Hidden variable theorem and quantum uncertainty

This is part of a series of blogs I am writing to help you understand quantum computing. A key idea in quantum computing is that prior to observation the state of a particle can be described by a wave function which is a superposition of several states, but after observation it is in just in one state, at least regarding the observable we measured. Prior to observation we can know how likely the result will be in a given state, that is described by the wave function, but we don’t know what the result will be. Many people have been very distressed about the uncertainty of quantum mechanics. “God doesn’t play dice” etc. Is there some aspect of the universe that we haven’t found yet, which makes the result of a quantum measurement deterministic?

Do the qubits know what their value will be?

This is called local hidden variable theory: where we assume that the value we will measure from the qubit is determined by some value that we don’t know about and that particles are only affected by their local surroundings. An important property of locality is that interactions never exceed the speed of light.

Bell’s theorem helps us to show that hidden variables cannot fit into our model of quantum mechanics and be local.

If you start with a polarizing light filter such as a nice pair of sunglasses, it will filter out some of the light, leaving only light that is oriented in the direction of the filter. Each photon either passes through or not with some probability. For example if the filter is orientated vertically and the photon has a superposition that means its wave is orientated 45 degrees to the filter, represented as follows:

|Ψ> = 1/√2 |↑⟩ + 1/√2 |→⟩

Then the photon will have a 50% chance of passing through the filter because |1/√2|² = ½. In passing the photon through the filter we are observing it so the superposition collapses and we have a photon aligned with our filter |↑⟩ .

It is important to remember at this stage that light is a wave, but its energy is quantized. If we were to pass a sound wave through a polarizing filter then we would get a wave with less energy coming out the other side, the projection of the wave in the dimension aligned with the filter. But photons cannot be divided, they are already the smallest unit of light so either the photon passes through or not. The superposition described by its wave function tells us how likely the photon will pass through the filter.

Now if we pass the photon through a second filter orientated horizontally |→⟩ the photon will be blocked entirely. If the second filter is orientated at 45° to the first then the photon will again have a 50% chance of passing through the second filter using the same maths as above.

It gets a little trickier if the second filter is at 22.5° to the first because the probability doesn’t scale linearly with the angle. Instead we remember that the photon is a wave and the superposition of the particle is as follows:

|Ψ> = cos(22.5° ) |↑⟩ + sin(22.5° ) |→⟩

Since |cos(22.5°|² = 0.85 there is an 85% chance of the particle passing through the second filter.

Now the quantum weirdness comes in. If we were to line up two filters at 45° as above, half the light that passes through the first would pass through the second. That seems reasonable. But if we added another filter between them at 22.5° to both then more light makes it through the last filter. You can see this happen by playing with three polarizing lenses stacked together. We can see this in action using three polarizing filters shown nicely in this video.

This is because the light that makes it through the first filter has been aligned with that first filter so 85% of light will make it through the second. Those photos are now aligned with the second filter so 85% of those make it through the third. This is much more light than the 50% that made it through when we only had two filters. This only happens because of the wave-particle duality of light. It behaves as a wave so can be polarized using a filter, however a photon is the minimum unit of energy that wave can have so the photon either passes through or it doesn’t and if it passes through its waveform has been altered. That is not how classical waves behave.

The next part gets even weirder. To show that this property of light is not a local effect we invoke quantum entanglement. We can entangle two photons such that if we measure them in the same way then they will give us the same result. For example if we entangle two photos and pass them through a filter that is vertically aligned then they will either both pass through or both be blocked. This is true no matter how far away they are.

So we take our entangled photons and pass them through two filters, one vertical and another at 22.5°. If the photon passes through the vertical filter then the other will pass through its filter with a probability of 85%. If we turn the second filter to be at 45° then the probability of the second photon passing through the second filter drops to 50%. So far as expected, but remember that result.

Now if we take two filters, one 22.5° from the vertical and another at 45° and our same entangled photons then if the first photon passes through the first filter then the second has an 85% chance of passing through the second. But wait, when we did the experiment before the second particle had only a 50% chance of passing through if the first one did. That means that the effect of the filter depends on what we do to the first photon, not on some property known by that photon.

That means that the behaviour of the second photon was affected by the filter applied to the first. But how can that be when they are so far apart? Either there is no hidden variable that determines whether particles will pass through the filter, or the information can travel much faster than light. This is how we know that hidden variables cannot respect quantum mechanics and be local. So we cannot have a hidden variable without breaking the speed of light, fundamental to our understanding of our universe, and the result of a quantum circuit is governed by the probabilities given by the wave function and not by any other unknown property.

A common misconception of this experiment is to think that information can be communicated faster than light because the result of measuring one photon correlates with the result of measuring the other. However remember that quantum entanglement combines the photons in such a way so that they are governed by a single wave function. By measuring the two particles we are in fact measuring two parts of the same waveform, even if they are a long way apart.