Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications

TL;DR

This paper derives closed-form formulas for projections related to fidelity and Bures distance, with applications in quantum information theory, including quantum channels, measurements, and state recovery maps.

Contribution

It introduces unified closed-form expressions for fidelity-based projections and a novel prior-channel decomposition of CP maps, extending the Choi isomorphism.

Findings

Closed-form fidelity projections for bipartite PSD matrices and ensembles.

A new prior-channel decomposition of CP maps generalizing the Choi isomorphism.

Recasting the pretty good measurement and Petz recovery map as fidelity projections.

Abstract

We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, the Bures and purified distances) onto several sets of interest. These include projections of bipartite positive semidefinite (PSD) matrices onto the set of PSD matrices with a given marginal, and projections of ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases corresponding to projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of measurements. 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-Jamiolkowski 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 - associated with a weighted ensemble - is the fidelity projection of the ensemble onto the set of measurements, and that the Petz recovery map - associated with a reference state and forward channel - is the projection of a CP map (constructed from the channel-state pair) onto the set of reverse quantum channels, thereby recasting 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 formalism [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].