Many articles have been written on the EPR paradox and Bell’s inequality. I want to write down, for my own reference, what the crux of the paradox is, how it relates to a counterfactual definiteness, what the various philosophical resolutions are, and why I feel that Everett’s many worlds interpretation is the least objectionable. By and large, I will be following Guy Blaylock’s paper “The EPR paradox, Bell’s inequality, and the question of locality”, and you probably ought to be reading that paper instead of this blog post.

Counterfactual definiteness is the claim that experiments that were not performed but could have been performed would have had definite outcomes if they had been performed. For most people, counterfactual definiteness is intuitive, after all, science is all about making prediction about the outcomes of experiments that may or may not actually be performed. However, counterfactual definiteness is problematic in the face of predictions made by quantum mechanics and special relativity as we shall see.

Let us set up a standard EPR thought experiment. Suppose Alice and Bob are placed very far apart from each other and at rest relative to each other and have synchronized their clocks. They are each sent a stream of photons entangled with the other party’s stream; let us say a thousand pairs of entangled photons. While this experiment could be analyzed with just a single pair of entangled photons, the paradox is more clear with a stream of entangled photons. Alice and Bob simultaneously choose an angle to measure their photon streams at, and then simultaneously measure the polarization of their stream of photons at their chosen angle. Let us say they end up choosing the same angle, which we will label as measuring at 0°. We postulate Alice and Bob are far enough apart that all of Alice’s measurements are preformed in a space-like separated manner from all of Bob’s measurements. Alice and Bob record the results of their measurements and travel to meet up afterward to compare notes.

Let us say Alice tabulated the following results for her measurements: `+---+--+-+--+--+…`

, where `+`

means the photon was measured as parallel to the alignment of her detector and `-`

means the photon was measured perpendicular to the alignment of her detector.
Bob will have recorded the following result: `+---+--+-+--+--+…`

.
Their results are identical because the both performed measurements of entangled photons at the same measurement angle.
Nothing surprising here.

But suppose, counterfactually, Alice had decided to measure her photons at an angle of 41.4° instead.
What would have happened at Alice and Bob’s meeting?
Presumably since Bob’s experiment has not changed, and his experiment was space-like separated from Alice’s experiment, his results do not depend on what experiment Alice decides to perform, so his notes would still record the result: `+---+--+-+--+--+…`

.
Quantum mechanics predicts that counterfactual Alice and Bob’s results should differ in about 25% of the entries in this counterfactual scenario.
So counterfactual Alice’s notes would have perhaps recorded something like `-+--+--+--+-+--+…`

, or perhaps something different.
But whatever she recorded, it would be something that differed in about 25% of the entries when compared to Bob’s result.
So far so good.

Now suppose, counterfactually, Bob decided to measure at an angle of -41.4° and it was Alice who kept her measurement at 0°.
What would have happened at Alice and Bob’s meeting in this case?
By the same logic, Alice’s measurements at 0° would still get the result `+---+--+-+--+--+…`

, and it is counterfactual Bob whose reported measurement differ in 25% of the entries.
Because counterfactual Bob measures at a negative angle, we don’t expect it to necessarily agree with the previous notes of counterfactual Alice.
Maybe counterfactual Bob’s notes would have recorded something like `++--++-+-+--+--+…`

, or perhaps something different, or perhaps it might even agree the notes of counterfactual Alice.
Everything is still okay, but maybe we are getting a little nervous.

Finally, let us suppose, counterfactually, Alice had decided to measure her photons at an angle of 41.4° and Bob had decided to measure his photons at an angle of -41.4°.
What would have happened at Alice and Bob’s meeting in this case?
Alice and Bob’s experiments are space-like separated so neither of their choices should influence the outcomes of each other’s experiments.
Presumably Alice’s notes would be the same as what we wrote above for counterfactual Alice’s notes: `-+--+--+--+-+--+…`

.
Similarly Bob’s notes would be the same as what we wrote above for counterfactual Bob’s notes: `++--++-+-+--+--+…`

.
Here is the crux of the EPR paradox.
Quantum mechanics predicts that counterfactual Alice and counterfactual Bob’s notes ought to differ in approximately 87.5% of the entries in this scenario!
But no matter how we rearrange counterfactual Alice and counterfactual Bob’s notes, they can only differ between 0% and 50% of their entries on average.
This is what it means for Bell’s inequality to be violated.

Clearly something is wrong in our naive description of the hypothetical experiments above. What are some proposed philosophical resolutions to this EPR paradox?

One possible resolution is that Alice’s choice in her measurement does somehow affect the outcome of Bob’s experiment! The problem with this is that Alice and Bob’s experiments are space-like separated. This implies that an observer traveling rapidly towards Bob and away from Alice will observe that Bob’s experiments conclude before Alice even begins her experiment when she makes her choice to whether to measure at angle 0° or 41.4°. According to this resolution, this observer sees Alice’s choices affecting the outcome of already completed experiments!

A symmetric possible resolution is that it is Bob’s choice in his measurement that affects the outcome of Alice’s experiment. But we have the same issue as above. There still exists an observer, this time traveling towards Alice and away from Bob, who observes that Alice completes her experiments before Bob begins his experiment.

Non-local interpretations of quantum mechanics, including the Copenhagen interpretation and hidden variable interpretations such as pilot wave theory, resolve the EPR paradox in the above manner. They suggest there is some special global reference frame that is used to absolutely decide which of Alice and Bob’s experiments are performed first and whichever experiment comes first is this special reference frame is the one whose outcome affects the other’s experiment. They suggest that the rest of the laws of physics conspire to keep all agents in the dark about which reference frame is this special global reference frame, as there are no experiments that can determine which reference frame is the special one. In particular, we cannot acutally perform an experiment where we go back in time to see would have happened if Alice or Bob had choosen a different angle of measurement.

Furthermore, in general relativity, I suspect it is more difficult, and probably impossible, to come up with any globally consistent universal reference frame to resolve the order of all events.

Of course, it could also be the case that both Alice and Bob’s choice affect the outcome of each other’s experiments. But this only makes the problem worse as it would mean that in every reference frame there are future events affecting past outcomes.

Another resolution to the EPR paradox is that Alice and Bob could not have chosen different angles of polarization; if they both measure at angle 0° then that is the only choice they could have made and Alice and Bob do not have free choices in the matter. This resolution is called superdeterminism. We can make our thought experiment more extreme by taking Alice and Bob’s free choice out of the picture. Instead we have Alice first measures the polarization of a CMB photon coming from the constellation Leo and chose her measurement setting, 0° or 41.4°, based on the outcome of that measurement. We have Bob measure the polarization of a CMB photon from Aquarius on the opposite side of the visible universe to chose his setting, 0° or -41.4°. Now superdeterminism requires that the universe has been conspiring since near the beginning of time so that the plasma of the early universe would cause two photons photons to be released and travel for 13 billion years to a point where life developed and would be setting up a quantum correlation experiment and pass through their measuring devices in such a way to force them to align their measurement settings to get exactly the correlation in their records that is predicted by quantum mechanics.

Furthermore, in a superdeterminstic world there could be arbitrarily extreme violations of Bell inequalities, even beyond the violations predicted by quantum mechanics. Yet the cosmic conspiracy chooses never to produce statisitical results that exceed the Bell-style voilations predicted by quantum mechanics for some reason.

A third resolution to the EPR paradox is to say that the question of what would have happened if Alice or Bob had done a different experiment is not a well-formed question. This is the resolution captured by the “shut-up and calculate” interpretation of quantum mechanics. There is not much else to say about this resolution beyond saying that I do not find the rejection of the very question to be particularly satisfying.

Lastly we come to Everett’s many world interpretation.
This interpretation resolves the EPR paradox by saying that all possible experimental outcomes of Alice and Bob’s experiment all happen and they exist together in a superposition.
The phase of Alice and Bob’s superposition changes based on the polarization they choose to make their measurements with, but no matter their measurement choice, all 2^{1000} possible outcomes of Alice experiment happen and exist together and similarly for Bob.
Later, when Alice and Bob meet to compare their notes, the superposition of Alices and the superposition of Bobs interfere with each other and split up in such a way that "most" of the Alices meet up with a version of Bob whose recorded outcomes have the correlations predicted by the quantum mechanics (or in the case of perfect phase alignment "all" of the Alices meet up with the corresponding Bob who has identical recorded noted).

This resolution violates counterfactual determinism because it does not predict any specific outcome for counterfactual Alice. It predicts a similar superposition of Alices but in a different phase. In that situation, the various Bobs in superposition could have met up with any number of possible different counterfactual Alices when they interfered. Furthermore, the different phase that the counterfactual Bobs would have been in would definitely influence this interference when meeting up with the superpositions of counterfactual Alices. It is not the case that Bob’s experimental choices affects Alice’s results, but his choices does affect the interference that happens when Alice and Bob meet, and does influence which version of the superposition of Alices he (or rather they) meet up with.

The many worlds interpretation is not without its own problems. If multiple words are all equally as real, why is it that we assign less probability to those worlds with the lesser probability amplitudes. After all, those words are, in some sense, just as real as the worlds associated with larger probability amplitudes. A better way of phrasing the problem might be: why is it rational to behave as if we expect outcomes with probability in accordance to the probability amplitudes of quantum mechanics?