Probability-Preserving Reparametrizations of Quantum States and Effects

The empirical content of quantum theory is entirely encoded in the Born probability pairing between states and effects. In finite dimension, we study the following structural problem: which reparametrizations of states and effects preserve all Born probabilities? We consider bijective affine maps on the state space and effect space that leave the Born pairing invariant for all state–effect pairs, and provide a complete classification of all such transformations. We prove that every probability-preserving reparametrization is necessarily induced by a Jordan *-automorphism of the matrix algebra B(H). In finite dimension, this restricts the transformation to the form A→UAU† or A→UA⊤U†, for some unitary operator U. In particular, equivalence classes of states under probability-preserving reparametrizations coincide with unitary orbits and are completely characterized by the spectrum of the density operator. The result shows that unitary and antiunitary conjugations exhaust all probability-preserving affine reparametrizations of the state–effect framework.