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

Abstract

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. Quantum error correction has a paradigm of its own, and we show that it is the counterpart of the random graph: the Pauli group on countably many qubits, taken modulo phases and with commutation as its relation, is the Fraïssé limit of the finite alternating spaces over the two-element field, and its form computes the syndromes of stabilizer codes. Seen through this correspondence, simplicity theory turns out to be a theory of error correction. Clifford encoding circuits implement the homogeneity of the Pauli group, forking recognizes the logical errors, and the independence theorem of Kim and Pillay is the gluing of syndromes. The two paradigms look alike as first-order structures, and they part company only when observables enter. Each defines a bipartite observable, and we prove that, once preparations are mixed, both acquire the tree property of the second kind in real-valued logic. They reach it by different roads, however, and we show that the probes decide which. The probes of the Pauli observable are the Pauli channels; for them the indicator of each syndrome sector is the difference of the responses of two uniform mixtures, and actual mixtures already witness the tree. The probes of the Rado observable see only marginals; conditions given by actual mixtures remain compatible on disjoint blocks, and the tree must be sought among idealized preparations.