The disagreement that one run settles

The two-particle argument has an uncomfortable shape. It says that a list of pre-agreed answers cannot make a certain combination of correlations exceed 2, and that the quantum state gets 2.828. To see that, an experiment has to collect a few hundred coincidences and compare an estimate against an error bar, and the conclusion arrives as a statement about how often things happen.

That is not a weakness in the argument. It is a weakness in how convincing the argument feels, and there is a version with the weakness removed.

What is being measured #

Three particles are prepared in the state $(|000\rangle + |111\rangle)/\sqrt{2}$, and each is sent to a separate analyser. Each analyser can be set to measure one of two things, called $X$ and $Y$ — for a spin-half particle these are its components along two perpendicular directions in the plane at right angles to the axis the state was written in, and for a photon they are two settings of a polarising analyser, whose classical law is a statement about the direction of the shaking that the single-photon version turns into a probability. Each returns $+1$ or $-1$.

There are eight settings of the three analysers. Four of them have the property this argument turns on: the state predicts the product of the three outcomes with certainty.

$$XYY = -1,\qquad YXY = -1,\qquad YYX = -1,\qquad XXX = +1.$$

No averaging. If $XYY$ is measured, the three outcomes multiply to $-1$; every single time.

The other four settings — $XXY$, $XYX$, $YXX$ and $YYY$ — predict nothing at all: their products come out $+1$ and $-1$ equally often, and no run of them tells anybody anything. That the certainties come in exactly the right combination to contradict each other is the content of the state, and it is worth noticing that a state chosen at random would not do it. Half of the eight settings are certain, and the four that are have products multiplying to $-1$ rather than $+1$, which is the single arithmetical fact the entire argument rests on.

Each individual outcome, meanwhile, is a fair coin. Any one analyser, watched on its own for as long as anybody likes, sees $+1$ half the time whichever setting it uses, and so does any pair. Everything the state has to say is in the three-way agreement, and nothing at all is visible from fewer than three vantage points at once — which is why no signal from one analyser to another could carry it: there is nothing at either end to send.

What is being measured needs two facts about a single analyser, and both are ordinary. A measurement returns one of a countable set of answers, so each of the three particles gives a definite $\pm 1$; and asking a second question after the first destroys the settled value of the first, so no particle can be asked both X and Y. Neither fact is peculiar to entanglement, and both are needed before the argument below can start.

Why no list of answers survives #

Suppose each particle carries instructions: a value it will give if asked $X$ and a value it will give if asked $Y$, decided when the three were prepared and carried unchanged to wherever the analysers happen to be. Call them $x_1, y_1, x_2, y_2, x_3, y_3$, each $\pm 1$.

Then the four predictions become four equations:

$$x_1 y_2 y_3 = -1,\quad y_1 x_2 y_3 = -1,\quad y_1 y_2 x_3 = -1,\quad x_1 x_2 x_3 = +1.$$

Multiply all four left-hand sides together. Every $x_i$ appears twice and every $y_i$ appears twice, and any of these numbers squared is $+1$, so the product of the four left-hand sides is $+1$ — for any list of answers whatever, without knowing a single one of them.

Multiply the four right-hand sides together and the answer is $(-1)(-1)(-1)(+1) = -1$.

That is the whole proof. It has no inequality in it, no probability, no assumption about how often anything happens. The four statements the state makes cannot all be true of a list of numbers, because they imply $+1 = -1$.

The enumeration in the hero figure says the same thing by brute force, and adds one detail worth having: since the number of disagreements must be odd, a list gets either three of the four or exactly one. There is no list that gets two, and none that gets four.

How this differs from the two-particle case #

Turning that gap into a decision means combining several angles, because at any single angle the two curves are close enough that no achievable error bar separates them.

A number is not yet a measurement, though, and the difference between 2.828 and 2 has to be established against statistics that start out far wider than it.

The three-particle version replaces all of that with a table. The price is that three particles have to be entangled at once and held that way, which is much harder than entangling two; the gain is that the conclusion no longer has an error bar attached to it in principle.

The single run, stated carefully #

The claim one run settles it needs care, and the care is where the argument is at its most interesting.

A single run of a single row proves nothing: measuring $XYY$ and getting $-1$ is consistent with any of the thirty-two lists that reproduce that row. What one run settles is a conditional. Take a list of answers — any list — that reproduces the three rows containing two $Y$s. The algebra then forces its prediction for $XXX$, with no freedom left: it must be $-1$. Measure $XXX$ once, get $+1$, and that entire family of lists is dead.

And that family is not a small one. It is every list consistent with the three measurements already made. So a single measurement of the fourth row eliminates, in one stroke, every explanation by pre-agreed answers that agreed with everything seen so far.

The single run has to be stated carefully, because the claim is unusually strong. It is not that the quantum prediction is more likely — it is that the two theories predict opposite outcomes for a measurement with only two possible results. Local hidden variables require the product of the three X measurements to be $+1$; quantum mechanics requires $-1$. One run distinguishes them, which no two-particle Bell test can do.

It is worth adding what is not being claimed about the particles. Nothing here says they are identical in the sense the exclusion principle uses, and nothing depends on their being far apart in any particular way. The three could be metres apart or millimetres; what the argument uses is only that each analyser’s setting is chosen freely and that each returns one of two values.

Twenty-five years between the two arguments #

Bell’s paper is from 1964. The three-particle argument is from 1989, and once seen it takes four lines.

The delay is worth a paragraph because it is not an accident of attention. Bell’s inequality was constructed for the simplest entangled system there is, and simplicity was the right instinct: a two-particle state was the only thing anybody could imagine making, and it took until 1972 for even that to be done convincingly. Nobody had a reason to ask what three particles would do, because three particles were not available and the two-particle answer was already the answer.

What changed was a shift in what the question was for. Through the 1970s the point of a Bell test was to test — to check experimentally whether nature obeys the inequality. By the late 1980s that had been settled to everybody’s satisfaction and the interesting question became how to state the conclusion most sharply, which is a question about arguments rather than about apparatus. Daniel Greenberger, Michael Horne and Anton Zeilinger asked it of a state nobody could yet build, and David Mermin’s exposition the following year reduced it to something that fits on a postcard.

The experiment followed a decade later, once three-photon states could be made, and confirmed what by then nobody doubted. That is the ordinary order of events for an argument of this kind: the theorem is the discovery, and the measurement is the receipt.

What has to be assumed #

Three things, and it is worth being explicit because the argument’s whole value is that it assumes so little.

There is also a quantum-mechanical fact behind why the four rows cannot be measured on one triple, and it is not an inconvenience of apparatus: X and Y on the same particle do not commute, so sharpness in one has to be paid for in the other. Three of the eight settings are certain and the other four are maximally uncertain for the same reason.

That each particle’s answers exist before it is asked. This is realism, and it is the assumption the argument destroys.

That a particle’s answer does not depend on what is asked of the others. This is locality. Both are needed; a theory that gives up locality can reproduce everything.

That the settings could have been chosen otherwise. If the choice of which row to measure were itself determined by whatever prepared the particles, no conclusion follows. Nothing here or anywhere else rules that out, and it is the reason the loophole has a name of its own.

What has to be assumed is the usual short list, and it is worth being explicit that the list has not got shorter. The analysers must be set independently of the source and of each other; the detected pairs must be representative of those emitted; and the settings must be spacelike separated from the outcomes. Every one of these has been the subject of a loophole and every one has been closed, but not by this argument — the three-particle version sharpens the logic and leaves the experimental requirements exactly where they were.

The same argument with more particles #

The three-particle case is the first member of a family, and the family has a shape.

The line’s slope is the useful part. Each additional particle multiplies the gap by $\sqrt{2}$, so the disagreement between the state and every possible list grows without bound. At two particles it is 41 per cent and statistical; at three it is a contradiction; at ten it is a factor of twenty-two.

The same growth is why such states are so hard to keep. A state whose predictions are exponentially far from any classical account is a state exponentially sensitive to anything that couples to it, and the two facts are the same fact seen from either end.

The argument extends to more particles and gets stronger as it goes. With four or more, the fraction of settings on which the two theories disagree rises, so the contradiction is not a special property of three — three is simply the smallest number for which a single run suffices. That is worth knowing because it rules out the suspicion that the effect is an artefact of some particular arrangement.

How far back the third assumption has been pushed #

The third assumption — that the settings could have been chosen otherwise — is the one that cannot be eliminated and can be made progressively harder to evade, and the effort spent on doing so is worth recording.

The first Bell experiments chose their settings with whatever was to hand, so a sceptic could suppose the choice and the particles shared a common cause. Aspect’s group in 1982 switched the analysers while the particles were in flight, which forces any such cause to act faster than light or to have been arranged before the run began. In 2015 three groups closed the last two loopholes at once — the settings chosen fast enough and far enough apart, and the detectors efficient enough that no assumption about the undetected events was needed.

What remains cannot be closed, only pushed backwards, and the pushing has become inventive. One experiment took its setting bits from the colours of photons arriving from two stars several hundred light years away, so any conspiracy fixing both the settings and the source must date from before that light was emitted. A later version used quasars, one of them at a redshift near four, which puts the required conspiracy some eight thousand million years in the past. Another recruited a hundred thousand people to generate the bits by pressing keys, on the grounds that human choice is at least a different kind of thing from a photodiode.

None of these is a proof, and none can be. What they establish is that the alternative to the theorem’s conclusion is a coordination between the source of the particles and the state of the distant universe, laid down before the solar system existed — which is not refuted, and is a considerably less comfortable position than the one it was invented to avoid.

The contradiction that needs no distance at all #

The GHZ argument removes the statistics and keeps the locality. There is a companion result that removes the locality too, and it is smaller.

Take two particles and nine observables arranged in a three-by-three square, chosen so that the three in each row can be measured together and the three in each column can be measured together. The quantum formalism then fixes the product of each row and of each column: five of those six products are $+1$ and one is $-1$.

Now try to write a number in each cell. Multiplying the six products together multiplies every cell’s value twice, so whatever numbers are written the answer must be $+1$. The operators say it is $-1$. There is no assignment.

That is Peres and Mermin’s square, and three things about it are worth naming. It uses no state — the contradiction is between the operators themselves, so it holds for every two-particle state there is. It uses no separation — both particles may sit in the same apparatus, and no assumption about signalling is made anywhere. And it is a nine-cell table with a sign in it.

What it destroys is slightly different from what GHZ destroys. It shows that no observable can be assigned a value independently of which commuting set it is measured alongside — a property called contextuality, of which Bell’s locality assumption is a special case where the context is “what somebody far away chose to measure”. So the impossibility is not fundamentally about distance. Distance is what makes the assumption compelling; the obstruction is there without it.

What it costs #

Three-particle entanglement is expensive. GHZ states of photons are made by post-selection: pairs are produced, mixed, and only those events in which one photon arrives at each of three detectors are kept. That is a real state and a real measurement, and the events that are thrown away are a loophole of their own unless the detection efficiency is high enough.

Perfect certainty is a limit, not a measurement. Every real apparatus has a visibility below one, so the four rows are not $\pm 1$ but something like $\pm 0.9$, and the contradiction becomes a statistical statement again — just a much sharper one. The 1999 and 2000 experiments reported agreement with the quantum prediction at a level no list of answers approaches, using thousands of triples rather than one.

The argument is about a particular state. It says nothing about entanglement in general and everything about this one, and the family of states for which so clean an argument exists is small.

What the state is good for besides an argument #

A GHZ state is not only a rhetorical device, and two of its uses depend on exactly the property the argument turns on.

Secret sharing. Three parties each hold one particle. Each measures $X$ or $Y$ at random and announces which, keeping the outcome. On the runs where the settings form one of the four rows, the three outcomes multiply to a known value — so any two of them together determine the third, and no one of them alone knows anything at all. That is a cryptographic primitive with a physical guarantee: a secret split three ways such that no proper subset can reconstruct it, enforced by the same algebra that makes the contradiction.

Error detection. The four rows are a set of measurements whose outcomes are fixed by the state. Measuring one of them therefore extracts no information about which state it is and does reveal whether something has gone wrong — a particle flipped, a phase acquired. That is the germ of a stabiliser code, and the operators in the four rows are precisely the stabilisers of the GHZ state.

And metrology. Three particles prepared this way accumulate phase three times as fast as three independent ones, so a measurement of a field with them beats the limit that independent particles set. The gain is the same $\sqrt{n}$ per particle that appears in the Mermin ratio, and it is bought with the same fragility.

Where the model stops #

Nothing here says what a measurement is. The four rows are predictions of a formalism; what happens when an analyser and a particle interact, and why one of the possible outcomes occurs, is untouched. The rung below this one states the same limitation from the other side.

The particles have been idealised as two-state objects. A real photon carries polarisation and also a frequency, a direction and a time of arrival, and every one of those is a wave property that has to be prepared identically for the three, or the correlation degrades before any measurement is made.

And the conclusion is negative. What has been shown is that a certain kind of explanation is unavailable. That is worth a great deal — it eliminates the whole class of theory most people reach for first — and it is not the same as an account of what is going on. Every interpretation on offer accepts the theorem and disagrees about what to give up.

The particles have been treated as separable objects with labels. They are three excitations of fields, and the tensor product that makes the argument possible is a structure of the formalism whose physical status is exactly what is in dispute.

And a tally is still not a fact. Every real experiment accumulates runs, so the “single run” of the argument is an idealisation about what the theories say rather than a description of what an experimenter does — the same double slit shows nothing after twenty arrivals and clear fringes after a thousand. What the three-particle version changes is the logic, not the statistics.

What the picture cannot show #

The hero figure is a table, and a table is an unusual thing for this collection to draw. It is drawn because the content is a logical impossibility rather than a quantity, and there is nothing to plot: the four rows have no continuous parameter between them, and the sixty-four lists are not ordered by anything. Sixty-four is small enough to enumerate on a page, which is the whole reason the argument can be made without the counting that turns enormous numbers of arrangements into a quantity anybody can print on a dial.

What no figure here can show is the state itself. Every drawing above is of outcomes and of what a classical account would have to say about them, and the object doing the work — a vector in an eight-dimensional space with no separate description of the three particles in it — has no picture. That is not a failure of draughtsmanship. A state with no factorisation into parts cannot be drawn as parts, and drawing it as parts is exactly the error the theorem refutes.

Where this ladder goes next #

Two rungs stand on entanglement. The first showed that the shape of a two-particle correlation is beyond any list of pre-agreed answers, by a measured 41 per cent. This one removes the statistics, and the removal took a third particle.

The habit worth carrying away is about where an argument’s strength lives. When a conclusion rests on a comparison of averages, ask whether the same conclusion has a version that rests on a single certain prediction — the second is not merely more convincing, it usually reveals a cleaner piece of structure underneath. Here that structure is an algebraic identity between operators, and it was invisible while the subject was about how often.

What is left on this ladder is what entanglement cannot be shared. Two particles maximally entangled with each other cannot be entangled with a third at all, and that constraint — monogamy — is what makes a shared key private and what a third party’s information about a correlated pair is bounded by.