Markovian dephasing has no entanglement-breaking threshold, and a fixed tolerance will report one anyway

Abstract

When modelling decoherence in a biological spin system it is tempting to seek a critical rate beyond which the channel is entanglement-breaking (EB) and quantum resources are gone. We show that for the two channel families in which such a model would be posed, no critical rate exists. For uniform dephasing at rate $\gamma$ the partial transpose of the Choi state has eigenvalue $-e^{-\gamma}/d$, so the channel is non-PPT -- hence not EB -- at every finite $\gamma$. Adding amplitude damping changes nothing: the qubit map's partial transpose stays negative at all finite rates. What does vanish at a finite rate is the coherent information, exactly where the qubit map turns antidegradable: the root of $c^{2}=p$, $\gamma_{\Ic}=0.668$ at $\kappa=0.1$. Antidegradability survives tensor products, so the single-letter and regularised thresholds coincide: of the three properties usually conflated here, two share one finite boundary and the third has none. Our main object is the numerical mechanism that manufactures a threshold where there is none. A quantity that decays exponentially to zero without reaching it, tested against a fixed absolute cutoff, yields a "threshold" set by the cutoff: the PPT test gives $\gamma=5.49$ at $\epsilon=10^{-10}$ and $6.58$ at $10^{-12}$, sliding by $0.55$ per decade. We reported the first number in three now-retracted preprints. Preparing this paper we made the same error three more times -- most seriously in $\gamma_{\Ic}$ itself, which a bisection against a $10^{-7}$ cutoff placed at $0.62$ -- and we document all four instances, with the closed forms and code that avoid them.