still use a probabilistic approach to make predictions.

So let's say an electron has the property that a measurement along z will yeld |+>. We say that the electron belongs to the set $z+$. After measuring it, we could say: "Ok! That electron already belonged to that set!". But what if we chose to measure along $x$? Turn back time and suppose we didn't measure along $z$. The electron must have a property that determines the outcome of a measurement along the $x$ axis! For example, it belongs to $x-$.

So the electron has two properties, which will determine the outcome if we decide to mesure along $x$, or along $z$. Therefore it belongs to $(z+, x-)$. However, a measurements still strongly modifies the system (the two measurements don't commute), so if we decide to measure $z$ and get $|+>$, the properties of the particle will change, and it is not true that if we measure $x$ we will get $|->$. That is what they mean with "Proponents of this model agree that it is impossible to determine $S_x$ and $S_z$ simultaneously". Summing up, $(z+, x-)$ doesn't mean that if we measure z and then x (or viceversa) we will surely get those two results, but only the first we decide to make.

Now to the last question: "If we measure $S_z$ and do not measure $S_x$ as is mentioned, how do we assign it to a type of the form (z+,x−)?"

Actually, we can not do that working with one particle, in this case of spin measurements. We can only say things like "That electron belonged to the set $z+$.", but not like "That electron belonged to the set $(z+,x-)$.", because we should do two measurements, procedure that as I said modifies the system after the first measurement.

What we can do, and this is what Sakurai presents, is to use the fact that the measurements of the two observers A and B are no more correlated! Let's see what has changed in this new, Einstein's world.

As before A and B possess one electron each. These two electrons are in the singlet state, but this state is now very different from quantum mechanics. Before, we had a couple of particles described by ONE state vector, which tells us the probability of projecting one of the electrons along z+ or z- (let's suppose it is 50-50 for semplicity, so $|\psi>=\frac{1}{\sqrt{2}}(|+->_z-|-+>_z)$). Now, instead, the couple possess a property that determines the result: suppose A's electron $\in z+$, since the couple is in singlet state, B's particle must be $\in z-$.

But QM succesfully predicts that if we measure a big set of A's particles, we get 50% $|+>$ and 50%$|->$. How do we recover this result? If we have N couples we assume there are $N/2$ couples in which A's electron $\in z+$ and $N/2$ where A's electron $\in z-$. So we must introduce two populations of couples: $$ \qquad A\qquad \qquad B $$ $$ N/2 \qquad z+\qquad \quad z- $$ $$ N/2 \qquad z-\qquad \quad z+ $$

And what if we decide to characterize the couples with two measurements instead of one? Measure along $z$ and $x$. The populations we obtain are 4, all made to ensure singlet state:

$$ \qquad \qquad A\qquad \qquad \qquad B $$ $$ a)\quad N_1 \qquad (z+,x+)\qquad \quad (z-,x-) $$ $$ b)\quad N_2 \qquad (z+,x-)\qquad \quad (z-,x+) $$ $$ c)\quad N_3 \qquad (z-,x+)\qquad \quad (z+,x-) $$ $$ d)\quad N_4 \qquad (z-,x-)\qquad \quad (z+,x+) $$

Finally, we can see what I meant with "using the fact that the two measurements are no more correlated". A measures z, gets $+$. So the couple was in a) or b). A now has not modified the state of B's particle! B measures x and gets $+$. So the couple belonged to b) from the beginning. The probability of getting this result is $N_2/\sum N_i$. Note that A and B can perform their mesurements at any time, this procedure does not depend on the order of measurements, while in QM the chronological order was important!

Bell's inequality is set assuming Einstein's way of seeing things: in certain conditions, it is not valid in the world of quantum mechanics. Its violation was tested experimentally and was the proof that QM was the right theory.

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