Qubits and P-bits
I have heard the refrain "everyone would understand quantum mechanics, if only it were explained thusly!" a surprising number of times recently, and I want to explore one such value of "thusly": The idea that quantum mechanics is just what falls out if you allow probabilities to be negative. (See e.g.)
I like the idea behind the Scott Aaronson link above -- and I think he's a crazy-smart guy with some great ideas -- but I think his explanation is still more technical than it has to be. He starts with the idea that the rules of quantum mechanics are just what probability would be, if you allowed negative probabilities; but then he immediately heads from probability into quantum, and starts talking about 2-norms and stochastic matrices.
I want to focus on going the other way: what if we took all the stuff we do in quantum, the same way we do it in quantum, but did it with ordinary probabilities instead? Thus, the p-bit -- "probabilistic bit" -- instead of the qubit.
Much like a qubit, which is 0 or 1 with some amplitudes, a p-bit is 0 or 1 with some probabilities. You can think of it as the result of flipping a biased coin, which comes up heads with some probability p, and tails with some other probability q. (You're jumping out of your seats trying to tell me that q = 1-p. Yes, I know, but there's a nice pattern which will fall out from not doing that just yet, so sit down.)
So adopting the same notation we use for qubits, in which |0> represents the zero-bit and |1> represents the one-bit, we can say that |0> is a p-bit, as is |1>, and so is 0.5|0> + 0.5|1>. Unlike qubits, of course, all the coefficients must be positive and real. Like qubits, p-bits have a normalization constraint: the coefficients, representing probabilities, must sum to 1.
When we put all this out in the open, the first interesting thing we see is that p-bits, just like qubits, display entanglement! Of course, in classical probability, we just call it correlation. Consider, for example, the system in which my friend flips a (fair) coin; then sets another coin next to it, so that both have the same side up; and doesn't tell me anything. If |0> is heads and |1> is tails, the coins are together in the state 0.5|00> + 0.5|11>: they are perfectly correlated, or "entangled" if you will. This kind of entanglement looks a lot less creepy than the quantum version, though. If you reveal one of the coins, you do immediately know the value of the other one; but it's clear that this is because they were the same all along. In physics terms, they share a "hidden variable". It appears, meanwhile, to be the case that quantum entangled states are more powerful than can be explained by a theory with purely local interactions, and hidden variables; informally, we can show that the states were not merely "the same all along".
P-bits also have the same property that qubits do, of having information "in the implementation" that can't be externally observed. If we look at a system that involves flipping coins to produce bits, this is the same as saying that, given just the results of the coinflips, we don't immediately know the probability distribution from which they were drawn. An interesting thing to note here is that the "no-cloning" rule in quantum physics appears to be outside of, and separate from, the math. Our p-bits here are purely mathematical, but we could imagine a physical implementation of them which also obeyed the "no-cloning" rule. In such a system, given an unknown "state" (i.e. an unknown probability distribution over bits), we'd be allowed to "observe" it once (i.e. take a sample from the distribution, and see what bits we get), but the result would not be enough to tell us what the original "state" (distribution) was.
Of course, p-bits do not demonstrate one other creepy phenomenon that qubits do, which is superposition. A p-bit has no phase: It can only be in |0>, or |1>, or some mixture of the two. This means no interference effects either. (Superposition is not as creepy when you consider that any superposition, is actually a pure state if you look at it in the right basis. But in any case, the only basis available for p-bits is the |0> |1> basis.)
I guess the conclusion I draw from all of the above is this: some of the difficulty traditionally associated with quantum mechanics comes from the complexity of the math, and some comes from unrelated parts of the physics. I don't know whether this approach has been taken before, but it seems to me that it could make a lot of sense, when teaching quantum, to start by teaching about what the world would look like if it were made of p-bits, and then introduce qubits as a revision of that world. What say you to this?
I like the idea behind the Scott Aaronson link above -- and I think he's a crazy-smart guy with some great ideas -- but I think his explanation is still more technical than it has to be. He starts with the idea that the rules of quantum mechanics are just what probability would be, if you allowed negative probabilities; but then he immediately heads from probability into quantum, and starts talking about 2-norms and stochastic matrices.
I want to focus on going the other way: what if we took all the stuff we do in quantum, the same way we do it in quantum, but did it with ordinary probabilities instead? Thus, the p-bit -- "probabilistic bit" -- instead of the qubit.
Much like a qubit, which is 0 or 1 with some amplitudes, a p-bit is 0 or 1 with some probabilities. You can think of it as the result of flipping a biased coin, which comes up heads with some probability p, and tails with some other probability q. (You're jumping out of your seats trying to tell me that q = 1-p. Yes, I know, but there's a nice pattern which will fall out from not doing that just yet, so sit down.)
So adopting the same notation we use for qubits, in which |0> represents the zero-bit and |1> represents the one-bit, we can say that |0> is a p-bit, as is |1>, and so is 0.5|0> + 0.5|1>. Unlike qubits, of course, all the coefficients must be positive and real. Like qubits, p-bits have a normalization constraint: the coefficients, representing probabilities, must sum to 1.
When we put all this out in the open, the first interesting thing we see is that p-bits, just like qubits, display entanglement! Of course, in classical probability, we just call it correlation. Consider, for example, the system in which my friend flips a (fair) coin; then sets another coin next to it, so that both have the same side up; and doesn't tell me anything. If |0> is heads and |1> is tails, the coins are together in the state 0.5|00> + 0.5|11>: they are perfectly correlated, or "entangled" if you will. This kind of entanglement looks a lot less creepy than the quantum version, though. If you reveal one of the coins, you do immediately know the value of the other one; but it's clear that this is because they were the same all along. In physics terms, they share a "hidden variable". It appears, meanwhile, to be the case that quantum entangled states are more powerful than can be explained by a theory with purely local interactions, and hidden variables; informally, we can show that the states were not merely "the same all along".
P-bits also have the same property that qubits do, of having information "in the implementation" that can't be externally observed. If we look at a system that involves flipping coins to produce bits, this is the same as saying that, given just the results of the coinflips, we don't immediately know the probability distribution from which they were drawn. An interesting thing to note here is that the "no-cloning" rule in quantum physics appears to be outside of, and separate from, the math. Our p-bits here are purely mathematical, but we could imagine a physical implementation of them which also obeyed the "no-cloning" rule. In such a system, given an unknown "state" (i.e. an unknown probability distribution over bits), we'd be allowed to "observe" it once (i.e. take a sample from the distribution, and see what bits we get), but the result would not be enough to tell us what the original "state" (distribution) was.
Of course, p-bits do not demonstrate one other creepy phenomenon that qubits do, which is superposition. A p-bit has no phase: It can only be in |0>, or |1>, or some mixture of the two. This means no interference effects either. (Superposition is not as creepy when you consider that any superposition, is actually a pure state if you look at it in the right basis. But in any case, the only basis available for p-bits is the |0> |1> basis.)
I guess the conclusion I draw from all of the above is this: some of the difficulty traditionally associated with quantum mechanics comes from the complexity of the math, and some comes from unrelated parts of the physics. I don't know whether this approach has been taken before, but it seems to me that it could make a lot of sense, when teaching quantum, to start by teaching about what the world would look like if it were made of p-bits, and then introduce qubits as a revision of that world. What say you to this?