# Jeffrey Bub & Tanya Bub: There are recipes for Pi. But quantum mechanics?

There’s a recipe for Pi, in fact quite a few recipes. Here’s one that dates to the fifteenth century, discovered by the Indian mathematician and astronomer Nilakantha:

For the trillions of decimal places to which the digits have been calculated, each digit in the decimal expansion of Pi occurs about one-tenth of the time, each pair of digits about one-hundredth of the time, and so on. Its still a deep unsolved mathematical problem to prove that this is in fact a feature of Pi—that the digits will continue to be uniformly distributed in this sense as more and more digits are calculated—but the digits aren’t totally random, since there’s a recipe for calculating them.

Quantum mechanics supplies a recipe for calculating the probabilities of events, how likely it is for an event to happen, but the theory doesn’t say whether an individual event will definitely happen or not. So is quantum theory complete, as Einstein thought, in which case we should try to complete the theory by refining the recipe, or are the individual events really totally random?

Einstein didn’t like the idea that God plays dice with the universe, as he characterized the orthodox Copenhagen interpretation of quantum mechanics adopted by Niels Bohr, Werner Heisenberg, and colleagues. He wrote to his friend the physicist Max Born:

I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist.

But Einstein was wrong. Consider this puzzle. Could you rig pairs of coins according to some recipe so that if Alice and Bob, separated by any distance, each toss a coin from a rigged pair heads up, one coin lands heads and the other tails, but if they toss the coins any other way (both tails up, or one tails up and the other heads up), they land the same? It turns out that if each coin is designed to land in any way at all that does not depend on the paired coin or how the paired coin is tossed—if each coin has its own “being-thus,” as Einstein put it—you couldn’t get the correlation right for more than 75% of the tosses. This is a version of Bell’s theorem, proved by John Bell in 1964.

What has this got to do with quantum randomness? The coin correlation is actually a “superquantum” correlation called a PR-correlation, after Sandu Popescu and Daniel Rohrlich who came up with the idea. Quantum particles aren’t correlated in quite this way, but measurements on pairs of photons in an “entangled” quantum state can produce a correlation that is close to the coin correlation. If Alice and Bob use entangled photons rather than coins, they could simulate the coin correlation with a success rate of about 85% by measuring the polarizations of the photons in certain directions.

Suppose Alice measures the polarizations of her photons in direction A = 0 or A′ = π/4 instead of tossing her coin tails up or heads up, and Bob measures in the direction B = π/8 or B′ = −π/8 instead of tossing his coin tails up or heads up. Then the angle between Alice’s measurement direction and Bob’s measurement direction is π/8, except when Alice measures in the direction A′ and Bob measures in the direction B′, in which case the angle is 3π/8. According to the quantum recipe for probabilities, the probability that the photon polarizations are the same when they are measured in directions π/8 apart is cos2(π/8), and the probability that the photon polarizations are different when they are measured in directions 3π/8 apart is sin2(3π/8) = cos2(π/8). So the probability that Alice and Bob get outcomes + or − corresponding to heads or tails that mimic the coin correlation is cos2(π/8), which is approximately .85.

Bell’s theorem tells us that this pattern of measurement outcomes is closer to the coin correlation pattern than any possible recipe could produce. So God does play dice, and events involving entangled quantum particles are indeed totally random!

Tanya Bub is founder of 48th Ave Productions, a web development company. She lives in Victoria, British Columbia. Jeffrey Bub is Distinguished University Professor in the Department of Philosophy and the Institute for Physical Science and Technology at the University of Maryland, where he is also a fellow of the Joint Center for Quantum Information and Computer Science. His books include Bananaworld: Quantum Mechanics for Primates. He lives in Washington, DC. They are the authors of Totally Random: Why Nobody Understands Quantum Mechanics (A Serious Comic on Entanglement).