Removing Trace Preservation from a Universal Spectral Bound for 2-Positive Maps

Vom Ende, Chruściński, Kimura and Muratore-Ginanneschi proved that every 2-positive trace-preserving map $Φ$ on $M_d=M_d(\mathbb{C})$ satisfies $\operatorname{Tr}Φ\le d\,m(Φ) + d^2 - d$, where $m(Φ)$ is the smallest real part of an eigenvalue of $Φ$, and deduced the universal relaxation-rate bound $\operatorname{Tr} L \le d\,m(L)$ for generators of 2-positive trace-preserving semigroups. They asked whether trace preservation is needed, and conjectured that every conditionally 2-positive map $L$, i.e. every $L$ for which $e^{tL}$ is 2-positive for all $t \ge 0$, satisfies $\operatorname{Tr} L \le d\,m(L) + (d^2 - d)\,s(L)$, where $s(L)$ is the largest real part of an eigenvalue of $L$. We prove this conjecture. The argument normalizes a 2-positive map by a positive definite eigenmatrix of its Hilbert-Schmidt adjoint, which turns it into a 2-positive trace-preserving map in the manner of a quantum Doob transform. The existence of such an eigenmatrix is guaranteed after adding an arbitrarily small multiple of the unnormalized depolarizing map, and the inequality then passes to the limit by continuity of the spectrum. No irreducibility, diagonalizability or faithful-invariant-state assumption is required. We also show that the constant $d$ is optimal within the class of non-trace-preserving completely positive semigroups, and we recast the result as a universal constraint on decay rates measured relative to the dominant spectral growth rate.