The quantum lifetime of a future-referential feedback loop: certified index, architecture floor, and thermal bonus

Abstract

A quantum feedback loop that returns information from a forward simulation to an earlier internal time induces a completely positive trace-preserving map on a message register; iterating that map eventually destroys its ability to carry entanglement. The entanglement-breaking index N = n_EB(Phi), the round at which this happens, is studied here along two axes: the interaction and the bath. On the interaction axis we prove that strict contraction onto a full-rank fixed point forces N finite in every dimension, via an explicit separable ball around the limiting Choi state; the qubit closed form overshoots the exact answer by the factor ln(2 sqrt 3) / ln 3 = 1.1309... on isotropic unital channels, and across 3997 sampled channels index and stability gap are one clock, N [1 - rho(A)] in [0.60, 1.50]. On the bath axis, replacing the zero-temperature ancilla by a thermal one at polarisation p = tanh(hbar omega / 2 k_B T) makes every channel quantity an exact quadratic polynomial in p, and the same one-sided certificates (no eigensolver) establish N = 3 at the reference circuit at every temperature, uniformly on a parameter box. The infinite-temperature index is an architecture floor with closed form and exact measure (46.427% of circuits break entanglement in one round however hot the bath). The floor is not a bound: certified circuits dip below it in interior temperature windows, unit-quantised in depth; over 4.1 million circuits their rate follows a Gaussian cutoff in epsilon = 1 - ||A||_2 with exponent 2.02 [1.90, 2.16], and 12 million more certified circuits show valleys surviving to epsilon = 0.125. Certified endpoints make comparisons exact: temperature moves the index by at most 21/14 = 3/2; the interaction moves it by 1213/3.