Greatest Quantum Channels with Prescribed Maximal Information

Two quantum channels can have identical information geometry while allowing different physical simulations. This paper asks whether a complete RLD information profile has a greatest realization: a channel from which every other realization of that profile can be produced by deterministic postprocessing. Equality of right logarithmic derivative (RLD) Fisher information for every faithful input, tangent and finite quantum reference determines a labelled von Neumann algebra, a normalized channel and a faithful normal state. The same equivalence relation is obtained from complete maximal f-divergences. Every finite-dimensional invariant has a greatest representative of minimum possible effective output dimension, without requiring the invariant state to be tracial. For separable trace-class outputs with a faithful normal tracial invariant state, an exact greatest member exists precisely when the algebra is a countable product of finite matrix algebras. Arbitrarily accurate greatest members within the same exact profile class exist precisely when the algebra is injective. Existence of a single sequence approximating every fixed target already implies existence of a sequence whose diamond-norm error is uniform over all targets. Spectral pinching and finite-dimensional Haar averaging connect these operational requirements to established operator-algebraic approximation theory. The paper gives diffuse binary examples, an explicit free-group class with a finite quantitative obstruction, polynomial-time exact profile comparison and polynomial-bit faithful witnesses for rational finite channels, and finite probe identification with conditioned stability. A fixed full-support family shows that capacity optimized over an exact profile class can be discontinuous even while every physical channel in the family is antidegradable and has zero quantum capacity. The infinite-output classification requires the stated tracial hypothesis. It supplies neither an effective general filtration nor an experimental convergence rate. The atomic/injective distinction, reverse-test and sufficiency theory, amenability machinery and finite identification methods are credited to their primary sources. RLD-profile equality is distinct from the SLD, Umegaki and Holevo comparison problems studied in Complete information orders of quantum channels: equivalence and undecidability.