Entanglement: Correlation Without Communication

Two entangled particles produce correlations that no ordinary physical model can reproduce: their results agree more tightly than any arrangement made in advance could manage. They also cannot carry a message from one end to the other. Both halves of that sentence are theorems, not opinions. Holding them together — the correlation is real, the channel does not exist — is the entire conceptual content of quantum entanglement.

Fourth in a series on spacetime, light and information: CERN, wormholes, gravitational lensing, entanglement, and ER=EPR.

Two detectors, two lists, and no message

Start with the instinct, because it is a good one. Two particles sit a kilometre apart. Measure one, and the other’s outcome is fixed instantly, with no delay anyone has ever found. That looks like a mechanism.

The reading is reasonable. Everything else in physics that links two distant places is a mechanism — a field, a wave, a wire — and mechanisms carry things. Einstein had a version of the same discomfort, and wrote a famous paper saying so.

So here is the apparatus. A source in the middle makes pairs of particles in a maximally entangled state — the strongest version of the link. One flies left to Alice’s lab, the other right to Bob’s. Each lab has a detector with a dial that selects the measurement; the detector answers +1 or −1.

They run this thousands of times. Alice ends up with a long column of +1s and −1s, Bob with his own. Three facts about those columns settle the question.

One: each column, taken on its own, is a fair coin. Whichever setting Alice picks, +1 comes up half the time and −1 half the time — no bias, no pattern, nothing to read. The same for Bob’s column.

Two: nothing Bob does changes Alice’s column. He can turn his dial to any setting, measure before her, after her, or walk out and not measure at all; her data stream is statistically identical in every case. That is not a limitation of today’s equipment but a theorem of quantum theory — the no-signalling constraints — which say that Alice’s odds of getting +1, ignoring Bob entirely, do not depend on Bob’s setting, and his do not depend on hers.

That second fact is what closes the door. To send a message you need something you can vary at one end and detect at the other. Bob has knobs; none of them move any quantity Alice can measure. There is no channel to modulate, and no better hardware will produce one.

There is one tempting escape here. When Alice measures and gets +1, she knows immediately what Bob will get. That is true, and it is not communication. To send Bob a single bit she would have to make her result be the value she wants to send, and she cannot, because her result is a fair coin. Knowing something the instant it happens is not the same as choosing it.

Three: the correlation is real, and it is not in either column. It is in the pairing — which of Alice’s entries sits next to which of Bob’s. Line the columns up row by row, with both dials on the same setting, and the entries are opposite every single time: where she has +1 he has −1, without exception. Neither column shows that alone.

And seeing the pairing means holding both lists at once. One of them has to travel: down a fibre, in an email, on a plane. That is ordinary classical communication, and it obeys the speed of light. The correlation is perfect and immediate; the evidence of it arrives at walking pace.

Panels 1 and 2 are everything either party can see alone: an even mix of outcomes, unchanged by whatever the far detector is set to — the no-signalling constraint, from Brunner et al. (2014). The correlation appears only in panel 3, after both records have crossed a classical channel.

What Einstein, Podolsky and Rosen actually argued

The 1935 paper is called “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, and the title is the argument.

Einstein, Podolsky and Rosen proposed a test for calling something real: if you can predict the value of a quantity with certainty, without disturbing the system that carries it, then something real corresponds to that quantity.

Apply it to a pair. A measurement on Alice’s particle predicts, with certainty, a quantity on Bob’s — and depending on which measurement you choose, you can predict either of two different quantities on Bob’s. Those two are non-commuting: measuring one scrambles the other, and quantum mechanics holds that they cannot both have sharp values at the same moment. Yet nobody went near Bob’s particle. By the EPR criterion, both should be real.

Their conclusion was a fork, and they left it open: either the wavefunction is an incomplete description — there is more in the particle than the theory writes down — or two non-commuting quantities cannot both be real.

That is a demand for extra machinery hidden inside the particles, not a claim that a measurement telegraphs across space: EPR would not give up locality. The popular caricature has Einstein believing in instantaneous influence at a distance; he was arguing the reverse — no influence, so the agreement must have been arranged beforehand.

Could the particles have agreed on a plan before they left?

That is the EPR proposal in plain words, and it is a good one. Suppose each particle leaves the source carrying an instruction sheet saying what to answer for every setting the dial might be turned to: at setting A answer +1, at setting B answer −1, and so on. The two sheets were written together at the source, which is why the answers agree later; nothing passes between the particles afterwards.

Physicists call the instruction sheet a local hidden variable. Hidden, because we never see it. Local, because each side consults only its own copy and its own setting.

This sort of explanation is ordinary and usually correct. Pack a pair of gloves into two boxes, ship one to Rome and one to Tokyo, and open the Rome box: you know instantly what is in Tokyo, and nothing travelled. But a glove has a definite handedness whether or not anyone looks; whether a particle has a definite answer before it is asked is the actual question.

Bell’s contribution, in 1964, was to make that question answerable by experiment. He showed that if such a plan exists, the correlations it can produce are limited: there is a ceiling on how well the two sides can agree, once you compare several combinations of dial settings rather than just one. Quantum mechanics predicts correlations above that ceiling, so no such plan can reproduce it. That is Bell’s theorem: local hidden variables are ruled out.

Clauser, Horne, Shimony and Holt turned the theorem into a number you can measure. Give each side two dial settings instead of one, run the pairs, and for each of the four combinations record how strongly the outcomes agree. Combine the four into a single quantity, S.

Three values of S carry the whole argument:

S ≤ 2. Everything a world of prior agreements can produce. Not just the obvious instruction sheets — every possible one.
S = 2√2 ≈ 2.828. Where quantum mechanics stops: on a maximally entangled pair it reaches 2.828 and goes no further. That ceiling is Tsirelson’s bound.
Anything in between. A measured value between 2 and 2.828 describes a world that made no prior arrangements.

There is a curiosity above the ceiling. Correlations stronger than 2.828 that still send no message can be written down, and as pure mathematics they are consistent. Nature does not produce them: quantum mechanics is more restrained than no-signalling alone would force it to be.

The green curve is the quantum prediction as one detector is rotated against the other, S = 3cos φ − cos 3φ; for photon polarisation, φ is twice the analyser angle. It peaks at 2.828 and never rises above it. Bounds from Brunner et al. (2014); measured point from Hensen et al., Nature 526, 682 (2015).

The two ways out, and how 2015 shut them

A Bell test rules out every prior-agreement model at once, but only if the experiment is clean. Every test before 2015 left a determined believer an exit, and there were two.

The detection loophole. Detectors miss things. If most pairs are never registered, the ones that are need not be a fair sample. A prior-agreement model can exploit that: let the instruction sheets also say when to be detected, and the detected subset shows S above 2 while the full set stays under 2. Taking the detected events to be representative is the fair-sampling assumption — long an assumption, not a result.

The locality loophole. If the two stations sit close together, or take too long to choose a setting and read out a result, a signal at or below the speed of light has time to carry Alice’s setting to Bob’s detector before his answer is fixed. His outcome could then depend on her dial by ordinary means.

Each loophole had been closed on its own before; closing both in the same run was the hard part, and it took an experiment widely separated, tightly timed and painfully slow.

At Delft, Hensen and colleagues used the electron spins of two nitrogen-vacancy centres in diamond — the name describes the defect, a nitrogen atom next to a gap where a carbon atom should be. The spin trapped there behaves as a single quantum bit, held in place by the crystal. Two of those, in two labs, 1,280 m apart.

The pairs were made by heralding. Each spin emits a photon; the two photons travel to a third station between the labs, where a joint measurement on them — a Bell-state measurement — announces which attempts succeeded. Only heralded attempts count as trials, so failures never enter the data.

Now the timing, the entire reason for the 1,280 m. That separation buys a 4.27 µs window: light leaving one lab cannot reach the other inside it, and everything at each station has to finish within it. Choosing the setting took 160 ns, reading out the spin 3.7 µs — they fit, with room left over. Neither lab’s choice could have reached the other in time, unless something outran light.

The detection loophole went with the readout: spin readout here catches essentially every event, so there is no unrepresentative subset to hide inside, and fair sampling stops being an assumption.

The price was speed. The whole run came to 245 trials. The result was S = 2.42 ± 0.20 — above the 2 that any world of prior agreements allows, below the 2.828 that quantum mechanics permits. Local realism was rejected with p = 0.039: the chance that a world of prior agreements throws up a result at least this extreme is about 4 in 100. A real violation carrying modest statistical weight.

Two photon experiments traded distance for statistics. The Vienna group reported a violation with probability at most 3.74 × 10⁻³¹ under local realism — a decimal point followed by thirty zeros before the first digit — an 11.5 standard deviation effect. The NIST-led group in Boulder reported p = 5.9 × 10⁻⁹, or 2.3 × 10⁻⁷ once setting predictability is allowed for: the dial settings come from random number generators, and a generator that is even slightly guessable gives a prior-agreement model a little room back. The weaker figure is what survives.

Treat this as established, in the strong sense of the word. Three independent groups, two different physical platforms, both loopholes shut in the same run, all within a single year.

The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger, “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”.

Teleportation, and the two bits it cannot do without

The cleanest refutation of the faster-than-light reading is the one protocol that genuinely moves a quantum state from one place to another.

Alice holds a qubit in some unknown state — the thing to be moved. She also holds one half of an entangled pair distributed beforehand; Bob holds the other half.

She then measures her two particles together, the unknown qubit and her half of the pair. The measurement is joint: it does not ask what either particle is on its own, but which of four possible relationships the two stand in. Alice gets one of the four.

Here is where the speed limit walks in. Bob’s half is now one correction away from being the original state, but which of four corrections he needs depends on which outcome Alice got. Naming one of four things takes two yes/no answers — two bits. Alice sends those two bits down an ordinary classical wire.

Until they arrive, Bob’s particle is not partly useful or awkward to use. It is useless: every measurement he can make on it returns exactly the statistics it would have returned before Alice touched anything — the no-signalling theorem again. He cannot even tell that the protocol has begun.

Teleportation therefore runs at the speed of the message. The entanglement was placed in advance and delivers nothing by itself; the two classical bits do the delivering, and they travel like everything else.

The same trick applied to half of an entangled pair is entanglement swapping, and it leaves two systems entangled that never interacted. It is how Delft built its spin pairs 1.3 km apart, and the heralding signal that told the labs a pair had been made was an ordinary classical message at ordinary speed.

On “a contraction of the spaces”

There is a natural next thought. If two things stay perfectly in step across a kilometre, perhaps the kilometre is not really there for them. Perhaps entanglement contracts the space between its ends, and that is why no signal is needed.

As a description of the experiments, that does not hold. Nothing about the 1,280 m between the Delft labs was any shorter for the spins. The distance was the instrument: the whole test was designed around it, and the 4.27 µs window is what made the result mean anything.

But the instinct that entanglement and geometry belong to the same subject is not naive. There is a serious research programme built on precisely that idea.

Maldacena and Susskind proposed ER=EPR: that two maximally entangled black holes are joined through their interiors by an Einstein-Rosen bridge — a wormhole. On that reading, entanglement and wormhole are two descriptions of one object, and the connection between two entangled systems is geometry. It is close to the instinct, reached from general relativity rather than from intuition.

Label it clearly: speculative. It is a quantum-gravity conjecture with no experimental test, and it sits in a different epistemic category from everything above it in this article.

And note what its authors put in their own opening paragraph: EPR correlations cannot be used to send information faster than light, and Einstein-Rosen bridges do not let a signal pass from one side to the other. Even the geometric version declines to be a channel. The next article takes ER=EPR on its own terms.

What to take away

Entanglement is nonlocal in a precise, tested sense: no model in which each outcome depends only on its own local setting and on a variable fixed at the source can reproduce the data. It is also local: no operation on one half changes anything observable about the other.

The wrong intuition is a natural one, and it is worth saying why rather than only saying that it is wrong. A correlation that holds instantly across a kilometre looks like a mechanism, and mechanisms carry things. Almost everywhere else in physics, that rule is sound. Entanglement is where it stops: you cannot steer the correlation, and you cannot read it from one end. You can only measure it twice, in two places, and compare afterwards — and the comparing travels at the speed of light.

Three numbers hold the argument. 2 is what a world of prior agreements allows. 2.828 is the ceiling quantum mechanics sets on itself. 2.42 ± 0.20 is what came out of a diamond in Delft.