The correlation, and the best a shared list of answers can do

At its defaults it draws the correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

bell-correlation is one function in lib/figures/quantum.js — the quantum of light and the wave of matter. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

The correlation between the two outer particles' measurements, against the angle between their analysers, for each of the four results the middle measurement can give. Every one of the four leaves the outer pair maximally entangled — each curve reaches one and minus one — so every outcome is as good as any other, and the outer parties have a perfect Bell pair whichever it was. What differs is which correlation they have, and the four are shifted and reflected versions of each other. The flat line is their average, which is zero at every angle, checked here at seven hundred and twenty angles to twelve figures. That vanishing is the whole reason the operation cannot be used to send anything: until the middle party's two classical bits arrive by an ordinary channel, the outer parties' data is indistinguishable from noise, and no correlation appears at all. The entanglement is created instantly and is useless until a message travelling no faster than light says which of the four it is.

Two quantities against the axis one half of an entangled pair is measured along. The correlation between the two outcomes runs from perfect anticorrelation through nothing to perfect correlation — a swing of two — as that axis is turned. The other half's own state, computed by summing over the first half's outcomes, does not move at all: its Bloch vector stays at 0.0e+0 of its maximum, which is machine zero, and its trace at one to 4.4e-16. So everything about the far measurement is present in the correlations and none of it is present locally. That is why entanglement carries no signal: seeing the correlation requires both sets of results in one place, and getting them there needs an ordinary message. It is also why a copier would break the argument — two copies of the local state could be measured along two axes, and the statistics would give the far setting away.

The CHSH value a party shares with a second, against the value the same party shares with a third. Every quantum state lies inside a quarter circle whose radius is Tsirelson's bound, 2.8284, because the sum of the two squared values cannot exceed eight. The classical limit is 2 on each axis, and the square that would hold both violations sticks out of the circle everywhere except at its corner: the best both can manage at once is exactly 1.999998, which is the classical value and no violation at all. So a party maximally entangled with one other is correlated with everybody else exactly as a classical object would be. Nothing about the measurement or the apparatus was assumed; this follows from the state alone.

The strongest correlation the outer pair can show, against how good the pairs that went in were, for a direct pair and for chains of one and three swaps. The pairs are Werner states — a perfect singlet mixed with noise in proportion — and a swap multiplies the parameters, so a chain of n swaps gives p^(n+1) and the imperfections compound rather than adding. The horizontal line at 2 is where a correlation stops being stronger than any list of pre-agreed answers could produce. A direct pair needs p above 0.7071; A chain of 1 swap needs p above 0.8409; A chain of 3 swaps needs p above 0.9170 — each located by bisecting the drawn curve rather than by rearranging the algebra. That compounding is the whole difficulty of building a quantum network: three swaps need pairs of ninety-five per cent fidelity to deliver anything a Bell test would recognise, and a chain long enough to be useful needs either better pairs than anyone can make or a way of improving them in the middle.

The rate at which entangled pairs can be delivered over a given distance of optical fibre, for a direct link and for chains of two to sixteen segments joined by swaps — a logarithmic rate against a linear distance. A photon survives a fibre with a probability falling exponentially, with an attenuation length of about twenty-two kilometres at telecom wavelengths, so a direct link over a thousand kilometres delivers 1.8e-10 pairs a second — one every 174 years, from a source firing ten thousand million times a second. Breaking the link into eight segments divides the exponent by eight rather than the rate, and delivers 4.3e+6 a second instead: a gain of 2e+16. That is what entanglement swapping is for and it is the only thing that works, because amplifying the signal is forbidden — a quantum state cannot be copied, so the classical repeater that rescues an ordinary optical link is not available. The lines assume a memory at each node that holds a half-pair until its neighbour is ready; without one the segments must all succeed at once and the exponential comes straight back.

physicscheck asserts something about bell-correlation that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.