Luders rule

In plain terms

Measure a quantum system and two things happen, as an outcome appears and the state afterwards is different from the state before. Textbooks handle this with a separate postulate, the update rule, bolted on beside the equation of motion, so you get two rules and no explanation of why the second exists or when it applies.

There's also a subtlety inside the update rule itself, and Luders is the one who caught it. Sometimes several distinct states share the same measured value, and such a measurement is called degenerate, so a detector that reports only which region a particle landed in, without resolving anything finer, is an everyday example. For a degenerate measurement the naive update, jumping to one particular state with the measured value, is wrong. The right update projects onto the whole block of states that share the value and disturbs nothing within the block, and if you get this wrong repeated measurements give inconsistent answers.

Von Neumann's original rule of 1932 got the degenerate case wrong. Luders fixed it in 1951, and the fix went largely unremarked for years, a correction to the foundations of measurement theory published quietly and absorbed silently.

In CSD

In CSD the update isn't a postulate at all. Records and the state update live on the same pointer arena, the space of states enlarged to include the pointer of the measuring device, and the same construction produces them. The record selects which region the trajectory realises, the Born weights come out of the volumes of the regions, and the state after measurement is the block projection, so one mechanism recovers the two textbook rules as its two faces.

The corpus treats the degenerate case as the general one and not as a footnote. Physics forces that ordering, as real detectors are almost always degenerate and report which region a particle landed in without resolving the states within the region. A companion result sharpens the point, as no fixed calibration at the level of individual states reproduces the update for every preparation, so the block form isn't optional.

The result carries a condition of its own, which should travel with it. It applies to outcomes whose weight isn't too small, and it says nothing about outcomes below that threshold.

Mathematically

On the pointer arena, the record-conditioned marginal reproduces the Born weights of the preparation, and the post-measurement state is the Luders block projection. The conditioning carries a smallness condition that should travel with the statement, as the outcome's weight must exceed the width of the corridor the smooth witness leaves unrecorded, so the result says nothing about outcomes weighted below it. Degenerate block measurements are the case treated, and rank-one projective measurements are the specialisation in which every block is a single ray.

Two companion results complete the picture. A swap-based apparatus reproduces the Luders marginal exactly when calibrated, and, as the sharper negative, no fixed ray- level calibration works for every preparation, so calibration must be context- dependent, and that negative is what forces the block treatment.

The name

Gerhart Luders (1920-1995) was born in Hamburg, trained in the Gottingen orbit, and held a chair at Gottingen from 1960. The update rule comes from a 1951 paper correcting von Neumann, in effect a graduate student's observation that the great man's measurement postulate mishandled repeated eigenvalues.

He's at least equally remembered for the CPT theorem, proved independently around 1954, which says that charge conjugation, parity and time reversal, composed, form a symmetry of every local relativistic field theory. Both results are structural, as they're about what quantum theories must do rather than what any particular one does.

Related entries

• CollapseIn CSD the quantum state never jumps, and that's a theorem about volume-preserving maps rather than a preference among interpretations.
• Measurement flowA deterministic, volume-preserving flow carries out a measurement, and the volumes of its pointer blocks are the Born weights, for every preparation.
• Pointer basisA measuring apparatus doesn't record what you ask it to, as it records what it can keep, and what it can keep is decided by its interaction with the surroundings and not by the experimenter.
• Naimark dilationEvery noisy, imperfect measurement made in a real laboratory is an ideal textbook measurement carried out on a larger system, and that single fact is what lets CSD's derivation of the Born weights cover actual experiments rather than idealised ones.
• Measurement contextA measurement isn't a question put to the quantum state, as it's a piece of apparatus, and the apparatus brings its own way of dividing the world into outcomes.