The entangled pair a list of answers can imitate

The correlation no instructions produce found the shape that separates quantum correlations from classical ones. Two particles sent from a common source to two analysers can be correlated in many ways by instructions they carry from the source — a list saying what to answer for each possible setting — and any such list keeps a particular combination of correlations, the CHSH quantity, at or below 2. A singlet reaches $2\sqrt2$. The excess is what Bell’s theorem is about, and since the 1970s it has been measured with increasing care and found every time.

The singlet is a pure state, and no real source makes one. A link between two that never met modelled the imperfection the simplest way: a singlet with weight $p$, mixed with complete noise — a random state of the two particles — with weight $1 - p$. The mixture, written $\rho = p\,|\psi^-\rangle\langle\psi^-| + (1-p)\,\mathbb 1/4$, is called a Werner state after the man who asked the question this essay is about. How much noise can the pair absorb before it stops being entangled, and how much before it stops violating a Bell inequality? The two answers are different, and the difference is not an inefficiency of the test.

Two thresholds on one line #

The first threshold is the entanglement itself. A two-particle state is unentangled, separable, if it can be made by handing out pairs whose two halves are in definite states of their own, chosen at random from some list. Testing that is hard in general, but for two two-state systems there is a complete test, the partial transpose: transpose the state’s matrix on one particle only, and the state is entangled if and only if the result has a negative eigenvalue. For a Werner state the partial transpose has a smallest eigenvalue of $(1 - 3p)/4$, negative exactly when $p > 1/3$. So the pair stays entangled until the noise is two-thirds of the mixture.

The second threshold is the CHSH violation. A singlet’s correlation between analysers set at an angle $\theta$ apart is $-\cos\theta$; noise contributes no correlation at all, so the Werner state’s correlation is $-p\cos\theta$, and the largest CHSH value is $2\sqrt2\,p$. That exceeds 2 only when $p > 1/\sqrt2 \approx 0.707$. Between $p = 1/3$ and $p = 0.707$ the state is entangled and the CHSH test does not see it.

The second figure shows where the two thresholds come from geometrically. A two-qubit state whose correlations are diagonal is fixed by three numbers, the correlations $t_1, t_2, t_3$ between the two particles’ spins along three axes, and the physical states fill a tetrahedron with the four maximally entangled states at its corners. Inside sits an octahedron of separable states, $|t_1| + |t_2| + |t_3| \le 1$. The states that violate CHSH form a different region: by a criterion the Horodecki family proved in 1995, those whose two largest correlations have squares summing to more than one. Werner’s states run along a line from the centre, which is complete noise, to the singlet’s corner. The line crosses the octahedron’s face at a third of the way and reaches the CHSH region at $1/\sqrt2$ of the way. The stretch between, almost forty per cent of the entangled part of the line, is entangled and invisible to the test.

A list of answers for an entangled pair #

The gap could have meant only that CHSH is a poor test, and that some cleverer inequality, with more settings or a different combination, would catch the rest. Werner showed in 1989 that for a large part of the gap no inequality can. He constructed a local model — a scheme in which each pair carries hidden instructions from the source and each analyser answers by consulting only its own setting and the instructions — that reproduces the statistics of the Werner state for every possible choice of measurement directions, for $p$ up to $1/2$.

The third figure runs a version of the model. Each pair leaves the source carrying a direction $\lambda$ chosen at random on a sphere. The first analyser, set along $\mathbf a$, answers $-1$ if $\lambda$ lies within ninety degrees of $\mathbf a$ and $+1$ otherwise. The second, set along $\mathbf b$, answers $+1$ with probability $(1 + \mathbf b\cdot\lambda)/2$, a coin weighted by how closely $\lambda$ lines up with its setting. Each analyser’s answers alone are an even mixture of $+1$ and $-1$, as the Werner state’s are. Their correlation, averaged over the random directions, is exactly $-\tfrac12\,\mathbf a\cdot\mathbf b = -\tfrac12\cos\theta$ — the Werner state’s correlation at $p = 1/2$ — for every pair of directions in space, not only for the coplanar settings drawn. The sampled dots fall on the quantum curve to within the scatter of forty thousand trials.

Nothing in this model is quantum, and nothing in it is entangled. It is a list of instructions of exactly the kind Bell’s theorem was built to rule out, and for this state Bell’s theorem does not rule it out, because the state’s correlations are too weak to exceed any bound such a list respects. Every experiment that measures one spin direction on each particle of a Werner state at $p = 1/2$ could, as far as its data can tell, have been a source of particles carrying directions.

The model was later extended. In 2006 Antonio Acín, Nicolas Gisin and Benjamin Toner connected the question to a constant from functional analysis, Grothendieck’s constant, and showed that local models exist for all projective measurements up to $p \approx 0.66$. Jonathan Barrett had shown in 2002 that even general measurements, not just measurements of spin along a direction, can be reproduced locally up to $p = 5/12$. Searches with more settings have since found Bell inequalities violated a little below $1/\sqrt2$, and local models a little above 0.66, and the exact value of $p$ at which a Werner state’s correlations stop being reproducible by instructions — somewhere near 0.68 to 0.70 — is not known.

Why noise helps a list of answers #

The model cannot be pushed to the perfect singlet, and seeing why shows what the noise does. A singlet gives perfectly opposite answers whenever the two analysers are set the same way, along any direction, and it does so while each answer alone is a fair coin. Bell’s original argument was that a list of answers cannot achieve perfect anticorrelation along every direction at once and still reproduce the cosine at intermediate angles; the disagreement that one run settles found that with three particles the conflict needs no statistics at all, only perfect correlations, which in its pure form is also what makes it fragile to noise. Werner’s model reaches only $-\tfrac12$ at zero angle, half the singlet’s perfect $-1$; its correlations are a scaled-down copy of the singlet’s, and the scaling is what lets them be faked. Noise dilutes every correlation by the same factor $p$, and a sufficiently diluted set of correlations — any correlations at all, provided they are weak enough — can be produced by instructions.

Acín, Gisin and Toner’s connection to Grothendieck’s constant makes that precise. For a fixed choice of measurements, a Bell inequality compares what a list of answers can achieve with what a quantum pair can achieve, and for correlations of the cosine type the largest possible ratio between the two, over every inequality anyone could write down, is a number Alexander Grothendieck bounded in 1953 in a paper about tensor products of normed spaces with no thought of physics: the constant $K_G$, here in three dimensions. Diluting a singlet by more than that ratio guarantees that no inequality of any kind is violated. The ratio is known to lie between about 1.44 and 1.52, which is where the uncertainty in the threshold comes from.

A third threshold, between the two #

There is a third threshold between the two, and it is exact. Schrödinger noticed in 1935 that a measurement on one particle of an entangled pair appears to steer the other into one of several states, depending on what was measured, and he called this steering. A local model of the Werner type can fake the correlations without either side’s particle being in any quantum state at all. A weaker kind of model fakes them with the second particle always in a genuine quantum state, chosen from a list at the source, and only the first carrying instructions. Howard Wiseman, Steve Jones and Andrew Doherty showed in 2007 that for Werner states this weaker model works for every projective measurement exactly when $p \le 1/2$, and fails above.

So the noisy singlet passes through three regimes as the noise is reduced. Above $p = 1/3$ it is entangled. Above $p = 1/2$ it is steerable: one side’s measurements demonstrably act on a quantum state at the other, a statement that can be checked by trusting one detector and not the other. Somewhere between 0.68 and 0.70 it becomes nonlocal in Bell’s sense, a statement that needs no trust in either detector. Each is strictly stronger than the one before, and the strength is paid for in purity. The steering threshold coincides with Werner’s own model, which is not a coincidence: his model is exactly of the kind that fails above one half for the steering test, and one half was where he could take it.

Entanglement that can be spent #

The entangled-but-local states are not useless. The fourth figure sets their CHSH value against what they can do in teleportation, the protocol in which a shared entangled pair and two classical bits transfer an unknown quantum state from one place to another. With a perfect singlet the transferred state is exact. With a Werner state its average fidelity is $(1 + p)/2$, and the best that can be done with no entanglement at all — measuring the state, sending a description and rebuilding it — is $2/3$, a limit set by the fact that the state that cannot be copied cannot be measured completely either. So the Werner state beats every classical method from $p = 1/3$, precisely where it becomes entangled. At $p = 0.6$ it teleports with fidelity 0.80 while its best CHSH value is 1.70, comfortably inside the local bound.

The noise in a Werner state is not an absence. A mixture can always be regarded as part of a larger pure state, and the $1 - p$ of noise is the trace of correlations the pair has shared with something else — the environment that scattered a photon, the stray light that a detector counted. What two have they cannot give a third found that entanglement is monogamous: the more a pair is entangled with the world, the less with itself. The Werner state is that trade frozen at one value, and the three thresholds measure how much of the pair’s correlation must remain its own for each kind of quantum behaviour to survive the sharing.

Teleportation uses the entanglement in a way a Bell test does not. A Bell test asks only for the statistics of a single measurement on each particle. Teleportation requires a joint measurement on the incoming state and one particle of the pair, followed by an operation on the other particle chosen by the result, a sequence that no local model of the single-measurement statistics needs to reproduce. The qubit that carries two bits found a different threshold for a different task: superdense coding with a Werner state beats a lone qubit only above $p = 0.748$. Each task draws on the entanglement differently, and none of the thresholds is the entanglement threshold except where the task is teleportation.

There are subtler ways to make the hidden nonlocality show. Sandu Popescu showed in 1995 that some Werner states of higher-dimensional systems, which admit local models for single measurements, violate a Bell inequality after a preliminary measurement on each side that filters out part of the noise. Carlos Palazuelos showed in 2012 that many copies of a local entangled state, measured together, can violate an inequality that one copy cannot. Nonlocality, unlike entanglement, is not a property a state either has or lacks; it depends on what is done with the state.

What it looks like on an optical table #

In the laboratory, $p$ is a number an experimenter reads directly. Polarisation-entangled photons are sent through two polarisers, one of which is rotated, and the rate at which both photons pass rises and falls with the angle between the polarisers. For a Werner state the contrast of those fringes, their visibility, is exactly $p$. A perfect source gives fringes that fall to zero; noise fills in the dips. The last figure draws three cases. At a visibility of 60 per cent the fringes are unmistakable, the photons are entangled, and no CHSH test on them can succeed.

This was the practical situation of the first Bell experiments. Visibility above 71 per cent is demanding when detectors are inefficient and sources emit pairs in many directions, and several early experiments produced correlations that were plainly quantum in origin and plainly interference between entangled photons, yet could not exceed the classical bound. The outcomes identical photons refuse found a similar boundary for two-photon interference, where a dip deeper than half proves the light is not classical, and a shallower dip proves nothing. Fringes are evidence of coherence; only a violated inequality is evidence that no list of answers made them.

Where the clean picture is cleaner than a laboratory #

The Werner state is a model of noise, and a very symmetric one: complete noise, equally likely to be any state of the two particles. Real noise has structure. Photons lose polarisation in a particular way in a fibre, detectors register dark counts, and a source emits accidental pairs that are uncorrelated rather than randomly polarised. Each produces a different mixture, with different thresholds for entanglement and for violation, and for many realistic mixtures the gap between the two is narrower or wider than Werner’s. The tetrahedron of the second figure is the space of only those states whose correlations are diagonal; a general two-qubit state has fifteen parameters, and the separable and CHSH-violating regions in that space have shapes the cross-section only hints at.

The local models also assume the measurements are chosen independently of the hidden directions the pairs carry and that the two analysers do not communicate. Those are the assumptions every Bell argument rests on, and the loophole-free experiments of 2015 were designed to enforce them. Here they are granted to the model rather than tested.

What a correlation curve does not record #

The figures show correlations and probabilities, averaged over many pairs, and the local model reproduces them in that sense: it gives the same statistics. It does not reproduce the state. A Werner state at $p = 1/2$ and a source of particles carrying random directions give identical results for every measurement of one spin direction on each particle, and different results for teleportation, for entanglement swapping and for joint measurements on several pairs. A figure of a correlation cannot show that difference, because the difference is in experiments the correlation does not describe. The questions that can be asked together located the line between what can and cannot be measured jointly; the local model succeeds precisely because a Bell test asks only questions that can be answered one particle at a time.

Still open: where the line falls #

The value of $p$ at which a two-qubit Werner state stops having a local model for projective measurements lies between about 0.68 and 0.70, and nobody knows where. The question has been turned into a problem about Grothendieck’s constant in three dimensions, whose exact value is unknown, and attacked with computer searches over local models and over Bell inequalities with hundreds of settings, each search narrowing the interval without closing it. For general measurements the interval is wider. That the boundary between quantum and classical correlations, for the simplest noisy state of the simplest entangled system, is not yet located is a measure of how different that boundary is from the boundary of entanglement.

The habit worth carrying away is to keep entanglement and nonlocality apart. A singlet diluted with noise to weight $p$ is entangled for every $p$ above 1/3 and violates CHSH only above $1/\sqrt2$; up to $p = 1/2$ — and by later work about 0.66 — a list of instructions carried by each pair reproduces every pair of single measurements on it, though the same pair teleports with fidelity $(1 + p)/2$, better than any classical scheme. A state can hold a resource that no test of correlations will reveal, and whether a test reveals it depends on what is asked of the state.