A link between two that never met

What two have they cannot give a third establishes how entanglement may be distributed: monogamously, with an exact trade, so that two parties strongly correlated with each other are weakly correlated with everything else. It says nothing about how it may be moved.

The answer is stranger than the constraint. Entanglement can be created between two particles that have never met.

Take two pairs — one shared between Alice and a middle station, one shared between the middle station and Bob — each of them a correlation no list of pre-agreed answers could imitate. Alice’s particle and Bob’s have no history in common: they were made in different places, from different sources, and have never been within any distance of each other. Now measure the middle station’s two particles jointly, in the basis of the four Bell states, and Alice’s and Bob’s particles are left entangled.

Four outcomes, one of which happened. 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. — Fig. 1 The correlation between the two outer particles, against the angle between their analysers, for each of the four results the middle measurement can give. Every one leaves them maximally entangled — each curve reaches one and minus one — and the four differ in which correlation it is. Their average is the flat line, which is exactly zero at every angle.

What the middle measurement does

The mechanism is worth stating carefully because the language of “transferring” entanglement suggests a picture that is wrong.

Before the measurement, the four particles are in a product of two pair states. Written out in the basis of Bell states of the middle two, that product is a sum of four terms, and each term has a Bell state of the middle pair multiplied by a Bell state of the outer pair. That identity is pure algebra; it involves no dynamics and nothing has happened yet. Writing one state as a sum over another basis is the move every superposition argument in the subject turns on, and here it is doing all the work.

Measuring the middle two in the Bell basis selects one of those four terms. The outer pair is then in whichever Bell state accompanied it — maximally entangled, in a definite state, with no operation ever having been performed on either of the outer particles and no interaction between them of any kind.

Nothing moved. The correlation between Alice and Bob was not carried from one to the other; it was created by an operation performed on neither of them, out of correlations that each separately had with a third place. That is the sense in which entanglement is not a substance, which monogamy makes precise from the other direction: a fixed total, allocated rather than possessed, and the allocation changed by a measurement somewhere else.

Why it cannot be used to send anything

The hero figure’s flat line is the whole of the answer and it is worth being exact about it.

Each of the four outcomes leaves a maximally entangled pair, so each is as good as any other for anything Alice and Bob might want. But they are four different Bell states, with correlations that are shifted and reflected versions of one another, and the outcomes are equally likely.

So if Alice and Bob simply compare their data without knowing which outcome occurred, they are averaging over the four — and the average is zero at every angle. No correlation of any kind appears. Their records look like independent noise, exactly as they would if no pairs had ever existed.

The two classical bits that say which outcome occurred are therefore not a formality. They are what makes the correlation usable, they travel by an ordinary channel at no more than the speed of light, and without them there is nothing. Something has been created between two particles that never met, instantly, and nothing has been sent — which is the shape every correct statement about entanglement has, and which is why the state that cannot be copied and the impossibility of signalling keep turning out to be the same prohibition.

One side changes everything, and the other side cannot tell. 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. — Fig. 2 The general statement the flat line is an instance of: what one party’s measurement statistics look like as the other party changes their analyser. Nothing moves. The marginal distribution at either end is independent of everything done at the other, for every state and every measurement, and that independence is a theorem rather than a coincidence of the case drawn.

What the swapped pair had to give up

There is a bookkeeping question the operation raises, and monogamy answers it exactly.

Before the swap, Alice’s particle is maximally entangled with the middle station’s first particle. Afterwards it is maximally entangled with Bob’s. By the monogamy constraint a maximally entangled particle can have no entanglement at all with anything else — so Alice’s particle must have lost every trace of its correlation with the middle station, and it has.

Two pairs, and only one of them may cheat. 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. — Fig. 3 The constraint that forces it: the CHSH value one party shares with a second, against the value the same party shares with a third. Every state lies inside the quarter circle, so two simultaneous violations sharing a party are impossible. A swap moves a party from one corner of that region to the other, and the transit is instantaneous because it is a change in a description rather than a motion of anything.

That is a useful way to see what the middle measurement accomplishes. It does not transport a correlation along the chain; it moves the whole four-party state from one allowed point of the monogamy region to another, and the outer pair’s gain is the middle pairs’ loss, exactly. Nothing is conserved here except the constraint, which is a different kind of bookkeeping from anything in classical physics and is why the operation has no classical analogue at all. Classical correlation is promiscuous — a hundred parties can hold copies of one random bit and nothing objects — so there is nothing for a classical swap to move and nothing it would cost.

What a chain of them costs

The operation composes: swap twice and the link stretches across three segments. What it does to the quality of the link is the practical question.

What a swap costs. 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. — Fig. 4 The strongest correlation the outer pair can show, against how good the pairs that went in were, for a direct pair and chains of one and three swaps. A swap multiplies the parameters rather than adding the errors, so a chain of n swaps gives p raised to n+1. The threshold for beating any list of pre-agreed answers rises from 0.707 for a direct pair to 0.946 for three swaps.

A realistic pair is not a perfect singlet but a mixture of one with noise, and the natural parameter is how much singlet there is. Swapping two such pairs gives a pair whose parameter is the product of the two, which is the important structural fact: imperfections compound rather than accumulate, and a chain of eight segments at ninety per cent each delivers 0.9⁸ = 0.43, which violates nothing.

The thresholds in the figure are the practical consequence. A direct pair needs $p > 1/\sqrt2$ to show any correlation stronger than a classical list permits. One swap needs $2^{-1/4} = 0.841$ . Three swaps need 0.946, which is better than most laboratory sources deliver and much better than anything after a hundred kilometres of fibre.

So a chain cannot simply be made longer. Something has to improve the pairs in the middle, and that operation exists: entanglement purification takes several poor pairs and, by local operations and classical communication, produces fewer better ones. Purification and swapping together are a quantum repeater, and the arrangement was worked out in the 1990s precisely because swapping alone does not scale. Purification’s own limit is the monogamy constraint again: entanglement cannot be broadcast, so several poor pairs can be traded for one good one and never the reverse.

Why anybody wants it

Why the link has to be broken into pieces. 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. — Fig. 5 The rate at which entangled pairs can be delivered over a given distance of fibre, for a direct link and for chains of two to sixteen segments. A photon survives twenty-two kilometres with probability 1/e, so a direct link over a thousand kilometres delivers one pair every hundred and seventy years from a source firing ten thousand million times a second. Eight segments deliver four million a second.

An optical fibre attenuates, and for a classical signal that is a solved problem: amplify it. For a quantum state it is not, because amplifying means copying, and a quantum state cannot be copied. The no-cloning theorem is usually introduced as a security guarantee; it is also the single largest engineering obstacle in the subject, and this figure is what it costs.

Three copiers, and what each of them costs. How well three copying machines reproduce a qubit, against the state's angle from the pole. The first is a linear machine built to copy the two pole states perfectly; linearity then fixes what it does everywhere else, and on an equal superposition it produces an entangled pair whose overlap with the two copies wanted is exactly 0.500. That is the no-cloning theorem written as a number rather than as an argument: no adjustment is available, because the machine's behaviour on superpositions was decided the moment its behaviour on the basis was. The second measures in a fixed basis and prepares two copies of what it found, which is perfect at the poles and averages 0.6667 over the sphere — two thirds, exactly. The third is the best machine there is, and it manages 0.8333 on every state alike: five sixths, and not one. — Fig. 6 Why there is no amplifier: what a hypothetical copier would have to do, and the contradiction it runs into. A machine that copies two particular states correctly gets superpositions of them wrong, because copying is not a linear operation and quantum evolution is. One line of algebra, and it removes the whole classical repairing strategy from a quantum channel.

The exponential is brutal. Every twenty-two kilometres costs a factor of $e$ , so every fifty costs a decade, and a thousand kilometres costs nineteen decades. A source firing ten thousand million times a second delivers one pair every hundred and seventy years.

Breaking the link into segments divides the exponent. Each segment is an eighth as long, so each survives with $e^{-45/8}$ instead of $e^{-45}$ , and although the chain needs all eight to work, they need not work at the same moment — each node holds its half-pair in a memory until its neighbour is ready, and the swaps are performed when everything is in place. The rate improves by sixteen orders of magnitude.

That memory is the hard part and it is what the subject is currently about. A quantum memory has to hold a state for long enough — against an environment that is continuously becoming entangled with it, which is what decoherence is and which the monogamy constraint prices exactly — for a classical signal to travel to the neighbouring node and back — several milliseconds over a hundred kilometres — with a fidelity good enough that the swap is still worth doing. Trapped ions, rare-earth-doped crystals and nitrogen-vacancy centres in diamond have each demonstrated the pieces; a full repeater chain outperforming direct transmission over a useful distance has been demonstrated over tens of kilometres and not over hundreds.

What has actually been done

The experiments are worth listing because the claims in this area are often loose and the record is specific.

Swapping itself was demonstrated in 1998 with photon pairs from parametric down-conversion, and the outer pair’s Bell violation was measured. It has since been done between memories, between distant nodes, and between systems of different kinds — a photon entangled with an ion at one place and with a different ion at another, swapped to leave the two ions entangled.

Over distance, entanglement has been distributed between nodes separated by about a kilometre with matter memories, and over twelve hundred kilometres by satellite — which evades the fibre’s attenuation by going through space, where the loss is a much gentler inverse square rather than an exponential, and which is a different solution to the same problem — and which works because a beam spreading into empty space thins out as a geometry rather than as an absorption.

And a chain of two swaps joining three segments, with memories, was operated in 2021 over tens of kilometres. The rate was low. What it established is that the pieces work together, which was not obvious.

The same trick, from the other side: teleportation

Swapping has a sibling that is usually taught first, and setting the two beside each other makes what each does clearer.

In teleportation, Alice holds an unknown state and shares a pair with Bob. She measures her unknown particle jointly with her half of the pair, in the same Bell basis, gets one of four results, sends Bob two bits, and Bob applies one of four operations. Bob’s particle is then in the state Alice’s was.

Swapping is that operation with the unknown state replaced by half of another entangled pair. Alice’s measurement then teleports her half of the second pair to Bob — and since that half was entangled with something else, what arrives at Bob is entangled with that something else.

So they are one operation, and which name it gets depends on what was handed to it. That is worth knowing because the resource accounting is then shared: one shared pair plus two classical bits moves one qubit, whatever that qubit happens to be correlated with. The pair is consumed, the bits are consumed, and nothing is copied — which is the same statement as the no-cloning theorem in a form that says what is possible rather than what is not.

Complete measurements, perfect memories, and a channel left out

The Bell measurement is taken as complete. Distinguishing all four Bell states of two photons with linear optics is impossible — only two of the four can be separated, so half the attempts are discarded and the effective success probability of a photonic swap is at most a half before any other loss. Matter qubits do not have that restriction, which is one of the reasons repeaters are built around memories rather than around photons alone.

The noise is modelled as a Werner state. Real imperfections are not isotropic: a dephasing channel, a loss channel and a detector’s dark counts each degrade the pair differently, and a single parameter is a summary rather than a description. The composition law survives in shape and not in detail.

The rate calculation assumes memories with unlimited lifetime. A memory that decoheres in a time comparable with the classical signalling time changes the whole accounting — the optimum number of segments becomes finite rather than growing with distance, and it is set by the memory rather than by the fibre.

The Bell basis is assumed measurable at all. It is a set of four states none of which is a product, so the apparatus has to ask a question about two particles jointly and refuse to ask anything about either separately. That refusal is what a measurement in quantum mechanics is, and building one that asks the right joint question is most of the experimental difficulty.

And the figures ignore the classical channel entirely. Every swap requires two bits to be sent and acted on, every purification round requires more, and in a long chain the classical traffic and its latency become the limiting resource rather than the photons. A repeater is a communications system with a quantum layer, not a quantum system with wires attached.

Four curves of which only one ever happens

The outcomes figure draws four correlation curves and hides that only one of them happens. Before the middle measurement there is no fact about which; afterwards there is exactly one, and the other three curves describe nothing. A figure showing all four at once is a picture of a probability distribution over histories, drawn as though the histories coexisted, which is the standard difficulty with drawing anything quantum.

The quality figure draws a composition law and cannot show what a swap physically is. The operation is a joint measurement of two particles in a basis whose states are all entangled — an apparatus that asks which of those four correlated states the pair is in and refuses to ask anything about either particle separately. That refusal is the whole mechanism, and nothing in a plot of parameters carries it.

And the reach figure draws smooth rates, which are averages over an intensely stochastic process. What actually happens is that segments succeed at random times, nodes wait, memories decohere while waiting, and the delivered rate is the tail of a complicated distribution. The smooth curve is the right order of magnitude and is not what any run looks like.

What the experiment has to be careful about

There is a criticism of the early demonstrations that is worth knowing, because it is the same criticism every Bell experiment has had to answer and it arrives here in a new form.

A swap is only interesting if the outer pair’s entanglement was created by the middle measurement rather than arranged beforehand. In an experiment where the middle measurement is performed before the outer particles are measured, a sceptic can say the outer particles were prepared appropriately — which is the ordinary hidden-variable objection in a new place.

The decisive version of the experiment therefore performs the middle measurement after the outer particles have been measured and their results recorded. That sounds impossible and is not: the outer detections happen, the records are stored, and only then is the Bell measurement made on the middle pair. The outer records, sorted afterwards according to which of the four outcomes occurred, show a Bell violation in each subset.

The entanglement, in other words, is being assigned to particles that no longer exist, on the basis of a measurement made after they were absorbed. That experiment has been done — the delayed-choice entanglement swap, 2012 — and the result is what quantum mechanics predicts. It is the same shape as the experiment where a single run settles the disagreement: the interesting content is in what a local model would have to have arranged in advance, and the arrangement it would need keeps becoming more absurd.

What it does not show is anything about causation running backwards. The sorting is done by an experimenter holding all the records, and no sorting of data already in hand can send a signal or change what happened. The honest statement is the one the flat line makes: the correlation is a fact about the joint records, it has no timelike direction in it, and reading it as an influence from the later measurement to the earlier one is reading a structure as a story.

Still open: whether a network is a chain or something else

Everything here treats a quantum network as a line of segments, because a line is what a long link is. A network is not a line, and the questions it raises are not settled.

Entanglement in a network is a shared resource with an unusual structure. A graph of pairs can be converted into other graphs by local operations — swapping is the simplest such conversion — and which final configurations are reachable from which initial ones is a question in a subject with no complete answer. Routing is not the classical problem either: a path that is used is consumed, several paths can be combined to give a better link than any of them alone, and the resource cannot be stored indefinitely or copied for retransmission.

There is also a question about what such a network would be for. Key distribution works over a single link and does not need a network, and its security rests on the monogamy constraint rather than on any network property. Distributed quantum computing does, and needs fidelities far beyond anything demonstrated. A network of atomic clocks entangled with each other would measure time better than any of them alone, and is the application closest to being useful. Which of these justifies the infrastructure is a judgement rather than a calculation, and the honest position is that the engineering is ahead of the case for it.

The habit worth carrying away is the one the hero figure’s flat line is. When something appears to be created instantly at a distance, look for what has to be sent before it can be used. The entanglement is genuinely there the moment the middle measurement happens. The correlation is unusable, and indistinguishable from noise, until two bits arrive by an ordinary channel. Every apparent conflict between quantum mechanics and relativity dissolves at exactly that seam, and it dissolves the same way each time.

A link between two that never met