Born volume ratio

In plain terms

Showing that probability equals volume for one contrived example proves little, as the claim earns its keep only at full strength, in every dimension, for every state, for every outcome, with no fine print. The Born volume ratio is the full-strength version, and it's the theorem that turns the volume story from an illustration into a derivation.

Take a quantum state in any finite dimension, that is, a system with any finite number of possible outcomes. Its Born weight for a given outcome, the number the textbook computes by squaring the size of that outcome's contribution to the state, equals the volume of an explicitly described region of the space of quantum states divided by the volume of the whole space. The volume is measured with the Fubini- Study measure, the natural notion of size on that space, the one that respects its symmetry. The equality is exact rather than approximate, and it holds for every state, including states in which some outcomes have zero weight.

The regions are barycentric cells over the probability simplex. The simplex is the set of all possible lists of outcome probabilities, a triangle when there are three outcomes, a tetrahedron when there are four, and so on, and a state's own list of probabilities is a point of it, so that point cuts the simplex into one cell per outcome. Nobody carved the cells to fit, as their sizes come out of the geometry.

In CSD

This is the derived side of the carving line in every finite dimension, the theorem that turned the volume story from realisation into derivation. The route runs through the moment map, which reads the Born weights off the geometry, and the Duistermaat-Heckman law, which says that pushing the symmetric measure on the state space forward through the moment map gives the flat law on the simplex, so the volume of a cell is an honest integral over the simplex, computed once and equal to the Born weight.

Downstream, the frequency companion turns the identity into an empirical statement, as independent draws from the Fubini-Study measure give observed frequencies that converge jointly to the full vector of Born weights.

Mathematically

fs_born_volume_ratio_N and its apex variant, in the unconditional form fs_born_volume_ratio_N_uncond (CsdLean4/LF4/BornRegionUncond.lean), say the Born weight of each of the N coordinates equals the Fubini-Study volume ratio of the corresponding barycentric region, unconditional in the preparation.

The proof composes the Gaussian representation of the Fubini-Study measure, the product-index bridge to exponential laws, and the joint Dirichlet pushforward, and the earlier positivity hypothesis was retired by a per-cell dichotomy on closed simplex faces.

Related entries

• 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.
• 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.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• LF1 main theoremIn a deterministic world with repeated preparations, the frequencies you observe converge to volume ratios with probability one, and the theorem is machine-checked with no randomness anywhere in the dynamics.
• 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.