Large-N Limit and Measure Concentration

Many-Copy Quantum States

Consider $N$ identical preparations of a quantum system in the same pure state

with $\{\ket{i}\}$ an orthonormal basis (e.g. the pointer basis selected by decoherence).

The joint state of $N$ copies is

This tensor product decomposes naturally into orthogonal subspaces labeled by occupation numbers

corresponding to sequences with $n_i$ occurrences of outcome $i$. Each subspace $\mathcal{H}_{\vec{n}}$ has dimension

Empirical Frequencies and Hilbert Space Statistics

Define empirical frequencies

The squared norm of the projection of $\ket{\Psi_N}$ onto $\mathcal{H}_{\vec{n}}$ is

Using Stirling's approximation for large $N$, this quantity behaves as

where $\vec{p}=(|c_1|^2,\dots,|c_k|^2)$ and

is the Kullback–Leibler divergence.

Thus, subspaces whose empirical frequencies deviate from $p_i=|c_i|^2$ are exponentially suppressed in Hilbert-space norm.

Concentration of Measure Theorems

The space $\mathcal{H}^{\otimes N}$ equipped with the natural (Haar-induced) measure exhibits strong concentration phenomena as $N\to\infty$. Standard concentration-of-measure results imply that, for any $\epsilon>0$,

for some constant $c>0$.

In words: almost all of the Hilbert-space measure of $\ket{\Psi_N}$ is concentrated in a thin shell of subspaces whose empirical frequencies $f_i$ are arbitrarily close to $|c_i|^2$.

This statement is purely geometric and measure-theoretic. No probabilistic postulate has been introduced. The result follows from the high-dimensional geometry of tensor-product Hilbert spaces.

Emergence of the Born Rule Without Probability

The large-$N$ limit thus enforces a unique, overwhelmingly dominant frequency structure:

Importantly, this convergence is not interpreted as a probabilistic law. Instead, it reflects the fact that all alternative frequency assignments occupy a vanishing fraction of Hilbert space.

Combined with Sections 3 and 4, the logic is as follows:

• Structural stability selects the squared-norm measure as the unique invariant measure on outcome subspaces (Appendix A, Lemmas A.1–A.2).
• Decoherence dynamically suppresses interference and stabilizes diagonal weights.
• Large-$N$ measure concentration renders all non-Born frequency patterns structurally negligible.

Therefore, the Born rule emerges as a geometric and dynamical necessity, not as an axiom or subjective probability assignment. Frequencies obey $|c_i|^2$ because any other assignment is unstable under both dynamics and measure concentration.

In this sense, quantum probabilities are not fundamental primitives but emergent features of high-dimensional structure and stability.