Illustrated Physics
physics, drawn
Series

Measurement — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

    The answer that was not there before

    Send a beam through an analyser and it splits in two. Send one half through a second analyser turned sideways, then through a third pointing the way the first did, and the property the first analyser removed has come back.

    part 1 · quantum
  2. A quarter of the time, an object is found without being touched. The outcomes of sending one photon into a balanced interferometer, with and without an opaque object in one arm, computed from the same amplitudes. With the arm clear, every photon leaves by the bright port and the dark port receives nothing. With the object in place, 50 per cent of photons are absorbed by it, 25 per cent reach the bright port and say nothing, and 25 per cent reach the dark port — which is impossible unless something is in the arm, and which happens with the object still sitting there unabsorbed and unlit. The photon that produced that click did not go through the blocked arm, because a photon that goes through a blocked arm is absorbed. So an object has been located by light that never met it. The price is that 50 per cent of attempts destroy the thing being looked for: an efficiency of 33 per cent, which the Zeno version of the apparatus takes as close to one as one likes.

    The measurement that never touched it

    A balanced interferometer sends every photon to one output and none at all to the other. Put an object in one arm and the empty port starts clicking — and a click there is caused by a photon that cannot have gone near the object, because a photon that goes near it is absorbed. The object has been found by light that never met it.

    part 2 · quantum
  3. 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.

    part 3 · quantum
  4. 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.

    part 4 · quantum

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