FRAME PHASE RETRIEVABILITY AND STATE DISTINGUISHABILITY OF QUANTUM CHANNELS

This survey explores the intersection of frame theory and quantum information, specifically focusing on the concept of phase retrieval. In quantum mechanics, pure states are represented by unit vectors up to a global phase, which corresponds to rank-one operators. The authors investigate when a quantum channel preserves enough information to uniquely identify all pure input states, a property termed phase retrievability. The approach uses the rank-one lift to map phaseless frame measurements to linear functionals of the lifted state, allowing the authors to analyze channel injectivity through the geometry of operator-valued frames and the kernel of the channel's adjoint.

The paper establishes that a channel is phase retrievable if and only if it is injective on the set of pure states. The authors provide several criteria for this property, including a secant criterion that relates the channel's kernel to the set of normalized rank-two operators. They demonstrate that while full informational completeness (tomography) implies pure-state injectivity, the converse is not true. The authors also distinguish between phase retrievability, perfect one-shot discrimination, and exact quantum error correction, showing that these tasks are governed by distinct operator spaces (the observable range, the noise operator system, and the Choi support).

Using representation theory, the authors analyze twirling channels, where the channel acts as an average over a group representation. In this structured setting, the commutant of the representation determines the channel's properties. The authors provide exact formulas for five operational indices: the independence number, the zero-error capacity, the orthogonality index, the subspace orthogonality index, and the correctability index. They show that for multiplicity-free representations, the phase-retrievability of a subspace reduces to the informational completeness of a POVM on the subspace coordinates.