Projective characterization of higher-order quantum transformations

Quantum 10, 1978 (2026).

https://doi.org/10.22331/q-2026-01-21-1978

Transformations of transformations, also called higher-order transformations, is a natural concept in information processing, which has recently attracted significant interest in the study of quantum causal relations. In this work, a framework for characterizing higher-order quantum transformations which relies on the use of superoperator projectors is presented. More precisely, working with projectors in the Choi-Jamiołkowski picture is shown to provide a handy way of defining the characterization constraints on any class of higher-order transformations. The algebraic properties of these projectors are furthermore shown to obey rules similar to $textit{multiplicative additive linear logic (MALL)}$, providing an intuitive way of comparing any two classes through their projectors. The main novelty of this work is the introduction to the algebra of the ‘prec’ connector. It is used for the characterization of maps that are no signaling from input to output or the other way around. This allows to assess the possible signaling structure of any transformation characterized within the projective framework. The properties of the prec are moreover shown to yield a normal form for projective expressions. This hints towards a general way to compare different classes of higher-order transformations.