Inevitability of the Petz transpose map

Abstract

Bayes' rule is central to classical statistical inference: given a prior and a stochastic model, it determines the corresponding retrodictive conditional probabilities. In quantum theory, however, the question of whether an equally general rule of retrodiction exists and, if so, whether it is uniquely determined remains nontrivial. Here we prove that a retrodiction assignment defined for all priors and channels, which takes each predicted output state back to its prior, must coincide with the Petz transpose map if it is involutive, respects sequential and parallel composition, and agrees with unitary inversion. The theorem holds for channels between finite-dimensional full matrix algebras, with the input and output spaces restricted to the supports of the corresponding reference states. In particular, affinity under probabilistic mixtures and the characteristic square-root dependence on the reference states follow from these requirements rather than being postulated. Without involutivity, the remaining requirements characterize a family of doubly rotated Petz maps. Our result thus explains why the Petz transpose map arises in seemingly different areas of quantum theory: it is the unique quantum analogue of Bayes' rule compatible with these consistency requirements.

The paper asks whether quantum Bayesian retrodiction is uniquely fixed by compositional consistency. A retrodiction assignment must map every channel $\Phi$ taking a prior $\pi$ to $\sigma=\Phi(\pi)$ to a channel taking $\sigma$ back to $\pi$; it need not invert $\Phi$ on other states. For channels between arbitrary finite-dimensional full matrix algebras, the main theorem proves that exactly one such assignment is involutive, reverses sequential composition, preserves parallel composition, and agrees with unitary inversion: the Petz transpose map, $\mathcal R[\Phi](X)=\sqrt\pi\,\Phi^\dagger(\sigma^{-1/2}X\sigma^{-1/2})\sqrt\pi$. Neither affinity under channel mixtures nor continuity is assumed. This resolves the stated axiomatic uniqueness question within this algebraic setting. The proof derives affinity on channels sharing a reference transition using controlled-swap circuits. Comparing an arbitrary assignment with Petz reversal establishes agreement at maximally mixed references. Compression and return channels then reduce the remaining freedom to reference-dependent unitary conjugations; involutivity and tensor compatibility eliminate these rotations. Without involutivity, the remaining axioms characterize doubly rotated Petz maps whose rotations satisfy unitary covariance and tensor compatibility. Requiring continuity in the prior reduces this freedom to $M_\rho=\operatorname{Ad}_{\rho^{it}}$, with one real parameter $t$ common to all systems; involutivity selects $t=0$. Singular reference states are covered only after restricting channels to the corresponding input and output supports, leaving reversal outside the predicted support unspecified. Extensions beyond full matrix algebras, including general infinite-dimensional $C^*$- and von Neumann-algebraic settings, remain open. The theorem characterizes retrodiction, not conditions for accurate error correction.