Illustrated Physics
physics, drawn
Concept

Entanglement — where it appears

A state of two or more particles that assigns no state to any one of them separately, so their outcomes correlate more strongly than pre-agreed answers allow. Two particles beat the classical bound by 41 per cent; three contradict it with certainty in one run.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

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.

The correlation no instructions can produce

A pair of gloves in two boxes agrees perfectly and needs no physics, because the answers were settled at packing. What no packing can imitate is the shape that appears as the two analysers are turned relative to each other, and the shape is a number — 2.828 where every list of pre-agreed answers is stuck at 2.

quantum · Entanglement Four predictions no list of answers can keep. The four measurements a three-particle GHZ state predicts with certainty, and what the best possible list of pre-agreed answers does with them. Each row is a choice of which quantity to measure on each of the three particles; the quantum value is not an average but a certainty, so a single run of any row has a determined outcome. Every list of answers assigns a value to X and to Y at each particle, which is sixty-four lists in all, and the enumeration finds that 32 of them get three of the four right and 32 get one. None gets four, and none can: multiplying the four left-hand sides together gives every X and every Y twice, so the product is +1 for any list whatever, while the product of the four quantum values is −1. The column on the right is one such list, which agrees with the first three rows and is then forced into the opposite of the fourth. That is the whole argument, and it needs no inequality, no average and no repetition: measure the first three rows on three copies, and the fourth is predicted with certainty and comes out the other way.

The disagreement that one run settles

Bell's argument is a statistical one — a correlation of 2.828 where a pre-agreed list of answers is stuck at 2, dug out of hundreds of coincidences. Add a third particle and the argument stops being about how often. Three measurements predict a fourth with certainty, every list of answers that gets the three right gets the fourth exactly backwards, and one run of the experiment is enough.

quantum · Entanglement What is lost is exactly what is recorded. Fringe visibility against the distinguishability of the record left in the environment, for six couplings between the interferometer and a marker. The points lie on the quarter circle V² + D² = 1, computed here to a part in 10¹² — the visibility from the output probabilities with the marker traced out, and the distinguishability from the overlap of the marker's two states, with nothing shared between the two calculations. The relation is the quantitative form of complementarity, and it is stronger than the usual statement: interference is not lost because something was disturbed, and not lost only when a measurement is made. It is lost exactly to the extent that the environment could in principle say which way the particle went, whether or not anybody looks at the environment. That is why decoherence is a matter of correlation rather than of disturbance: the coherence has not been destroyed but relocated, into a correlation between the particle and something else.

Where the interference goes

A superposition does not stop being a superposition when something interacts with it. What happens is that the coherence moves — out of the system and into a correlation between the system and its surroundings — and the interference disappears from any measurement made on the system alone. For a dust grain in air the move takes 10⁻²⁸ seconds, which is why nothing large has ever been seen in two places.

quantum · Measurement 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.

The state that cannot be copied

Every measurement in this collection disturbs what it measures, and the obvious way round that is to make a spare first. It cannot be done, and the reason is not a practical difficulty or a limit on how good an apparatus can be: a copier is a linear machine, so fixing what it does to two states fixes what it does to their superpositions, and what it then does is not a copy.

quantum · Measurement 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.

What two have they cannot give a third

Entanglement will not be shared. A pair that violates a Bell inequality is correlated with everything else exactly as a classical object would be, and the trade is exact enough to be drawn: two CHSH values must fit inside a circle of radius 2√2.

quantum · Entanglement 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.

A link between two that never met

Take two entangled pairs sharing no particle, measure the two inner particles jointly, and the two outer ones — which have never interacted, never been in the same place, and have no history in common — are entangled. Nothing travelled between them. What has to travel is two classical bits saying which of four results occurred, and until those arrive the outer parties see nothing at all.

quantum · Entanglement How much a block knows about the rest. The entanglement between a block of a one-dimensional chain and everything outside it, against how long the block is, for three states of the same number of particles. The straight line is a randomly chosen state, whose entanglement is the block's length times the logarithm of two — a volume law, and what almost every state in Hilbert space does. The flat curve is the ground state of a chain with a gap: it saturates, varying by less than a twentieth of a per cent from a block of eight to one of forty, because the boundary of a one-dimensional block is two points however long the block is. The middle curve is the ground state of a gapless chain, which grows as the logarithm of the size with a coefficient measured here as 0.333 against the third that conformal field theory gives. Both ground states are enormously less entangled than a random state, and that is not a detail about chains: it is why a ground state can be written down at all.

The corner of Hilbert space that is ever visited

Monogamy between three parties says how much of a correlation a pair may hold. Read across a boundary in a many-body system it says something much stronger: the entanglement between a region and the rest scales with the boundary rather than the volume, for the ground state of anything with local interactions. That is why such a state can be written down at all — and why almost every state in Hilbert space is one that nothing ever prepares.

quantum · Entanglement Four operations on one half, four states of the pair. The sender holds one half of an entangled pair prepared in the state |Φ+⟩ = (|00⟩ + |11⟩)/√2, the receiver the other. To send two bits she applies one of four operations to her half alone — do nothing, flip it (X), shift its phase (Z), or both — and sends it. The table gives, computed from the state vectors, the probability that the receiver's joint measurement of both halves in the Bell basis returns each of the four Bell states: one on the diagonal, zero everywhere else. Four perfectly distinguishable outcomes from one transmitted qubit is two bits. Sent without the pair, a qubit can carry at most one.

The qubit that carries two bits

A single quantum two-state system, sent from one person to another, can carry at most one bit of message — a theorem, not an engineering limit. If the two already share half each of an entangled pair, the sender can do one of four things to her half alone, send just that half, and the receiver can tell which of the four she did. Two bits arrive on one qubit. Nothing is free: the shared pair had to be delivered, and counted honestly the rate is still a bit per qubit. What entanglement buys is the freedom to send half the message before the message exists, and it also buys privacy, since the qubit in transit carries nothing at all on its own.

quantum · Entanglement A round uncertainty sheared into a thin one. The uncertainty in the direction of 100 atoms' combined spin, seen along the spin, drawn out to one standard deviation in each direction. Left: the atoms each in an independent superposition — a disk of variance N/4 = 25 in every transverse direction, so a rotation shows only once it moves the spin by √N/2 = 5.0 units. Right: after one-axis twisting, which turns the spin about the vertical axis at a rate proportional to its height, so that the upper half of the disk is pushed one way and the lower half the other (arrows). The disk becomes an ellipse with variances 1.22 and 535, its long axis tilted 12.2° from the horizontal. Its narrow direction, turned to lie along the rotation to be measured, is 12.0 dB quieter than the disk, allowing for the 11.8% the twisting costs in the length of the spin.

The atoms that measure better together

A clock read with a hundred independent atoms locates its phase no better than the scatter of a hundred coin tosses allows, because each atom answers the final question by itself. Let the atoms interact for a moment before they are read and that scatter can be shaped: the round uncertainty of their combined spin is sheared into a thin ellipse, and a phase error invisible against the coin-toss noise becomes four standard deviations wide. The gain grows with the number of atoms, and it is real only if the detector can count atoms nearly one by one.

quantum · Measurement Four regimes along one line of mixed states. The weight p of the singlet in Werner's mixture of a singlet and complete noise, from 0 (pure noise) to 1 (a perfect singlet), divided by what is known about it. Below p = 1/3 the state is separable: it can be made by handing out unentangled pairs. Above 1/3 it is entangled, by the partial-transpose test. Yet up to at least p = 1/2 — Werner's explicit model of 1989 — and, by a later argument through Grothendieck's constant, up to about 0.66, a local hidden-variable model reproduces the outcomes of every pair of projective measurements on it, so no Bell experiment of that kind can reveal the entanglement. Above 1/√2 ≈ 0.707 the CHSH inequality is violated. Between about 0.66 and 0.71 (and, by more recent searches, between about 0.68 and 0.70) it is not known whether any Bell inequality is broken.

The entangled pair a list of answers can imitate

Entanglement and the violation of a Bell inequality are usually spoken of as the same thing, and they are not. Mix a perfect singlet with random noise and the pair stays entangled until the noise is two-thirds of the mixture. It stops violating the CHSH inequality much sooner, when the noise passes three-tenths. In between lies a family of genuinely entangled states whose every measurement can be reproduced by particles carrying instructions fixed in advance — Reinhard Werner wrote the instructions down in 1989. The entanglement in those states is real and can be spent, on teleportation for one thing. It simply cannot be seen by asking two distant analysers what they find.

quantum · Entanglement

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementSuperpositionBell inequalityCorrelationDecoherenceQuantum stateDensity matrixLocalityNo-cloningHidden variablesSpinArea law

All concepts