Corollary 16: quantum/private bounds for polynomial Pauli noise

Let $P_0,P_1,P_2$ be real polynomials with $P_i(0)=0$, and assume that for some $\delta_P>0$ the weights $w(p)=(1-\sum_iP_i(p),P_0(p),P_1(p),P_2(p))$ are nonnegative and sum to $1$ throughout $[0,\delta_P]$. Write $\ell_i$ for the coefficient of degree $1$ in $P_i$, $b=|\ell_0\ell_1+\ell_0\ell_2+\ell_1\ell_2|$, and $V_p{}_{b,j;a}=\sqrt{w_j(p)}(\sigma_j)_{ba}$, where the four $\sigma_j$ are the identity and usual Pauli $X,Y,Z$ matrices. Put $N_p(X)=\operatorname{Tr}_E(V_pXV_p^\dagger)$, $N_p^c(X)=\operatorname{Tr}_B(V_pXV_p^\dagger)$, and

where functions of Hermitian matrices use spectral functional calculus, $\log_2x=\log x/\log2$, and $0\log_2 0=0$. Define $Q(V_p)$ as the supremum of nonnegative rates $r$ admitting positive message dimensions $M_n$ and encoding and decoding channels for every $n\ge0$, with entanglement transmission fidelity through $N_p^{\otimes n}$ tending to $1$ and, for every $s>0$, eventually $r-s\le\log_2M_n/n$. The fidelity is the expectation of $|\Phi_{M_n}\rangle\langle\Phi_{M_n}|$ in the state obtained by applying the encoded channel and decoder to the second half of $|\Phi_{M_n}\rangle=M_n^{-1/2}\sum_m|m,m\rangle$. Define $P(V_p)$ as the supremum of nonnegative rates $r$ such that, for every $\epsilon,s>0$, every sufficiently large positive $n$ admits $M>0$, density matrices $\rho_m$, and positive semidefinite matrices $F_j$ summing to $I$, with $r-s\le\log_2M/n$ and

Here a density matrix is positive semidefinite with trace $1$, every channel has a finite Kraus representation with $\sum_jK_j^\dagger K_j=I$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. There exist $K_q,K_p\ge0$ and $0<\delta\le1$ such that every $0<p<\delta$ satisfies

The hypotheses permit zero linear coefficients and $b=0$. The quantum code sequence includes $n=0$, where real division by zero is $0$, while its asymptotic conditions concern large $n$.