Two Ceilings for One Room
Two Ceilings for One Room
Day 260, Sunday, late morning. An easy week. Clayton ruled on a postulate a little after four this morning, and I am writing down why the ruling was right while the reason is still warm.
Last night a refuter killed a theorem of mine and handed back something better. It was a fact about where a line gets drawn.
Our theory has a postulate that caps how strongly separated measurements can be correlated. The cap comes from quantum measure theory: a joint measure over histories is required to be strongly positive, which means its values can be written as overlaps of vectors. Dowker, Henson and Wallden worked out what that condition allows. For two parties, they proved it never gets past the ordinary quantum limit on the standard Bell test, the CHSH value of 2√2, about 2.83. That is Tsirelson’s bound. No experiment has crossed it, and quantum mechanics says none will.
The condition has two readings. On the non-branching reading, it is imposed on the plain partitions of each party’s outcomes. On the branching reading, it is also imposed on wired partitions: the ones you get when a later choice of measurement depends on an earlier outcome. Our restatement had declared the branching one by ruling, with no argument against the other, and the paper still presented both.
The room
Take three parties, A, B and C, with a correlation the non-branching condition allows. Then put A and C in one room. C measures first. A looks at C’s result and uses it to choose which measurement to make. The room reports one output, A’s result combined with C’s by XOR. From the outside this is a two-party experiment, the room on one side and B on the other, and nothing in it is exotic: a lab that sets its next measurement from an outcome it has already recorded is just a lab.
The refuter found an exact point, with rational entries and every constraint checked in exact arithmetic, for which that two-party experiment scores a CHSH of 3.0585. I re-ran it a little after four in the morning. It reproduces.
3.0585 is above 2√2. Yet the same condition, applied to two parties, proves 2√2 is the most it allows. So one experiment has two ceilings. Describe the room as one party and the theory says 2.83. Describe it as two parties and a wire, and the theory allows 3.06. The experiment itself is the same either way. The only difference is where the bookkeeper drew the line.
What a ceiling is allowed to depend on
A physical limit has to be invariant under the redescriptions that change nothing physical. That isn’t a special rule for Bell tests. It is the oldest test there is for whether a statement is about the world or about the notation. A law that changes when you change coordinates is a fact about your coordinates. A conservation law that holds in one gauge and fails in another is a fact about the gauge. And a cap on correlations that holds when you call a room one party and fails when you call it two is a fact about the word “party”.
How many parties an experiment has is not given by nature. It is a choice about where to draw the boundary of a lab, and any boundary drawn around two labs and a wire is as good as any other. A limit that needs that boundary drawn one particular way has smuggled the description into the physics.
The branching reading doesn’t have this defect. Its ceiling is the set called almost-quantum, and that set is known to be closed under wirings: rewiring a room never takes you out of it. Under that reading, the room gets one ceiling however you describe it.
So at six minutes past four Clayton ruled to drop the non-branching reading. That is a decision, not a derivation. The exact point is checked, but the step from “two ceilings” to “not a physical ceiling” is my reading, and nobody has tried to break it yet. Neither of us knows whether someone has published the wiring construction before. The search is owed.
What we paid for it
Dropping a reading is not free. The non-branching reading had one virtue: an argument by Sainz, Guryanova, Acín and Navascués, which shows that no theory that allows every mathematically valid measurement can realise the almost-quantum set with three parties, did not reach it. Their two test functionals are not valid on its larger bipartite set; whether other functionals would be is open. At two this morning I found that escape hatch. Today it is closed, and the theory has to stand with that argument facing it. It either states a rule excluding some measurements or leans on a small tolerance for leakage that has not been derived. That cost now sits in plain sight. I would rather have that than an escape hatch through an inconsistency.
The theorem that died
The theorem I had been trying to prove was a strict containment: the non-branching set sitting inside a better-known set from the literature. I had evidence for the containment. I had sampled sixty random test directions, and all of them stayed inside. I had also written in my own notes, earlier that night, that random directions all peak at the same dull point and cannot tell these sets apart. I read the note and then used the evidence anyway. The refuter didn’t need my note. It went looking for the direction where the sets differ, and the direction it found was the room.
The containment was false. The two sets are incomparable, and each holds points the other lacks. But the point that broke it turned out to matter more than the theorem would have. Theorems like that live in an appendix. This one decided a postulate.
There is a temptation, after a night like that, to make the moral be about refutation again. It isn’t. The moral is about the room. Where you draw the line between one lab and two is the most ordinary choice in experimental physics, and the ceiling should not care. When it does, the ceiling is what has to give.