The Einstein-Podolsky-Rosen (here after EPR) paper published in 1935 argues that the wave function in quantum theory cannot give a complete description of a quantum system, and some "local hidden variables" must be added. The title of their paper "Can quantum mechanical description of physical reality be considered complete?" is treated somewhat as a philosophical question on the nature quantum mechanics that is answered in the negative.

We'll follow each step of the argument to make it clear what the paradox contains and then we'll briefly discuss Bell's inequality theorem which is a formal refutation of the argument. EPR open their paper by stating a "criterion of reality" they state "in a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system" this statement seems true. If an experimentalist measures a quantum system but does not disturb it then the system acts as it would have acted in the absence of any measurement.

This criteria sounds obvious but it's also important which makes it worth stating. EPR go on to make a second assumption they assume that quantum mechanics must be local. The principle of locality is maintained in every non-quantum theory and is taken for granted by EPR because again this condition seems obvious, special relativity requires that no thing supersede the velocity of light. Back in the 1930s physicists understood that one day quantum theory and general relativity had to be combined into one complete description of reality so we can't have two incompatible frame works.

With that explained lets get onto the argument. Consider an entangled pair of particles which obey a momentum conservation law such that they have equal and opposite momentum and equal and opposite displacement from a common origin. These particles will obey the Heisenberg uncertainty principle:

If I measure the position of particle one I can calculate the position of particle two with certainty. Similarly if I measure the momentum of particle one instead I can calculate the momentum of particle two with certainty. EPR's criterion of reality is violated unless we consider the wave function to be an incomplete description of the quantum system. As these properties of the system were not given in the wave function and EPR reject the notion that measurements on particle one can influence particle two as these can be made at arbitrarily large distances. EPR state "No reasonable definition of reality can be expected to permit this" the authors believe that the particles contained some hidden variables which pre-determined a particles properties independent of measurement and these are not described by the wave function.

There were a number of responses including from the typically very difficult to read Neil's Bohr but in 1964 it was John Bell who formally refuted the EPR argument in a famous inequality known as "Bell's inequality theorem". Let's consider a modern variation of Bell's argument to make the logic easier to follow. Consider a pair of photons which can be polarised veritically, horizontally or at an angle. If we send the photons on opposite direction towards two vertical polarisers we know one photon will get through and the other wont because each member of the pair posses an opposite polarisation. When you perform the experiment you find a perfect anti-correlation. When Alice at one end tallies a one each time the photon goes through the polariser, Bob at the second polariser will note down a zero indicating that photon did not get through and vice versa.

So far there's nothing "quantum mechanical" about the experiment Alice and Bob performed. Perfect correlations can be replicated in classical physics as the famous example of Bertlmann's socks illustares. However, if we use a diagonal polarisation such that the photon passes Alice's polariser with some probability, then the corresponding photon at Bob's detector will enter with a corresponding probability obeying the unitarity rule

Now our discussion is exclusively quantum mechanical, these probabilities cannot be replicated in classical mechanics. When you perform the experiment the naive estimate you obtain for a "local hidden variable theory" when you measure three properties of the particle is given by

All the particles with property a but not b plus all the particles with property b but not c are greater than or equal to the number of particles with property a but not property c. Again, this should be obvious. Particles with property a but not property c already fall into at least one of the two categories on the left hand side. But neither category on the left hand side necessarily implies being included in the category on the right hand side.

The only way to derive this inequality is to make the same assumptions as EPR. So any violation of the inequality implies that one or more of these assumptions is wrong. In a famous experiment Alain Aspect found that the inequality was indeed violated so local hidden variables are at odds with the predictions of quantum mechanics.

So it remains still bizarre that the photon at Bob's detector "knows" what happened at Alice's detector. These photons travel in opposite directions at the speed of light so there's no way to send a signal between them without violating relativity. But the physics is even stranger because according to special relativity there is no "correct" ordering of space-like separated events. Events can only be temporarily ordered if they can exchange a light signal. Therefore to different observers the answer to "who collapsed the wave function?" will have different answers. The only way in which both Alice and Bob can agree on who first collapsed the wave function is if there's a preferred frame of reference with is prohibited by quantum field theory. In an upcoming post I'll talk about how the Consistent Histories interpretation resolves the paradox.

We'll follow each step of the argument to make it clear what the paradox contains and then we'll briefly discuss Bell's inequality theorem which is a formal refutation of the argument. EPR open their paper by stating a "criterion of reality" they state "in a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system" this statement seems true. If an experimentalist measures a quantum system but does not disturb it then the system acts as it would have acted in the absence of any measurement.

This criteria sounds obvious but it's also important which makes it worth stating. EPR go on to make a second assumption they assume that quantum mechanics must be local. The principle of locality is maintained in every non-quantum theory and is taken for granted by EPR because again this condition seems obvious, special relativity requires that no thing supersede the velocity of light. Back in the 1930s physicists understood that one day quantum theory and general relativity had to be combined into one complete description of reality so we can't have two incompatible frame works.

With that explained lets get onto the argument. Consider an entangled pair of particles which obey a momentum conservation law such that they have equal and opposite momentum and equal and opposite displacement from a common origin. These particles will obey the Heisenberg uncertainty principle:

If I measure the position of particle one I can calculate the position of particle two with certainty. Similarly if I measure the momentum of particle one instead I can calculate the momentum of particle two with certainty. EPR's criterion of reality is violated unless we consider the wave function to be an incomplete description of the quantum system. As these properties of the system were not given in the wave function and EPR reject the notion that measurements on particle one can influence particle two as these can be made at arbitrarily large distances. EPR state "No reasonable definition of reality can be expected to permit this" the authors believe that the particles contained some hidden variables which pre-determined a particles properties independent of measurement and these are not described by the wave function.

There were a number of responses including from the typically very difficult to read Neil's Bohr but in 1964 it was John Bell who formally refuted the EPR argument in a famous inequality known as "Bell's inequality theorem". Let's consider a modern variation of Bell's argument to make the logic easier to follow. Consider a pair of photons which can be polarised veritically, horizontally or at an angle. If we send the photons on opposite direction towards two vertical polarisers we know one photon will get through and the other wont because each member of the pair posses an opposite polarisation. When you perform the experiment you find a perfect anti-correlation. When Alice at one end tallies a one each time the photon goes through the polariser, Bob at the second polariser will note down a zero indicating that photon did not get through and vice versa.

So far there's nothing "quantum mechanical" about the experiment Alice and Bob performed. Perfect correlations can be replicated in classical physics as the famous example of Bertlmann's socks illustares. However, if we use a diagonal polarisation such that the photon passes Alice's polariser with some probability, then the corresponding photon at Bob's detector will enter with a corresponding probability obeying the unitarity rule

Now our discussion is exclusively quantum mechanical, these probabilities cannot be replicated in classical mechanics. When you perform the experiment the naive estimate you obtain for a "local hidden variable theory" when you measure three properties of the particle is given by

All the particles with property a but not b plus all the particles with property b but not c are greater than or equal to the number of particles with property a but not property c. Again, this should be obvious. Particles with property a but not property c already fall into at least one of the two categories on the left hand side. But neither category on the left hand side necessarily implies being included in the category on the right hand side.

The only way to derive this inequality is to make the same assumptions as EPR. So any violation of the inequality implies that one or more of these assumptions is wrong. In a famous experiment Alain Aspect found that the inequality was indeed violated so local hidden variables are at odds with the predictions of quantum mechanics.

So it remains still bizarre that the photon at Bob's detector "knows" what happened at Alice's detector. These photons travel in opposite directions at the speed of light so there's no way to send a signal between them without violating relativity. But the physics is even stranger because according to special relativity there is no "correct" ordering of space-like separated events. Events can only be temporarily ordered if they can exchange a light signal. Therefore to different observers the answer to "who collapsed the wave function?" will have different answers. The only way in which both Alice and Bob can agree on who first collapsed the wave function is if there's a preferred frame of reference with is prohibited by quantum field theory. In an upcoming post I'll talk about how the Consistent Histories interpretation resolves the paradox.

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