The complete deterministic and asymptotic comparison chain

For any finite nonempty types $A,B$, each with decidable equality, and any positive semidefinite complex matrix $\rho$ indexed by $A\times B$ with trace $1$, define $X^\Gamma_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, let $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$, and let $\|\cdot\|_{\mathrm{op}}$ be the Euclidean operator norm. Define $F_X(k)$ as the real supremum of $\operatorname{Re}\operatorname{tr}(XQ)$ subject to $k>0$, $0\preceq Q\preceq I$, and $-k^{-1}I\preceq Q^\Gamma\preceq k^{-1}I$, where $\preceq$ means positive semidefinite difference. Define $D(X)=\sup\{\log_2 k:k\in\mathbb R,\ k>0,\ F_X(k)=1\}$, $D_{\mathrm{reg}}(X)=\sup_{m\ge1}D(X^{\otimes m})/m$, and $\Gamma(X)=\sup\{r\in\mathbb R:F_{X^{\otimes n}}(2^{nr})\to1\}$, with no nonnegativity restriction on $r$. Tensor powers have entries $\prod_i X_{(a_i,b_i),(a'_i,b'_i)}$, with the zeroth power equal to the scalar $1$. Finally, set $W(X)=\log_2\sup_R\operatorname{Re}\operatorname{tr}(X^\Gamma R)$, where $R$ is Hermitian, $\|R\|_{\mathrm{op}}\le1$, and $R^\Gamma\succeq0$, and set $N(X)=\log_2\|X^\Gamma\|_1$; all suprema here are real-valued. Then