The EPR Paradox

The Paradox

I only recently realized how philosophical the original EPR paper was. It starts out by providing a sufficient condition for something to be an “element of reality”, and proceeds from there to try to show the incompleteness of quantum mechanics. Let’s walk through this argument here:

The EPR Reality Condition: If at time t we can know the value of a measurable quantity with certainty without in any way disturbing the system, then there is an element of reality corresponding to that measurable quantity at time t. (i.e. this is a sufficient condition for a measurable property of a system at some moment to be an element of the reality of that system at that moment:)

Example 1: If you measure an electron spin to be up in the z direction, then quantum mechanics tells you that you can predict with certainty that the spin in the z direction will up at any future measurement. Since you can predict this with certainty, there must be an aspect or reality corresponding to the electron z-spin after you have measured it to be up the first time.

Example 2: If you measure an electron spin to be up in the z-direction, then QM tells you that you cannot predict the result of measuring the spin in the x-direction at a later time. So the EPR reality condition does not entail that the x-spin is an element of the reality of this electron. It also doesn’t entail that the x-spin is NOT an element of the reality of this electron, because the EPR reality condition is merely a sufficient condition, not a necessary condition.

Now, what does the EPR reality condition have to say about two particles with entangled spins? Well, suppose the state of the system is initially

|Ψ> = (|↑↓ – |↓↑) / √2

This state has the unusual property that it has the same form no matter what basis you express it in. You can show for yourself that in the x-spin basis, the state is equal to

|Ψ> = (|→← – |←→) / √2

Now, suppose that you measure the first electron in the z-basis and find it to be up. If you do this, then you know with certainty that the other electron will also be measured to be up. This means that after measuring it in the z-basis, the EPR reality condition says that electron 2 has z-spin up as an element of reality.

What if you instead measure the first electron in the x-basis and find it to be right? Well, then the EPR reality condition will tell you that the electron 2 has x-spin right as an element of reality.

Okay, so we have two claims:

  1. That after measuring the z-spin of electron 1, electron 2 has a definite z-spin, and
  2. that after measuring the x-spin of electron 1, electron 2 has a definite x-spin.

But notice that these two claims are not necessarily inconsistent with the quantum formalism, since they refer to the state of the system after a particular measurement. What’s required to bring out a contradiction is a further assumption, namely the assumption of locality.

For our purposes here, locality just means that it’s possible to measure the spin of electron 1 in such a way as to not disturb the state of electron 2. This is a really weak assumption! It’s not saying that any time you measure the spin of electron 1, you will not have disturbed electron 2. It’s just saying that it’s possible in principle to set up a measurement of the first electron in such a way as to not disturb the second one. For instance, take electrons 1 and 2 to opposite sides of the galaxy, seal them away in totally closed off and causally isolated containers, and then measure electron 1. If you agree that this should not disturb electron 2, then you agree with the assumption of locality.

Now, with this additional assumption, Einstein Podolsky and Rosen realized that our earlier claims (1) and (2) suddenly come into conflict! Why? Because if it’s possible to measure the z-spin of electron 1 in a way that doesn’t disturb electron 2 at all, then electron 2 must have had a definite z-spin even before the measurement of electron 1!

And similarly, if it’s possible to measure the x-spin of electron 1 in a way that doesn’t disturb electron 2, then electron 2 must have had a definite x-spin before the first electron was measured!

What this amounts to is that our two claims become the following:

  1. Electron 2 has a definite z-spin at time t before the measurement.
  2. Electron 2 has a definite x-spin at time t before the measurement.

And these two claims are in direct conflict with quantum theory! Quantum mechanics refuses to assign a simultaneous x and z spin to an electron, since these are incompatible observables. This entails that if you buy into locality and the EPR reality condition, then you must believe that quantum mechanics is an incomplete description of nature, or in other words that there are elements of reality that can not described by quantum mechanics.

The Resolution(s)

Our argument rested on two premises: the EPR reality condition and locality. Its conclusion was that quantum mechanics was incomplete. So naturally, there are three possible paths you can take to respond: accept the conclusion, deny the second premise, or deny the first premise.

To accept the conclusion is to agree that quantum mechanics is incomplete. This is where hidden variable approaches fall, and was the path that Einstein dearly hoped would be vindicated. For complicated reasons that won’t be covered in this post, but which I talk about here, the prospects for any local realist hidden variables theory (which was what Einstein wanted) look pretty dim.

To deny the second premise is to say that in fact, measuring the spin of the first electron necessarily disturbs the state of the second electron, no matter how you set things up. This is in essence a denial of locality, since the two electrons can be time-like separated, meaning that this disturbance must have propagated faster than the speed of light. This is a pretty dramatic conclusion, but is what orthodox quantum mechanics in fact says. (It’s implied by the collapse postulate.)

To deny the first premise is to say that in fact there can be some cases in which you can predict with certainty a measurable property of a system, but where nonetheless there is no element of reality corresponding to this property. I believe that this is where Many-Worlds falls, since measurement of z-spin doesn’t result in an electron in an unambiguous z-spin state, but in a combined superposition of yourself, your measuring device, the electron, and the environment. Needless to say, in this complicated superposition there is no definite fact about the z-spin of the electron.

I’m a little unsure about where the right place to put psi-epistemic approaches like Quantum Bayesianism, which resolve the paradox by treating the wave function not as a description of reality, but solely as a description of our knowledge. In this way of looking at things, it’s not surprising that learning something about an electron at one place can instantly tell you something about an electron at a distant location. This does not imply any faster-than-light communication, because all that’s being described is the way that information-processing occurs in a rational agent’s brain.

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Measurement without interaction in quantum mechanics

In front of you is a sealed box, which either contains nothing OR an incredibly powerful nuclear bomb, the explosion of which threatens to wipe out humanity permanently. Even worse, this bomb is incredibly unstable and will blow up at the slightest contact with a single photon. This means that anybody that opens the box to look inside and see if there really is a bomb in there would end up certainly activating it and destroying the world. We don’t have any way to deactivate the bomb, but we could maintain it in isolation for arbitrarily long, despite the prohibitive costs of totally sealing it off from all contact.

Now, for obvious reasons, it would be extremely useful to know whether or not the bomb is actually active. If it’s not, the world can breathe a sigh of relief and not worry about spending lots of money on keeping it sealed away. And if it is, we know that the money is worth spending.

The obvious problem is that any attempt to test whether there is a bomb inside will involve in some way interacting with the box’s contents. And as we know, any such interaction will cause the bomb to detonate! So it seems that we’re stuck in this unfortunate situation where we have to act in ignorance of the full details of the situation. Right?

Well, it turns out that there’s a clever way that you can use quantum mechanics to do an “interaction-free measurement” that extracts some information from the system without causing the bomb to explode!

To explain this quantum bomb tester, we have to first start with a simpler system, a classic quantum interferometer setup:

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At the start, a photon is fired from the laser on the left. This photon then hits a beam splitter, which deflects the path of the photon with probability 50% and otherwise does nothing. It turns out that a photon that gets deflected by the beam splitter will pick up a 90º phase, which corresponds to multiplying the state vector by exp(iπ/2) = i. Each path is then redirected to another beam splitter, and then detectors are aligned across the two possible trajectories.

What do we get? Well, let’s just go through the calculation:

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We get destructive interference, which results in all photons arriving at detector B.

Now, what happens if you add a detector along one of the two paths? It turns out that the interference vanishes, and you find half the photons at detector A and the other half at detector B! That’s pretty weird… the observed frequencies appear to depend on whether or not you look at which path the photon went on. But that’s not quite right, because it turns out that you still get the 50/50 statistics whenever you place anything along one path whose state is changed by the passing photon!

Huh, that’s interesting… it indicates that by just looking for a photon at detector A, we can get evidence as to whether or not something interacted with the photon on the way to the detector! If we see a photon show up at the detector, then we know that there must have been some device which changed in state along the bottom path. Maybe you can already see where we’re going with this…

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We have to put the box in the bottom path in such a way that if the box is empty, then when the photon passes by, nothing will change about either its state or the state of the photon. And if the box contains the bomb, then it will function like a detector (where the detection corresponds to whether or not the bomb explodes)!

Now, assuming that the box is empty, we get the same result as above. Let’s calculate the result we get assuming that the box contains the bomb:

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Something really cool happens here! We find that if the bomb is active, there is a 25% chance that the photon arrives as A without the bomb exploding. And remember, the photon arriving at detector A allows us to conclude with certainty that the bomb is active! In other words, this setup gives us a 25% chance of safely extracting the information about if the bomb is active!

25% is not that good, you might object. But it sure is better than 0%! And in fact, it turns out that you can strengthen this result, using a more complicated interferometer setup to learn with certainty whether the bomb is active with an arbitrarily small chance of setting off the bomb!

There’s so many weird little things about quantum mechanics that defy our classical intuitions, and this “interaction-free measurements” is one of my new favorites.