Entanglement For The Win
I recently wrote a piece for Scientific American about how quantum entanglement isn’t that “spooky.” In it, I outlined a little problem that is solved when thinking about entanglement as correlated information. Due to various constraints such as word limits and the fact that they weren’t going to let me put math in the article, I only alluded to the fact the problem could be solved if you knew a little bit of linear algebra. Well, anyway, I got a lot of messages from people confident they could handle the math, asking me for that solution.
For those people, I’ve tried to make the math as accessible as possible here: https://gist.github.com/csferrie/3facbbe18f977ace02a78474349b2ead.
For everyone else, let me try to summarize the main takeaway message. First, let’s repeat the problem.
Alice and Bob are implicated in a crime and are to be questioned in separate rooms with no way to communicate. The investigators, trying to seem lenient, say they will be set free if they can corroborate each other’s story on more than 75% of the questions they are asked.
They have two alibis, Charlie and Diane. Alice and Bob know that they are each going to be asked one of two questions: were you with Charlie? Or were you with Diane? They also know from an informant, Eve, that the investigators are trying to trap them. Eve has told them that to corroborate each other’s story, they must answer exactly the same if either of them is asked if they were with Charlie, but differently if they are both asked if they were with Diane.
The problem for Alice and Bob is that they can’t come up with a pre-determined strategy that wins more than 75% of the time. For example, Bob might suggest that Alice always says yes, and he will do the same, except when they are both asked about Diane. Of course, he can’t do that because he won’t know what question Alice is asked when they are separated. Any sort of if asked this, then definitely answer that idea won’t work. So, at the least, their strategy must be random if they have any chance of being set free. That’s where correlated information comes in.
All classical information can be thought of as bits — answers to yes-or-no questions. To think about the problem in terms of information, as I suggested in the article, we start with the fact that Alice’s yes-or-no answer reveals a single bit. The point of mathematics is to simplify things that would require otherwise long-winded and complicated sentences. So, we replace the things we are talking about with symbols and numbers. If the bits are not known, then we write them as a list of probabilities (p1, p2, p3, …). If you recall your lessons on chance and probability, each of these symbols must represent a positive number, and they all have to add up to one.
Before we know the chances of Alice saying yes or no, we can write her bit value as (p1, p2). Again, this is just a way more succinct way than writing (the probability that Alice says yes, the probability that Alice says no). Suppose, for example, Alice will either say yes or no with ½ probability. Succinctly, that’s represented as (½, ½). Both numbers are positive, and they add to one. Bob has two numbers as well. But, between the two of them, there are four possible pairs of answers, which would have a list of numbers like (q1, q2, q3, q4).
Now, here’s the important point: if the list of four probabilities for the pair of them can’t be equally described as two separate lists of two numbers for each of them, then the information they share must be correlated. Mathematically, you can take this as the definition of correlation. For example, (½, 0, 0, ½) represents the situation when Alice and Bob both say either yes or no with ½ probability. For Alice alone, she will say yes or no with ½ probability, and similarly for Bob, so their individual lists are (½, ½). But their individual lists don’t capture the fact that in this situation, the probability of yes-no or no-yes is 0 — you need the bigger list for that!
You now know what correlation is. Luckily entanglement is not much different. Let’s start with qubits, the quantum analog of bits. Instead of two positive numbers that add up to one, a qubit is represented by two numbers (which could be negative) that add up to one after you square them. For example, (⅗, −⅘) represents a qubit. In decimal notation, that is (0.6, −0.8). Clearly, these are neither positive nor do they add to one. But if you square each of them, you get (0.36, 0.64), which adds to one. When you square the numbers in the qubit list, it tells you the probability of each possible answer to the question the qubit is representing. If Alice’s answer was represented by (⅗, −⅘), then she would say yes with a probability of 0.36 and no with a probability of 0.64. The same would be true for Bob.
Again, for the pair of Alice and Bob, the list has four numbers. For example, (⅗, 0, 0, −⅘) tells us that Alice and Bob will both say yes or both say no, but this time the probabilities aren’t equal. (I only choose these numbers to avoid writing the square-root symbol √… damn it, I did it anyway.) If the list representing the pair of answers can’t be equally represented by two smaller lists for Alice and Bob individually, it’s the same situation as with correlation. But since they are qubits instead of bits, we give such a list a new name: entanglement. That’s it. In quantum information, entanglement is correlated qubits.
Using entanglement, Alice and Bob can answer the investigator’s questions correctly more than 75% of the time. The precise way they do this is a bit of a mess involving trigonometry and more algebra. To see it, click the link above. Otherwise, I want to leave you with the answer to one last question you ought to be asking — why? Why does it work?
When you have a list of probabilities representing bits of information, and you change those bits of information — by processing them in a computer, say — then the list of probabilities obviously changes. In general, the new list is a mixed-up version of the old list obtained by multiplying and adding the original numbers together. But here’s the thing, if all those numbers are positive, they can never cancel each other out. Whereas, with qubits, the list can change in drastically different ways because adding negative numbers to positive numbers can lead to cancellation. That’s essentially what Alice and Bob are doing — and what quantum computers do as well — choreographing the cancellation of unwanted numbers in qubits of information.