# There’s No Cloning in Football

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.

1. #1 Anon
November 17, 2009

Can Emmy tell me which alternate universe I need to go to in order to watch my beloved Browns actually not suck?

2. #2 Lord
November 17, 2009

Whether it increased their chance of winning is highly dependent on which priors one wants to consider. Simply considering fourth downs is not a very astute play. Fourth downs in the last two minutes, ones deep in ones own territory, ones considering the relative team performance, could shift these, and while some have been considered, there are innumerable others that could bear heavily on it.

3. #3 Jonathan Vos Post
November 17, 2009

Very nice thread. In my 38-page To Do list is to expand and rewrite my partial draft of Astronomy and Baseball, Physics and Football, and other Sports/Science essays.

4. #4 Garbage
November 18, 2009

Nonsense. What he did was plain stupid and there is no statistics to compare with! No other coach would or have done that, and forgive me, but it wasn’t a masterpiece, even if it had worked out. It took the colts those 28 yards almost the whole 2 minutes!
Sorry, you can defend him if you wish, but he just f*ed up :)

Go Steelers!

5. #5 Rob Jase
November 18, 2009

I’ve always been horrible with statistics but as a long time Pats fan I agree with the coach’s decision.

Who wants to watch a dull game?

6. #6 Maxwell
November 18, 2009

Just like a Steelers fan to throw science out the window. Similar to the article, there’s no way of knowing what any other coach would do in that situation without them being in the exact situation.

7. #7 doug bennion
November 18, 2009

A not dissimilar situation is this. There are seconds left on the clock (this will be effectively the last play). You have just scored a touchdown and are now one point behind. You have the choice of (1) a virtually guaranteed single point conversion, to tie the game, which will send you into overtime with ~50% chance to win or (2) an option to attempt a two-point conversion, which has an X% chance of working, which will settle the issue right then and there.

You should try for the two-pointer if you think X > 50% given the circumstances (it’s a close call). I’ve never seen that choice made, or discussed. I’m quite sure if it was tried, the decision would be either ‘idiotic’ or ‘brilliant’, depending on the result.

8. #8 Matt Springer
November 18, 2009

Interestingly, the percentage of successful 2 point conversions is slowly creeping up to the 50% mark. From a purely mathematical perspective, this trend continuing should make the PAT extinct except in rare situations. Will it in practice? Beats me, but it’s a question coaches may have to start asking themselves.

BTW: Who dat!

9. #9 Captain C
November 19, 2009

“It took the colts those 28 yards almost the whole 2 minutes!”

To be fair, that was partly because they were trying to run out the clock. If the Pats had punted and sent the Colts back to their own 25 yard line, they (Colts) would have tried to move the ball a lot more quickly.

10. #10 Jonathan Vos Post
November 20, 2009

One excerpt on the origin of football, from my novelette-in-progress “Great White North”:

I came near the Shaman’s mammoth-skin tent with my transistor radio on. It was playing a Green Bay Packers game being played in a snowstorm, with die-hard fans huddling and shivering under layers of blankets.

It was about 30 below zero, Fahrenheit, here in the white corner of Nunavut, but clear in the dark night, lit by a fingernail-clipping Moon and coruscating Northern Lights.

These people living on latitude 65 degrees North can expect to average 240 to 250 nights of northern lights a year, significantly more than most Canadians, who live in areas of 50 to 100 auroras per year.

The radio hissed and squealed as the lights writhed. Sometimes I could hear a hiss or crackle from overhead.

“Turn that crap off,” said the Shaman. “What do you white men know about football, anyway?”

“We invented it,” I said. “Sorry, Master.” I turned off the radio.

“That’s an insult to all Northern people,” he said, grumpy. “Even you white people who’ve looked into the matter end up asking the question this way: Who really invented football — Native Americans? South American Indians? Was it Mexicans, Florentines, Chinese, Japanese or Eskimos? Screw the Eskimos with a Florentine burrito. Back before
football and soccer and rugby and Canadian Football had spring from common loins, the first inter-continental soccer match took place in Greenland in 1586 between an English explorer, John Davis, with his crew, versus the inhabitants of Godthab. But, as usual, everything noble in Inuit culture came from us. WE invented football.”

He sent me back to the main settlement, with a message, and soon a couple of dozen of the tribesman arrived. Among them, I recognized Bye, with his ironic green-dyed mohawk, Anticknap, Sherk, Deyo, Irniq (which means ‘son’ in Inuktitut, he’d told me, but “ravisher” in the old Mammoth People’s language), Quatchi (who preferred hockey, but was eager to play most any game), Smutylo, Greyeyes, and compact energetic Tuller. Tuller was Deyo’s cousin from the territorial capital Iqaluit, on Frobisher Bay. That name derives from Inuit for “place of many fish.” The others sometimes called Tuller “sardine.”

They identified themselves to me as Arsarnerit, or “the football players.” They made food and tobacco offerings to the Shaman, put on glowing headbands and rainbow belts and began to play their kind of prehistoric football — with a Walrus skull for the ball. Sherk invited me to play “and get warm” but I politely declined.

“I like to watch,” I said.

“That’s not what his sister says,” joked Anticknap, adjusting his tie-dyed scarf.

The aurora gleamed and glistened on the football pitch, ice marked off into a rectangle with walrus-hide-wrapped stones.