Bound on thermal-valley contrast

Prove or refute that, for the three-qubit thermal qubit collision-loop architecture defined by the controlled feedback rotation, leak interaction, partial swap, and message-qubit rotation, the contrast of every thermal valley is bounded by a quantity of order a few times 10^{-3}, and relate any such bound to the valley’s distance from the entanglement-breaking set.

Background

The paper studies thermal valleys, in which the entanglement-breaking index of a qubit collision channel is smaller at intermediate bath polarisation than at infinite temperature. The valley contrast measures the smallest signal separating the interior dip from its surrounding non-entanglement-breaking behavior at the relevant iteration depth.

A gated random search over 33 million circuits found no valley with contrast at least 0.05. The strongest certified true return had contrast approximately 2.85 × 10{-3}, while the strongest cliff had contrast approximately 3.06 × 10{-3}. These observations motivate a conjectured architecture-dependent upper bound, but the search does not establish a non-existence theorem. The authors connect the problem to the geometry of the entanglement-breaking set and explicitly state that either a proof or a counterexample would be significant.

References

We record the observation as a conjecture: over this architecture, valley contrast is bounded by a few times $10{-3}$. A valley is a boundary phenomenon, and the conjecture says that its visibility cannot exceed its distance from the entanglement-breaking set; a proof or a counterexample would be a statement about the geometry of that set for the collision family.

— Pulsed control of an entanglement-breaking index: thermal valleys, a certified valve, and its open-close asymmetry  (2610.13135 - Kopel, 9 Oct 2026) in Section 3.2, “The shape of the wall” (paragraph following Table 1); revisited in Section 8, “Discussion and outlook”