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I was reading this introduction to quantum computing that says that when one photon in a pair of entangled photon is measured, the other one will have the opposite result when measured in the same way. But how do we know that all the responses for all the possible measures are not decided for each qubit when the pair is created?

The first photon of the entangled pair would have a surface of half a sphere for which measurements give a positive result, and the other photon would have the other half of the sphere. If you measure both photons on the same axis, then they will never give the same result, as it's impossible for a single point to be in the two halves of the sphere at once, but as you increase the distance between the axes of measurment, the probability for one point to be in the top half of the sphere and the second in the bottom half of the sphere increases.

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marked as duplicate by ACuriousMind, Kyle Kanos, user10851, John Rennie, Ryan Unger Aug 18 '15 at 14:11

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An important thing to understand is that Bell's Inequality is about how certain (incorrect) theories make predictions. It tells you absolutely nothing about other theories, not matter how similar the theories sound.

For instance Ball assumed that there is a hidden variable that combines with the orientations to tell you the results but that the hidden variables are powerless to affect the orientations even though the hidden variables should be enough to explain absolutely everything.

Keep in mind that what we call measurements are interactions that change the state in question.

Yet, the hidden variables for Be weren't even allowed to do basic things like know what find of device will be measuring (i.e. interacting with) them. Whereas there are many devices for measuring the spin of a single particle that interact with it differently (they for instance produce different probability currents and deflect different parts of an incoming beam to become a spin up beam in a continuous way (sending the rest of the wave in a different direction to become a spin down beam).

Literally you can make a device that sends the left part of the beam to become spin up and sends the right part of the beam to become spin down. And the sizes of the parts depends on the spin and the two devices always get the size (and hence the frequencies for identically prepared states). But they do it in different ways.

Bell assumes your hidden variables can't take into account the exact things you need to even model what the Schrödinger equations says must happen in the early stages of actually measuring the spin with a real device.

I'll repeat that, you need to know what kind of device you have to accurately model the actual dynamics. Some devices produce the same frequency of different results and so are considered equivalent for single particle measurements. Bell assume the hidden variables have to treat those the same.

Bell assumes that the hidden variabkes can't even have access to the wavefunction itself since the wave is nonlocal. The initial conditions of an entangled state is written in a a nonlocal way and Bell doesn't allow you to have access to it.

So Bell assumes you don't know exactly what you have and don't know exactly what it is interacting with, then outs enough constraints so you cannot possibly reconstruct this information, which you need, and those shows that such a handicapped theory does a bad job if agreeing with experimental results.

Now let's look at yours, on the face if it you are assigning results to orientations but you don't care about which device you use. Which means you've already lost any ability to accurately describe correlation between the own particles positions and spin since some devices have the left part become spin up and other devices have the right part become spin up and your description of the state can't model that.

So you already fail to explain correlations between a position measurement that happens inside a partially completed spin measurement all on one particle. Let alone be able to get correlations between two particles. So you threw away information about the state, for no reason. Because unlike Bell you aren't trying to handicap a theory to force it to give bad results you were trying to ask whether things are determined.

So you through away information about the state and then ignores the details of which device you use even though this matters to accurately describe what happens under arbitrarily measurements. Then you assume that even though a measurement interacts and changes things that you have zero explanation about how the state is supposed to change when you measure the spin.

How are you to explain that if you measure the x component and then the z component then the x component you get don't always get the same result for the two x results? Unless you say how your state changes you just failed to describe three sequential measurements on one particle and no entanglement how can you possibly expect to model two entangled particles when you a t as if the word measure means you aren't changing it.

Measuring x then z then x and measuring x then x then z give different results since the two x measurements give the same result in that second option and only agree half the time in the first option.

So if you want to say whether things are determined you need something that is capable of reproducing the known results.

And it isn't that hard. I'll do spin on an electron just because I'm more familiar with it.

So electrons have a spatial wave and a spin state. You can model both by having at every location in configuration space have an overall magnitude and phase (for the wave) and for every electron have a unit vector for its spin Have it point in the direction $(n_x,n_y,n_z)$ if the spin is an eigenvector of $$n_x\sigma_x+n_y\sigma_y+n_z\sigma_z,$$ with positive eigenvalue. This leaves it undetermined up to phase and magnitude but there is only one phase and magnitude for the whole configuration (wavefunctions are defined on configuration space).

Now a spin measurement must change the state into a state that points in the right direction corresponding to the new matrix that it will give a positive eigenvalue. This is mandatory to explain correlation between multiple measurements done in series on the same particle and to explain how the order matters.

Next we look at actual devices that measure spin for instant if the magnetic field points in the y direction and has an inhomogeneity in the y direction that has it increase in the $\pm\hat y$ then it will have beams deflected to split in the y direction with detectors in one side finding spin up and with detectors in the other side finding spin down.

We don't have to guess. Knowing the actual configuration of the magnetic field we know the actual Hamiltonian and we know how the actual Schrödinger equation evolves the actual incoming beam into two beams deflected in different directions and we know that it must and does polarize the spin of the incoming electron so that the value of the spin vector is in the $\pm\hat y$ direction. And it must polarize the beams between the two $\pm\hat y$ directions based on which deflected beam it is and based on which device we have, it is deflected the same direction you expect a force to deflect a classical magnetic moment in the $\pm\hat y$ direction in that specific device (whether the magnetic field pointing in the $\hat y$ direction is getting larger or smaller as you move in the $\hat y$ direction).

It must do this to agree with how a position measurement cones out if you measured position after passing though a series of inhomogeoeus magnetic fields.

So the beam must deflect and polarize this way for these devices becsuse that is what the Schrödinger equation says it must.

The wavefunction is the minimum amount of information you need to predict what is going on, so we start with that as a real thing that should determine what is going on.

Now what is left. The size of those deflected beams. Each beam, again according to the Schrödinger equation, must split into two beams one deflected $\pm\hat y$ based on whether the spin gets polarized to $\pm\hat y$ (or $\mp\hat y$ for the oppositely calibrated device). But the dividing plane that determines which streamlines of the Schrödinger equation current depends on the incoming spin state (that had to come in at some point) and the dynamics of the actual Schrödinger equation for the actual Hamiltonian for the actual device says exactly where that plane is. And since tiu know we could follow with a position measurement we know where it ends up, it ends up so that when you send in something with a particular spin the ratio of the L2 projections onto the eigenstates is the ratio of the L2 measure of the size of the two beams.

So specifically if it spin is all up then the whole beam is deflected instead of split and if the spin incoming and outgoing are orthogonal then the plane is in the middle of the beam (if the beam is symmetric) so the left goes left and the right goes right.

This is essential. For a single particle with no entanglement we see ... jst by looking at the Schrödinger equations dynamics (which is needed to predict position measurements) in the presence of magnetic fields we find that whether you get spin up for the second measurement of a spin x then spin z series of measurement will depend 100% on whether you were in the upper half of the beam that comes out of the spin x device ... and on which of the two differently calibrated devices you choose to use. And depends on nothing else. And if anything else happened to the evolution of the wavefunction then position measurements would come out wrong, and the Schrödinger equation would be violated which would be unacceptable if we are trying to agree with what we see in the lab.

So here we have a realistic model of what is going on in a spin measurement for a single particle with no entanglement. It tracks the correlations between position and measurement and has the right correlations between different spin measurement done in different orders over time becsuse it accurately let's spin states change by using the actual Schrödinger equation.

So now let's do entanglement. What does the Schrödinger equation say? It assigns a two particle spin state at every position in configuration space. That a function from $\mathbb R^7$ into like projective $\mathbb C^5$ which is a bit hard to visualize. But we know it splits beams and polarizes spin.

So let's have the x axis represent the width of one beam (B1) and the y axis be the width of the other beam (B2). Then splitting one beam looks like getting wider and then splitting into two rectangles. Splitting the other beam makes it get longer and then splitting into two rectangles. Like a horizontal or vertical wall forms for a dividing cell.

So what happens when you measure the spin of one of the particles? The beam gets wider and the spins polarize to the point that when they split into two they each have a spin vector for the first particle be $\pm\hat y$ depending on if it went left or right for the calibration of the device that measured it (yes it affects the literal evolution of the wavefunction) and the other particles spin becomes polarized to point the opposite direction. The spin of the first particle is now entangled with its position. By monogamy it is not entangled with the other electron anymore. The square (representing the width of both beams) got wider until it was twice the original width and then split into two equal squares. If later the other particle has the spin measured in that same direction then the left square gets deflected one way and the right one gets deflected the other way. If the second particle is measured in a different direction then those two left and right rectangles each gets longer and splits into complementary sized pieces.

There is actually zero freedom about this since this is what the Schrödinger equation says. And the statistics from repeated experiments for identically prepared systems comes solely from the size of those different regions in configuration space (weighted by the magnitude of the wave there) as determined by eventual position measurements.

If you have position be passively revealed by position measurement then every other observable is fixed, spin, momentum, energy, total angular momentum, z components because the position measurements have to track the Schrödinger equation so you have to keep and evolve the Schrödinger equation.

Anyone that wants to agree with quantum mechanics has to have access to the full wavefunction because it is the minimum information needed. And there is almost no freedom left for realism for anything once you have realism for position (or even just probability current, and that is what affects the rate detectors fire so you have to give that a reality if you have something that can read a clock and count how many times a detector goes off).

So that is a lot that is determined. But you need more than just saying yes no for every orientation. You have to say how things change because of interactions with measurement devices. But almost all of that is fixed for you. You can need the wavefunction becsuse you are going to be wrong without it, and you have to let it evolve according to the Schrödinger equation because otherwise you will be wrong. No freedom there.

Then you can lick nothing else. Or you can pick a hidden variable like position and get dBB. But no one would use Bell's theory that was designed to be wrong.

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I found the answer to my question on the Bell's theorem Wikipedia page:

With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables could agree with a linear dependence of the correlation in the angle but, according to Bell's inequality (see below), could not agree with the dependence predicted by quantum mechanical theory, namely, that the correlation is the negative cosine of the angle. Experimental results match the curve predicted by quantum mechanics.

If each photon in the pair had a surface of half a sphere for which measurments give a positive result, then the probability that the two points end up in the two different hemispheres increases linearly with the angle.

However, in reality, the correlation is the negative cosine of the angle, which does not match the correlation that would exist if the hidden variable theory was correct, as shown in this graph:

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