Record light cone

In plain terms

CSD holds that measurement outcomes are written into a hidden part of the arena called the fibre, an extra set of coordinates attached to every point of the visible state space. A sceptic should ask how robust the writing is, and whether a distant disturbance, some operation performed far from the apparatus, could reach in and rewrite the record, because if it could, records would be too fragile to ground anything.

The answer is a locality theorem aimed exactly at that worry. Take a record written into the fibre by a local interaction, take any disturbance confined to a region far away, with distance measured by the number of interaction links separating it from the apparatus, and let everything evolve. The theorem bounds how far the record cell can move by a quantity that decays factorially in that distance, the same light-cone suppression the Lieb-Robinson bound provides for ordinary observables.

Distant meddling doesn't slightly corrupt records. It fails to reach them, by a margin that shrinks factorially with the distance.

In CSD

This is where the record layer and the field track meet. The arena with its fibre attached carries record strokes, shifts of the fibre controlled by observables of the base, the visible part of the arena, and the Lieb-Robinson machinery transports its light cone through a bridge into that setting. Disturbances that don't overlap the recording interaction commute with record writing exactly, meaning the order in which they happen makes no difference, and distant disturbances steer the written cell by at most the cone bound.

The scope is stated where it's proved. The theorem covers the stroke shape of fibre activity, base-dependent shifts of the fibre, which is the corpus's own record mechanism, and it doesn't cover arbitrary base-coupled fibre velocities, which nothing in the record layer needs.

Mathematically

The theorem record_lightcone states that for a record stroke driven by an observable A and a disturbance outside distance d, the displacement of the recorded fibre cell after time t is bounded by the Lipschitz data of the stroke times 2 (2 norm(S) t)^d / d! times norm(A), the rigid part of the fibre rotation cancelling between the disturbed and undisturbed histories.

Companion results give exact commutation of disjoint kicks with record writing, and the arena-level cone the bridge transports.

RecordInfluence.lean turns the bound into a relation between the regions two contexts read: one can influence the other within n interacting periods exactly when the second region lies in the coupling graph's n-ball around the first, which is reflexive, grows with the period count, and composes with the periods adding. Two regions whose cones have not met have commuting observables, so their records can be assigned outcomes together; a disturbance in either region leaves the other's record untouched exactly, with no error term, because in whole periods the evolved observable stays inside the cone; and the joint law of two records written into the two channels of the fibre is the product law of the medium, so the medium adds neither correlation nor bias. The relation is the causal shape the assumed coupling graph defines, with no metric and no signature, and the factoring is about the medium rather than about the state.

Related entries

• Lieb-Robinson boundEven without relativity, influence in a quantum system with local interactions has a speed limit, and on CSD's finite field arena that limit is a theorem with the velocity written out.
• Record layerA measurement is over when the world itself carries which outcome happened, and the record layer is where that fact lives, on the arena and not in anyone's notebook.
• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.
• Interaction priceSwitching on an interaction between field modes makes local quantities spread at a rate proportional to the coupling, and the corpus proves the rate is exact, at most twice the coupling times the elapsed time, and at least a fixed fraction of it.