Sunday night, the Patriots lost a heartbreaker to the Colts 35-34. The talk of the sports world yesterday was Bill Belichick’s decision to go for it on fouth-and-two on his own 28 yard line when he was up by six with just over two minutes to play. They didn’t get the first down, and turned the ball back over to the Colts, who went on to score a touchdown and win the game.
Yesterday’s discussion was a low point even by the standards of sports talk radio, with one idiot after another holding forth about how stupid Belichick’s decisions was, and how he “disrespected his defense,” and various other dumb sports cliches. In actuality, people who know how to do math know that he was playing the odds, and had a higher probability of winning by going for it than he would’ve had if they had punted.
Belichick’s problem is one that’s well known to quantum mechanics. His decision to go for it increased his team’s chances of winning, but the actual outcome of the game was still probabilistic– no matter what he did, the result would come down to chance. And there’s no way to get information about probability from a single measurement. The only way to determine probabilities is through many repeated measurements on identically prepared systems, but the rules of football do not allow this, no matter how satisfying it would be to stick it to jackass sports radio yappers.
In quantum terms, what Belichick faced was a superposition of winning and losing states, like so:
|Game> = W|Win>> + L|Lose>
In this state, there’s some probability of winning (W2), and some probability of losing (L2), but the actual outcome of the game is indeterminate until the final whistle sounds, and the state is observed to be in either the |Win> or |Lose> states. Determining the probabilities of the two outcomes requires measuring the two coefficients W and L, but this can’t be done in a single measurement– the outcome of one game is either |Win> or |Lose>, and all subsequent measurements of the same game will have the same result. You can’t make multiple measurements of the single game to determine both, any more than you can re-play the last two minutes on your DVR and see the Patriots win.
“Well, ok,” you say, “But surely you can make many copies of the state, and measure those, and construct the probability distribution from the results of all the measurements.” It’s a nice thought, but it can’t work, any more than the Pats can ask for a hundred replays of the final two-plus minutes, to prove to the sports world that Belichick isn’t an idiot. In the football case, this is forbidden by the referees, who have families to get home to. In the quantum world, it’s forbidden by the no-cloning theorem, which says that it’s impossible to make a faithful copy of a single quantum state.
The proof of the no-cloning theorem is remarkably simple, and can be demonstrated with a minimal amount of mathematical slight-of-hand. It’s a proof by contradiction, which means we first assume that we can make a perfect quantum copy, and then show that it leads to a contradiction. So, let’s imaging that we have some quantum operator Û that acts on the |Game> state to produce a perfect copy:
Û|Game> = |Game>|Game>
Now, we can just plug in our superposition state from above, to find the total state:
|Game>|Game>=(W|Win> + L|Lose>)(W|Win>> + L|Lose>)
which becomes, with a little eighth-grade algebra:
|Game>|Game>=WW|Win>|Win>+WL|Win>|Lose> + LW|Lose>|Win> + LL|Lose>|Lose>
That might look a little scary, but if you just look at the states included, you’ll see that this is a superposition of all four possible outcomes: winning both games, winning the first and losing the second, losing the first and winning the second, and losing both games. Each of those outcomes has some probability, as you would expect.
But– and it’s a proof by contradiction, so there must be a “but”– this isn’t the only way we could find our final state. The initial state is a superposition itself, and the clone operator should work on the two pieces individually:
Û|Game> = WÛ|Win>> + LÛ|Lose>
This means that each piece gets its own duplicate:
Û|Game> = W|Win>|Win> + L|Lose>|Lose>
This is a very different state than what we found above– this state is a superposition of only two possibilities: winning both games, and losing both games. There’s no term here for a split decision, which we expect logically, and saw in the first method.
Logic demands that these two methods should give the same answer– they’re two different ways of calculating the same thing– but they don’t. There’s no way to make these two equal for any random choice of W and L– if one or the other is 0, it can work, but that corresponds to a 100% chance of either winning or losing, and that never really happens in sports or quantum physics. Since there’s no way for this to work in general, there must be some cotradiction in our set-up of the problem, and since the only thing we assumed was that we could make a cloning operator Û, then it must be impossible to make that operator. And thus, it is impossible to make a perfect copy of a single quantum state, in exactly the same way that it is impossible to re-play the last few minutes of a football game over and over, to demonstrate the actual probabilities involved.
The only way to determine the probabilities is to repeat the state preparation over and over again, in exactly the same way, either by repeating the operations on a whole bunch of different quantum systems, or by playing a multi-game series. You can use that to get an aggregate probability over the whole ensemble, but that’s a different game– specifically, baseball. In football and quantum physics, there’s just no way to get probabilities from a single measurement, meaning that quantum mechanics have to resort to “quantum teleportation,” while football coaches need to wait until the next time the two teams play, either in the playoffs, or next season.