Joint count law and correlations

The joint number law depends on $A_{12}$ only through its modulus. A local phase transformation changes its phase without changing any number-diagonal probability. Number counts therefore do not determine the complete joint state.

Figure fig:counts records what the generating function determines and what it does not. Orthogonal packets are independent only when $A_{12}=0$. Equal moduli with different phases share the number law and are separated only by a phase-sensitive observable.