Illustrated Physics
physics, drawn
Concept

No-cloning — where it appears

The impossibility of building a machine that copies an arbitrary unknown quantum state, which follows in one line from the linearity of the evolution. It underlies the security of quantum key distribution and the monogamy of entanglement.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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

Named alongside it

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

EntanglementMeasurementSuperpositionBell inequalityCorrelationDensity matrixLocalityQuantum stateDecoherenceEavesdroppingFidelityHidden variables

All concepts