Factorizable Rational Quantum Channels at an Explicit Distance from Finite Tracial Baths

A quantum channel can always be implemented by a unitary interaction with an environment that starts in a pure state. With a finite, maximally mixed environment, the implementable channels are the noisy operations. A tracial von Neumann algebra as environment gives the factorizable channels, and by a theorem of Haagerup and Musat combined with MIP* = RE, some of these lie outside the closure of the noisy operations. Version 1.0 of this record wrote down explicit rational channels of this kind, with no value for their distance from the closure. This version gives explicit lower bounds. Using Wang and Zhi's dimension-uniform reflection estimate, it constructs a factorizable unital channel on twelve qubits with Kraus entries in {0, 1, ±3/5, 4/5}, whose normalized Choi and diamond distances from the closure are at least 2^-(2J+50), with J = 2^(2^1122000). The bound comes from a compiler theorem: group relations that are rigid uniformly in the matrix dimension give a rational channel with an explicit distance bound. For the channels of version 1.0, a new spectral certificate for the root groups of their presentation gives explicit gaps: 2^-(2J+41) for the fifteen-qubit channel and its two variants and, by the bound of version 1.0, 2^-(2J+36) in Choi distance and 2^-(2J+21) in diamond distance for the fully referenced channel. The numbers are tiny because Wang and Zhi's tolerance is 2^-J. The results depend on recent preprints of Wang and Zhi, Thom, and Alekseev, Liu and Thom, which are not yet refereed. This archive includes the paper, the channel data, the certificates, and programs that rebuild and check them; python verify_all.py ends with OK ALL.