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.

33 pages, 1 figure, 9 tables. Companion to arXiv:2608.13764. Ancillary files in two subfolders: zero-temperature/ and thermal/

arXiv:2609.27908 [quant-ph]

(or arXiv:2609.27908v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.27908 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

thermal/README.md

thermal/cert3/certify_denominators.py

thermal/cert3/certify_scan3.py

thermal/cert3/certify_scan3_stats.py

thermal/certified_index.py

thermal/certified_model.py

thermal/certify_theorem3.py

thermal/closeout.py

thermal/counterexamples.json

thermal/feedback_model.py

thermal/floor_measure.py

thermal/profile_valleys.py

thermal/render.py

thermal/strat/scan3_analyse.py

thermal/strat/valley_scan_smallfloor_v2.py

zero-temperature/LICENSE.txt

zero-temperature/README.txt

zero-temperature/audit.json

zero-temperature/audit_mpmath.py

zero-temperature/audit_report.txt

zero-temperature/certificate.json

zero-temperature/certificate_report.txt

zero-temperature/choi_certified_region.py

zero-temperature/choi_slice_bound.py

zero-temperature/combined_certificate_calc.py

zero-temperature/combined_margins.py

zero-temperature/combined_radius.py

zero-temperature/falsifiability.py

zero-temperature/independent_crosscheck.py

zero-temperature/make_fig1_index_landscape.py

zero-temperature/nEB_crosscheck.py

zero-temperature/partial_swap_extension.py

zero-temperature/requirements.txt

zero-temperature/safe_interval_certificate.py

zero-temperature/symbolic_restricted.py

zero-temperature/verified_certificate.py

zero-temperature/verify_analytic_family.py

zero-temperature/verify_analytic_standalone.py

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