Afham

We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, Bures and purified distances) onto several sets of interest. These include projecting bipartite PSD matrices onto the set of PSD matrices with a fixed marginal, and projecting ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases given by projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of POVMs. We introduce 'prior-channel decompositions' of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi--Jamiołkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel--state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the 'pretty good measurement' is the fidelity projection of an ensemble onto the set of POVMs, and that the Petz recovery map is the projection of a CP map—constructed from a prior state and a forward channel—onto the set of reverse quantum channels, which recasts the well-known identification of the Petz map with quantum Bayes’ rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer--Spekkens quantum state over time [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].