Since i just finished a course in Quantum Information, which studies quantum cryptography, quantum computers and other such things, i thought i might write a few articles about these things as they are interesting, highly useful and seem to have captured the interest of the general public fairly well. I hope i can manage to explain this in a way that a lay-person could read this and get the gist of it, if i fail for some reason, feel free to leave a comment and i’ll try to elaborate on points i was unclear on.

To start with, i wanted to do a short article on something called Bells inequality, which shows us that the fact that outcomes of measurements in quantum systems occur with a certain probability (as opposed to being able to precisely predict the outcome of an experiment in classical physics) can not be explained away by simply saying that the theory of quantum mechanics is incomplete and if only we knew about those factors we have not found yet, we could perfectly predict everything.

To begin with, i’ll have to introduce one of the weirdest results of quantum mechanics, quantum entanglement. It more or less means that it is possible to have two quantum systems (for example single photons (particles of light) or single atoms), that are somehow connected (entangled) with each-other, so there is no way to describe one particle without including the other. So for example lets say we have two atoms that have their spin entangled. A simplified way of imagining what the spin of an atom is, is that it’s the equivalent of the earth rotating about it’s axis. The two possible spins that the atom can have is spin-up, or spin-down, and using the analogy from before about the earths rotation, you can imagine spin-up being a clockwise rotation and spin-down being a counter-clockwise rotation. So what does it mean that they are entangled? Well it means that if you were to take atom #1 and measure it’s spin along the x-axis and get some result, then you know for certain, that atom #2 has the exact opposite spin. So say #1 measures spin-up, then you know that #2 has spin-down. This holds true even if the two particles are at the opposite ends of the galaxy!

Now if you think this is weird, don’t feel stupid, great minds were (and are) boggled by this as well, and in fact Einstein was sure that this could not be true. He argued that if you have two systems, A and B, that are completely separated by space, then the measurement of A must not modify the description of B. But when doing the math in quantum mechanics, this is exactly what you get! So Einstein maintained that the theory of quantum mechanics was incomplete, and that if you DID have the complete theory then the results of a measurement of a quantum system would not depend on probabilities, but you should be able to completely predict your results (like you can in classical everyday physics).

Phew! Ok that was a lot of backstory and explaining in order to get to what i really wanted to talk about, Bells inequality. So on one hand we have a theory giving us some very unintuitive answers, and on the other we have Einstein saying that this is only because the theory is incomplete, and if we had the full picture then all the weird things in quantum mechanics would make sense to us. But how can we test which one is true? This is where Bells inequality comes into play!

What it basically states, is that if you have three coins, and throw them all into the air at once, then you are 100% sure that at least two of them will be identical (heads/heads or tails/tails). So of P(1,2) is the probability of coin 1 and coin 2 being the same, then

P(1,2)+P(2,3)+P(1,3) >= 1

So the sum of all the probabilities should be equal to or more than 1 (100% probability). Seems pretty straight forward and obvious yes? Obviously 3 coins will have a 100% certainty to have 2 outcomes be the same, at least in the classical world we live in, and that’s where the catch is. If Einstein was to be right, and quantum mechanics was indeed deterministic like the classical world we know (that is to say, the current model of quantum mechanics is wrong), then this inequality HAS to be fulfilled for quantum mechanics.

So lets see what happens in the quantum picture. We’ll return to our entangled atoms from before, where Alice has one of them, and Bob has the other. The coins in this picture, are the spins of the atoms along different axis, so for example the spin along the x-axis can be either up or down, and each one has a 50% probability of being measured (remember that they can only do one measurement on each atom). So lets say that Alice measures the spin in the x-direction, and gets spin-up, she calls Bob and tells him the good news, so now Bob KNOWS that if he measures in the x-direction as well, he’ll get spin-down with 100% certainty (because they are entangled). So what happens now if Bob measures the spin along the y-axis? What is the probability of him getting the same result as he would have got along the x-axis (the equivalent of getting heads/heads, tails/tails)? Well without going into the math of quantum mechanics, i can tell you that it is 1/4. So looking back at the equation from Bells inequality, P(1,2)=1/4, and in fact P(2,3) and P(1,3) is also 1/4 (it doesn’t matter which axis you measure, it’s always the same math), so you see that

P(1,2)+P(2,3)+P(1,3)=3/4

which is obviously less then 1, and Bells inequality is therefore NOT fulfilled, and Einsteins explanation was wrong. This is also heavily backed by experiments done to test the Bells inequality, so it would seem that the current theory of quantum mechanics lives to see another day!

Now if you think about it, it’s not really that surprising that these things are completely counter-intuitive. After all, we humans never experience any quantum mechanical effects (they can only be observed at extremely small levels, like single atoms or single photons), so our brains have simply not evolved to be able to comprehend these things intuitively.

Lastly! Phil Plait over at the Bad Astronomy blog actually did a similar article a few weeks ago, where he discusses quantum cryptography (which also heavily depends on entanglement), i might write an article on that as well, assuming people didn’t fall sleep over this one and are interested in hearing more, but until then, you can check out his article.