Projective characterization of higher-order quantum transformations

TL;DR

This work develops a projective (superoperator–projector) framework to characterize higher-order quantum transformations, translating type-theoretic constructions into constraints on CJ operators. By mapping base types to operator-system state structures and defining type connectives as algebraic projector operations (bar, tensor, and transform), the authors recover the known type theory and gain a Boolean–lattice view with a new one-way signaling connector, prec. The algebra of projectors provides normal forms that enable direct comparison of higher-order maps and clarifies signaling structures, showing in particular that quantum combs and quantum networks are isomorphic under this projector formalism. The approach also generalizes to alternative base state structures (e.g., biased quantum theories) and delivers a systematic framework to distinguish no-signaling, one-way, and two-way signaling maps, while highlighting when ICO (indefinite causal order) can or cannot arise. Overall, the projector formalism offers a unifying, computable route to analyze causal and signaling properties of complex higher-order quantum processes, with implications for the interpretation and realization of quantum networks, combs, and beyond.$

Abstract

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-Jamiolkowski 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 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.

Figures (12)

• Figure 1: Picturing types graphically. In the following, the notion of a type like $A$ will be subsumed by the one of a state structure $\mathscr{A}_{}$, but the diagrammatic notation will remain the same for simplicity.
• Figure 2: Picturing the characterizing equations of an effect state structure, Proposition \ref{['theo:det_fctal']}. Fig. \ref{['fig:Tr_A_notA']} is the graphical representation of the defining relation \ref{['eq:fctal_def']}. Fig. \ref{['diag:A_notA']} is a diagrammatic representation of how the Hilbert space splits between the operator system $\mathscr{A}_{}$ (pink) and $\overline{\mathscr{A}}{}$ (green). (In the diagram, the regular fonts $A$ and $\overline{A}$ have been used instead of script-style because of software limitations; 'Im' means the image of a linear map). Together they span the full space (whole line), yet they only intersect at identity (central dot).
• Figure 3: Quasi-orthogonality condition \ref{['eq:QO']} is equivalent to the factorization \ref{['eq:fctal_indep']} of the inner product. Graphically, the element of a state structure proportional to $\mathds{1}$ is by convention represented by the symbol /// (this is to single it out from arbitrary elements, which are represented by a half-circle). This diagram should then be read as $\forall V \in \mathscr{A}_{}, \forall N \in \overline{\mathscr{A}}{},\, \mathrm{Tr}_{}\left[ N \cdot V \right] = \mathrm{Tr}_{}\left[ \frac{\mathds{1}}{c_A} \cdot V \right] \mathrm{Tr}_{}\left[ N\cdot \frac{\mathds{1}}{c_{\overline{A}}} \right]$. The diagrammatic translation of quasi-orthogonality between two state structures is thus the possibility of 'cutting' the wire between them; this disconnection conveys the idea that these two state structures cannot influence each other.
• Figure 4: Diagrammatic representation of the image of the tensor product of projectors, as in Def. \ref{['prop:tensor']}. The full wheel represents $\mathcal{L}\left( \mathcal{H}^{A}\otimes \mathcal{H}^{B} \right)$ while its parts represent its tensor factors, Eq. \ref{['eq:decomposition_II']}. (Like in Fig. \ref{['diag:A_notA']}, 'Im' means 'Image' and e.g. $D$ is a shortcut for $\mathrm{Im}\{\mathcal{D}\} = \text{Span}\left\{ \mathds{1} \right\}$). By convention, the projectors acting on $\mathcal{L}\left( \mathcal{H}^{A} \right)$ are always on the left-hand side of tensor products and vice-versa for $B$, so that subscripts can be omitted. Note that the intersections are well defined; for example the line '$A \otimes D$', which represents $\mathrm{Im}\{\mathcal{P}^{}_{A} \otimes \mathcal{D}_B\}$ is indeed the intersection $A \otimes B \cap A \otimes \overline{B}$.
• Figure 5: Picturing the characterizing equations of a transformation between state structures, Proposition \ref{['theo:det_map']}. Fig. \ref{['fig:A_to_B']} is the graphical representation of the defining relation \ref{['eq:A_trasnformsinto_B']}. Fig. \ref{['diag:A_to_B']} is a diagrammatic representation of how the Hilbert space splits between the span of the state structure of transformations, $\mathscr{A}_{} \rightarrow \mathscr{B}_{}$ (blue), and the span of the deterministic functionals on it, $\mathscr{A}_{}\otimes \overline{\mathscr{B}}{}$ (yellow). As in Fig. \ref{['diag:A_notA']}, they span the full space together (i.e., they cover the whole disk), yet they only intersect at the identity (central dot). Notice how the transformation does not contain the boundaries $\Im{\mathcal{D}_A \otimes \overline{\mathcal{P}}^{}_{B}}$ and $\Im{\mathcal{P}^{}_{A}\otimes \mathcal{D}_B}$ as they belong to $\mathscr{A}_{}\otimes \overline{\mathscr{B}}{}$ already.

Theorems & Definitions (31)

• Definition 1: Operator system ChoiEffros1977.
• Definition 2: Projector on operator system
• Definition 3: State structure
• Definition 4: Resolution of a state structure
• Proposition 1: Functional
• Definition 5: No signaling (bipartite) composition
• Definition 6: Structure-preserving map
• Proposition 2: Transformation between state structures
• Lemma 1
• proof