CHAPTER 3The clever experiment

In which Bell asks one extra question, counts very carefully, and discovers a wall that every sealed-envelope theory in the universe has to obey — but that the universe itself does not.

§3.1Bell's idea — use three dials

Here's where Bell's small genius shows up. He noticed that asking "do they match?" with only two dials — Alice's and Bob's — never quite catches a clever sealed-envelope theory cheating. With only two angles in play, there's always enough wiggle room to invent a hidden rule that fits.

So he added a third dial setting. For the clean counting version, use three directions evenly spaced around a clock face, so that every pair of directions is 120° apart. Let's name them:

• A — pointing straight up, at 0°
• B — pointing 120° clockwise from A
• C — the remaining direction, 120° from both A and B

On every trial, Alice picks one of these three for her dial. Bob, independently, also picks one of these three. Each writes down their choice and their flash. After many thousands of trials they have two big tables, with three possible dial settings on each side.

§3.2The eight possible envelopes

Now suppose, for the sake of argument, that the sealed-envelope explanation is right. Each particle leaves the source carrying a complete instruction sheet: what to flash if the dial is at A, what to flash if at B, what to flash if at C. Three boxes to fill in, each containing either ▲ or ▼.

How many possible sheets are there? Well — three boxes, each with two choices, so 2 × 2 × 2 = eight possible sheets, listed below. (Alice's sheet says one thing; Bob's, by the same anti-correlation rule we saw in chapter 1, says the exact opposite of Alice's at every angle.)

§3.3The counting argument

Now look closely at any one of those eight sheets, and notice something. Pick any two of its three columns. Compare them. Either they show the same symbol, or they don't. Across all three possible column-pairs (A&B, A&C, B&C), how often do they match?

• For sheets 1 and 8 (all-▲ and all-▼): all three column-pairs match. Match rate across pairs: 3 out of 3 = 100%.
• For each of the six mixed sheets: exactly one column-pair matches and two don't. Match rate: 1 out of 3 ≈ 33%.

The point is that every sheet, no matter which one the source picks, has at least one matching pair. So at least 1/3 of all column-comparisons must match, no matter how the source weights the eight sheets.

Now translate that into Alice and Bob's table. Whenever Alice's dial is at X and Bob's is at Y (with X different from Y — that's what we care about for cross-dial trials), they're effectively comparing two of the three columns of one sheet. Alice's flash is what's in column X; Bob's flash is the opposite of what's in column Y, because Bob's particle carries the inverted sheet.

So Alice's flash matches Bob's flash exactly when the original sheet's columns X and Y are different. Across all the cross-dial trials, the fraction where they match is the fraction where the corresponding column-pair of the original sheet differed. By the counting argument above, this fraction is at most 2/3.

Notice what this is. It's not a fact about any specific sealed-envelope theory. It's a fact about all of them, simultaneously, by virtue of arithmetic alone — no matter what rule the envelope uses, no matter what weights the source uses, every theory in this whole family must respect that ceiling. The 2/3 isn't a guess; it falls right out of "at least one of the three column-pairs must match."

§3.4What the universe actually does

Now let's see what quantum mechanics — or, equivalently, what the actual experiment — predicts for this same setup. We go back to the curve in chapter 1 and read off the values for the relevant angles.

With Alice and Bob each picking randomly from {A, B, C}, the angle between their two directions, when they pick differently, is always 120°. From the curve:

• Dials 120° apart → match rate 75%

The average across all six different-dial combinations is therefore 75%. That is above the sealed-envelope ceiling of 67%. The simple three-dial setup catches the violation.

Bell's own 1964 paper writes the inequality in a slightly different algebraic form, but the logic is the same: locality plus hidden instructions imposes a bound, and the quantum prediction crosses it. The cleanest modern version uses four angles and is called CHSH.

§3.5The version experimentalists actually use

In 1969, four physicists — John Clauser, Michael Horne, Abner Shimony, and Richard Holt — re-derived Bell's argument with a small but important tweak. Instead of three dial settings, use four. In the photon-polarizer version used in many labs, Alice's dials are at 0° and 45°, and Bob's are at 22.5° and 67.5°. Compute a specific weighted score from the cross-dial correlations, called S.

The CHSH bound for any sealed-envelope theory: |S| ≤ 2. Ideal quantum mechanics, with the angles above, predicts |S| = 2√2 ≈ 2.83. Real experiments do not always land exactly at the ideal value, because sources and detectors are imperfect, but they do land above 2 in the way quantum mechanics predicts. The ideal violation is roughly 41% above the wall.

The puzzle is solved, but the answer is unsettling. The local sealed envelope cannot exist. Some other story has to be true.