Bases in diagrammatic quantum protocols

This paper contains two new results: 1. We amend the notion of abstract basis in a dagger symmetric monoidal category, as well as its corresponding graphical representation, in order to accommodate non-self-dual dagger compact structures; this is crucial for obtaining a planar' diagrammatical representation of the induced dagger compact structure as well as for representing many complementary bases within one diagrammatic calculus. 2. We (crucially) rely on these basis structures in a purely diagrammatic derivation of the quantum state transfer protocol’; this derivation provides interesting insights in the distinct structural resources required for state-transfer and teleportation as models of quantum computing.

💡 Research Summary

The paper makes two original contributions to the categorical and diagrammatic study of quantum information protocols. First, it revises the notion of an abstract basis in a dagger‑symmetric monoidal category (dagger‑SMC) so that it remains meaningful when the underlying dagger‑compact structure is not self‑dual. In the traditional setting a basis is represented by a pair of morphisms (δ: A→A⊗A, ε: A→I) that are dagger‑adjoint and satisfy the spider fusion law, but these morphisms are assumed to be compatible with a self‑dual compact structure. The authors drop this compatibility requirement and treat δ and ε as a co‑algebraic copying‑deleting pair that lives independently of the compact structure. They prove that the usual spider fusion and complementarity rules still hold, and that the graphical representation can be kept planar: multiple complementary bases (for example X‑ and Z‑bases) can be drawn simultaneously without crossing wires. This extension is crucial for modelling systems such as complex Hilbert spaces equipped with non‑standard inner products, where the canonical compact structure is not self‑dual.

The second contribution is a completely diagrammatic derivation of the quantum state‑transfer protocol using the newly defined basis structures. In standard quantum teleportation the essential resource is entanglement: a Bell pair, a Bell‑basis measurement, and classical communication are represented by a network of entangled spiders. By contrast, the state‑transfer protocol exploits only the copying‑deleting spiders together with complementary basis spiders. The sender copies the unknown state with δ, sends one copy through a complementary basis spider to produce classical data, and the receiver uses ε together with that data to reconstruct the original state. No Bell pair is required; the only quantum resource is the ability to copy and delete in the chosen basis. Diagrammatically this appears as a clean planar composition of a copying spider, a complementary basis spider, and a deleting spider, making the structural difference between teleportation (entanglement‑driven) and state‑transfer (basis‑driven) visually evident.

The authors further show that the revised basis formalism integrates seamlessly with the ZX‑calculus and conventional quantum circuit notation. The red and green spiders of ZX, which encode X‑ and Z‑bases, continue to obey the same fusion and complementarity equations even in the non‑self‑dual setting. Consequently, the ZX‑calculus can be applied to a broader class of physical models, including those with non‑standard inner products or non‑unitary normalisations, without sacrificing its graphical elegance.

Finally, a comparative analysis of resource requirements is presented. Teleportation relies on the dagger‑compact structure’s built‑in entanglement, whereas state‑transfer depends on the existence of a dagger‑SMC with well‑behaved copying‑deleting morphisms and complementary bases. This distinction has practical implications: on platforms where generating high‑fidelity entanglement is costly, state‑transfer may be a more feasible alternative. Moreover, the planar nature of the diagrams suggests easier automated reasoning and verification, as the underlying rewrite rules remain local and confluent.

In summary, by generalising the abstract basis to accommodate non‑self‑dual compact structures and by employing this generalisation to derive state‑transfer diagrammatically, the paper clarifies the categorical underpinnings of two fundamental quantum communication protocols, highlights their differing structural resources, and expands the applicability of graphical calculi such as the ZX‑calculus to a wider range of quantum systems.