Theorem: Conditional reduction and Born weights

Equation (eq:collapse-SDE) has a global norm-preserving solution. Its ensemble density solves (eq:collapse-generator).

Suppose the vectors

are pairwise distinct. Then the populations $p_a(s)=\text{abs}{⟨a,{\psi }_s⟩}^2$ converge almost surely to a vertex of the probability simplex. The probability of the vertex $a$ is $p_a(0)$.

If some ${\mathbf{\ell }}_a$ coincide, only reduction to their indistinguishable groups is guaranteed.

Proof. Itô's rule gives zero drift and zero stochastic increment for $\text{norm}{{\psi }_s}^2$. The coefficients are smooth and bounded on the unit sphere. Local finite-dimensional existence, norm preservation, and compactness of that sphere give a global solution.

For $P_s=|{\psi }_s⟩⟨{\psi }_s|$, Itô's rule gives

The nonlinear mean terms cancel from the drift. The stochastic integrands are bounded, so their means vanish. Taking expectations proves the ensemble equation.

The populations satisfy

They are bounded martingales and hence converge almost surely. Let

Since $\mathrm{d}{m}_{\alpha }=2{\sum }_{\beta }{C}_{\alpha \beta }\mathrm{d}{W}_{\beta }$, Itô's rule gives

Therefore

The integral is finite almost surely.

The convergent populations make $C(s)$ converge. If the limiting $V$ were positive, the limiting $\mathrm{Tr}(C^2)$ would be positive, contradicting integrability. Hence the limiting population law is supported on a set of identical vectors ${\mathbf{\ell }}_a$. With pairwise distinct vectors, that set is a singleton. Finally, $\mathbb{E}{p}_a(\infty )=p_a(0)$ by bounded martingale convergence, which gives the stated weights.