We present an infinite number of construction schemes for quantum structures, including unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on the type structure of biunitary connections, 2-categorical structures which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.We present an infinite number of construction schemes for quantum structures, including unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on the type structure of biunitary connections, 2-categorical structures which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.

Leibniz international proceedings in informaticsJan 1, 20172 citations

Quantum Constructions

Mark A. Reed

ScienceOct 8, 19937 citations

Comment on "On the emergence of preferred structures in quantum theory" by Soulas, Franzmann, and Di Biagio

Ovidiu Cristinel Stoica

This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian $\mathsf{H}$ and the unit vector $|ψ\rangle$ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from $\mathsf{H}$ and $|ψ\rangle$ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between $|ψ(t)\rangle$ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian $\mathsf{H}$ and the unit vector $|ψ\rangle$ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from $\mathsf{H}$ and $|ψ\rangle$ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between $|ψ(t)\rangle$ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.

quant-pharXiv:2603.07674Mar 10, 20260 citations

Quantum Structures

L. A. Lugiato, A. Gatti, K.I. Petsas

European Quantum Electronics Conference (EQEC)Jan 1, 19980 citations

This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian

\mathsf{H}

and the unit vector

|ψ\rangle

only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from

\mathsf{H}

and

|ψ\rangle

either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between

|ψ(t)\rangle

and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.