Preparation region

In plain terms

No laboratory procedure controls a system completely. Cool an atom, align a polariser, fire the same laser pulse, and what you fix is everything your instruments can see, while the world keeps the rest. So preparing the same state a thousand times can't mean putting the world in the same complete condition a thousand times, and the preparation region is the honest version of that phrase, and it's where quantum statistics begin.

The preparation region is the set of all points of the arena, the space of complete world-states, that are compatible with what the procedure fixed. Each run of the experiment is one point drawn from that region, so the statistics of many runs are the statistics of the region, and how often an outcome occurs is how much of the region leads to it.

Once the matter is put this way, quantum probability stops being mysterious in kind. It's the same ignorance over a region of possibilities that classical statistical mechanics runs on when it describes a gas by its temperature rather than by the position of every molecule. The work is in proving that the numbers come out right, and that's where the theorems about measures and volumes earn their keep.

In CSD

The preparation region is a primitive of the corpus's first formal layer, a measurable subset of the arena with positive volume under the arena's natural measure, the Liouville measure. Repeated trials draw from it independently, each draw governed by the arena's measure restricted to the region and rescaled so the region as a whole has total weight one, and every frequency theorem in the corpus is stated relative to it, so the weight of an outcome is the volume of the part of the region that leads to that outcome, divided by the volume of the whole region.

The refinement to pure states carries its own honesty note. When the measure is passed to the continuous setting, the set of arena points lying over a single quantum state has zero volume, so restricting the ambient measure to it isn't defined. Pure preparations therefore carry a posited law for trials on that set rather than one derived from the ambient measure, and the corpus states this wherever the posit is used.

Mathematically

prepMeasure conditions the Liouville measure on the preparation region, repeated trials are i.i.d. draws from it, and the LF1 main theorem sends empirical frequencies to volume weights almost surely.

The conditional-measure formula and the weight-as-volume-ratio identity, the weight of an outcome region O being the Liouville volume of O intersected with the preparation region divided by the volume of the preparation region, are the exported infrastructure the higher layers consume.

Related entries

• Outcome regionAn outcome is a region, and the apparatus alone draws its boundary, so the preparation decides where the trajectory lands and doesn't get to draw the map.
• Epistemic measureCondition on what a preparation actually fixes, and the ignorance left over is a measure, and that measure is a theorem rather than a stipulation, as the arena's own volume disintegrates into exactly it.
• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.
• Typicality volumeAlmost every way the experiment could have started gives the quantum statistics, and that isn't an assumption about probability, as it's a measured volume.