Large-N Limit and Typicality

Many-Copy Quantum States

To connect single-system weights with empirical frequencies, we consider $N$ identical copies of a quantum system prepared in the same pure state

where $\{\ket{i}\}$ is the pointer basis selected by decoherence. The composite state on the tensor product Hilbert space $\mathcal{H}^{\otimes N}$ is

This state encodes all possible sequences of outcomes across $N$ repetitions of an experiment, without introducing probabilities at the fundamental level. The structure of $\ket{\Psi_N}$ is purely combinatorial and geometric, determined entirely by the amplitudes $c_i$.

Frequency Operators and Empirical Statistics

For each outcome $i$, one may define a frequency operator $\hat{f}_i$ acting on $\mathcal{H}^{\otimes N}$ by

which measures the relative frequency of outcome $i$ across the $N$ copies. The expectation value of $\hat{f}_i$ in the state $\ket{\Psi_N}$ is

independently of $N$. However, this equality alone does not yet explain why observed frequencies concentrate around this value. That explanation arises from the geometry of high-dimensional Hilbert space.

Concentration of Measure Theorems

As $N$ becomes large, the state space $\mathcal{H}^{\otimes N}$ exhibits strong concentration-of-measure phenomena. The overwhelming majority of the norm of $\ket{\Psi_N}$ is supported on branches in which the empirical frequencies $f_i$ are extremely close to $|c_i|^2$. Deviations of order $\epsilon$ are exponentially suppressed in $N$:

for some positive constant $\alpha$. This result is a geometric statement about volume in Hilbert space, not a probabilistic postulate. Branches with highly atypical frequencies occupy an exponentially small fraction of the total measure defined by the squared norm.

Typical Outcomes Without Fundamental Probability

From the structural stability perspective, typicality replaces probability. In the large-$N$ limit, sequences of outcomes with frequencies matching the Born weights form a structurally stable set: small perturbations of the state or of the measurement process do not move the system out of this set. Atypical branches, while not forbidden, are structurally unstable and negligible in measure.

Thus, empirical frequencies arise without assuming fundamental randomness. The Born rule appears as a law of large numbers for Hilbert space geometry: almost all branches, in the sense of measure concentration, exhibit frequencies $f_i \approx |c_i|^2$. Observed statistical regularities are therefore consequences of typicality in high-dimensional state space, grounded in structural stability rather than primitive probability.