Classification of bipartite observables III: simplicity, mixtures and quantum error correction

The paradigm of a simple unstable theory in the sense of Shelah is the theory of the random graph, the Fraïssé limit of the finite graphs. Its counterpart in quantum error correction is the Pauli group on countably many qubits, taken modulo phases and with commutation as its relation: the Fraïssé limit of the finite alternating spaces over the two-element field, whose form computes the syndromes of stabilizer codes. We show that simplicity theory reads as a theory of error correction: Clifford encoding circuits implement the homogeneity of the limit, a logical error is an undetectable error whose type over the stabilizer group and the logical operators forks over the stabilizer group, and the independence theorem of Kim and Pillay is the gluing of syndromes. Both structures define bipartite observables, and for both, mixing preparations produces the tree property of the second kind. The two paradigms reach it by different routes, dictated by their probes. The probes of the Pauli observable are the Pauli channels; the indicator of each syndrome sector is the difference of the responses of two uniform mixtures, and actual mixtures witness the tree. The probes of the Rado observable see only marginals, conditions given by actual mixtures are compatible on disjoint blocks, and the tree lives among idealized preparations.