Entanglement: correlation you cannot fake

The gloves that prove nothing

A pair of gloves is separated. One goes in a box to Sydney, the other in a box to Reykjavík. The traveller in Sydney opens her box, finds the left glove, and knows instantly that the box in Reykjavík holds the right one.

Nothing strange has happened. The gloves were a pair when they were packed. The answer was fixed on the packing bench, and opening the box only revealed what had been true all along.

This is what almost every popular account of entanglement actually describes. It is ordinary correlation, and it is not surprising in the slightest.

Entanglement is the case where nothing was packed — and the results still match. The problem is that the two look identical from the outside. So the real question, and the whole content of this item, is: how could you ever tell?

The test that separates them

The trick is not to keep asking the same question. It is to ask different ones.

Set up two particles, one sent each way. Each measuring station has a dial with three settings — call them A, B and C. Each run, both sides pick a setting, measure, and write down a 0 or a 1. Afterwards the two lists are compared.

Two facts come out of the experiment:

1. When both sides happen to choose the same setting, the answers always agree. Every time, with no exceptions.
2. When the two sides choose different settings, the answers agree about a quarter of the time.

Fact 1 looks like it settles the matter in favour of the gloves. If the answers always agree when you ask the same question, surely the answers were packed in advance?

Follow that thought all the way and it breaks.

Why the gloves cannot do it

Suppose the answers really were decided at packing. Then each particle leaves carrying a little table: what to say for A, what to say for B, what to say for C. And because same-setting always agrees, both particles must carry the same table.

There are only eight possible tables — 000, 001, 010, 011, 100, 101, 110, 111.

Now take any one of them and count. Each table has three entries, and each entry is a 0 or a 1, so at least two of the three entries must be equal. There is no way round this; with two options and three slots, something repeats.

There are six ordered ways to pick two different settings. If at least two entries match, at least two of those six comparisons agree. So the agreement rate on different settings is at least 2 in 6 — at least 33.3%.

That is not a claim about clever packing. It is a claim about every conceivable packing, including ones nobody has thought of, including ones that cheat. Any pre-decided answer gives at least 33.3%.

The experiment gives 25%.

What that actually means

Twenty-five is less than thirty-three, so no table was packed. The answers did not exist before they were measured. And yet they are perfectly correlated when the same question is asked.

Notice carefully what has not been shown. Nothing travelled from one station to the other. You cannot use this to send a message, because each side alone sees nothing but a random string of 0s and 1s — the correlation only appears when the two lists are brought together and compared, which takes an ordinary phone call at ordinary speed.

So entanglement is not communication. It is a shared state that the two particles are in together, which no description of them separately can capture. Two entangled qubits are not two things with a link. They are one thing with two ends.

What it buys the machine

This is why the memory cost from the first item exploded the way it did. If qubits could be described one at a time, describing fifty of them would take fifty descriptions. Entanglement is precisely the fact that they cannot, which is why it takes 2⁵⁰ numbers instead — and why an ordinary computer chokes while a quantum one does not.

Entanglement is not a feature bolted on top of quantum computing. It is the reason there is anything to compute with.

Holding the idea honestly

There is a discipline in this argument worth keeping. Nobody proved entanglement by looking closer at the particles. They proved it by finding a number that every possible “the answer was already there” story must respect — and then measuring past it.

That is a rare and valuable move: instead of arguing about which explanation is right, work out what all explanations of a certain kind would have in common, and go and check that. It rules out a whole family at once, including the ones you were not clever enough to imagine.