Measurement flow

In plain terms

Saying that measurement is de-isolation dynamics is a picture, and the picture carries a debt, which is to exhibit actual dynamics, one deterministic map that respects the arena's volume, couples a system to a pointer, and sorts trajectories into outcome blocks of the right sizes. The measurement flow is the theorem that pays that debt.

The flow exists, it preserves volume, and it's provably not the identity, so it genuinely does something. After it runs, the arena decomposes into pointer blocks, one per outcome, which are regions of starting points that end with the pointer showing that outcome. The volume of each block is exactly the Born weight of its outcome, and if you repeat the preparation many times the fraction of runs landing in each block converges to that weight, with probability one.

There's no collapse anywhere. There's one deterministic map, run once per trial, plus ignorance of the starting point.

In CSD

This is the tier of the corpus that realises von Neumann's measurement scheme dynamically. In that scheme a measurement is an ordinary evolution of a larger system, obtained by adjoining a pointer to the system measured, an enlargement called a dilation. The dilation abstract measurement theory cites is here an actual flow on the enlarged arena, and that closes the gap between the volume results, which are purely kinematic, and the de-isolation picture. The outcome map for each individual starting point, saying which block a given microstate lands in, is delivered by the theorems that partition the arena into pointer blocks, and that discharges a debt the corpus had carried explicitly, the need for an outcome function assigning a result to each microstate.

Its boundary is equally explicit. The flow realises measurement dynamics at the level of the space of quantum states, and whether a physical de-isolation interaction generates a pointer whose readings carry the Born measures on the ontic fibre is the open question that remains. The earlier form of that question, deriving the fibre mechanism from a mixing flow, was retired as mis-specified, as the fibre entry explains.

Seen from the system alone, the measurement flow is a channel. Trace out the pointer after the flow has acted on a preparation that starts with the pointer ready, and the system's density operator is the de-isolation channel applied to its density operator before the measurement, so the operator is kept on the pointer basis and its coherences between different pointer readings are removed. No representative of the rays has to be chosen for this to hold, because the corpus supplies a canonical measurable one. The channel keeps entropy from decreasing, and the amount of classical information it transmits is exactly one bit per qubit, the pointer bit and nothing else.

Mathematically

measurement_flow_born_frequency (LF5 capstone, CsdLean4/LF5/Capstone.lean) says the de-isolation flow is Fubini-Study measure-preserving with pointer-block volumes equal to the Born weights, and empirical block frequencies converge almost surely, for every unit preparation. The earlier genericity hypothesis was retired by the unconditional volume engine.

The disjointness of the pointer blocks and the outcome map per microstate are separate, reusable theorems.

measurementFlow_traceRight_barycenter_unitSection (LF6/MeasurementFlowChannel.lean) says the environment marginal of the flowed barycentre is deisolationChannel applied to the system's barycentre, with Projectivization.unitSection as the representative, so no section hypothesis remains. deisolationChannel_apply_eq_pinch identifies the channel's action with the pinching of Thermo/SecondLaw.lean, deisolationChannel_holevoCapacity (LF6/DeisolationCapacity.lean) gives log 2 as the greatest element of its single-letter Holevo range, and measurementFlow_vonNeumannEntropy_le is the entropy statement.

Related entries

• De-isolationNothing collapses, as a measured system stops being isolated, and measurement is what de-isolation dynamics looks like from inside.
• Stinespring dilationEvery noisy quantum process is a reversible process on a larger system with part of that system ignored, and in CSD the larger system is the constraint surface itself, so every channel the programme uses is the environment's marginal of a volume- preserving flow.
• 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.
• Holevo boundA quantum system can be prepared in infinitely many distinct ways, yet no reader can extract more classical information from it than from a classical system of the same size, and the measurement channel of CSD is proved to pass exactly one bit per qubit.
• 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.