Supplement V.B, Eqs.(187)–(190): operational key-capacity bound

For every natural number $d\ne0$ with $d\ge2$, and every matrix map $\mathcal E$ on $\mathbb C^d$ admitting a finite Kraus representation $\mathcal E(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$, use the following definitions. A density matrix is positive semidefinite with trace $1$; a separable density matrix is a finite convex combination of products of density matrices. For matrices on a finite bipartite space, let

$$\mathcal R(\rho)= \left(\inf_{\sigma\text{ separable density}}D_2^+(\rho\|\sigma)\right)_{\mathrm{real}},$$

where $D_2^+=+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\rho$, and otherwise

$$D_2^+(\rho\|\sigma) = \frac{\max\!\left\{0,\operatorname{Re}\operatorname{tr} \bigl(f(\rho)-\rho\log\sigma\bigr)\right\}}{\ln2}, \qquad f(x)=x\ln x.$$

These matrix functions use functional calculus and the scalar convention $\ln0=0$; the infimum is in $[0,+\infty]$, and the subscript “real” sends finite values to their real values and sends $+\infty$ to $0$. For any map $\mathcal F$ on $d$-dimensional matrices, define

$$\mathfrak A(\mathcal F) = \sup_{\substack{a,b\in\mathbb N\\ \rho\text{ density on }(\mathbb C^{a+1}\otimes\mathbb C^d)\otimes\mathbb C^{b+1}}} \bigl[\mathcal R(\mathcal S_{\mathcal F}(\rho))-\mathcal R(\rho)\bigr],$$

with real differences embedded in the extended real line, where

$$\mathcal S_{\mathcal F}(\rho)_{(x,(y,z)),(x',(y',z'))} = \mathcal F\!\left((i,j)\mapsto \rho_{((x,i),z),((x',j),z')}\right)_{y,y'}.$$

Thus the output of the distinguished input register belongs to the second party. Write $J(\mathcal F)=(\operatorname{id}_d\otimes\mathcal F)(|\Phi_d\rangle\langle\Phi_d|)$, where $\Phi_d=d^{-1/2}\sum_i|i,i\rangle$, and let $J_M(\mathcal E)$ be the grouped $M$-fold product, explicitly

$$J_M(\mathcal E)_{(a,b),(c,t)} =\prod_{i=0}^{M-1}J(\mathcal E)_{(a_i,b_i),(c_i,t_i)}.$$

Here identity amplification applies a map to each reference-indexed matrix block. For each $M\ge2$, define a port-based map $\mathcal T_{d,M}$ as follows. Put $\mathcal A=\{0,\ldots,d-1\}^M$, define

$$F_i((x,a),(y,c)) =\frac1d\mathbf1_{\{x=a_i,\ y=c_i,\ a_j=c_j\ (j\ne i)\}}, \quad S=\sum_iF_i,\quad R=f(S),\quad f(t)=(\sqrt t)^{-1}, \quad H_i=RF_iR+\frac{I-RSR}{M},$$

using functional calculus and scalar inverse $0^{-1}=0$, and set

$$\beta_M((a,r),(c,t)) =\prod_j\frac1d\mathbf1_{\{a_j=r_j,\ c_j=t_j\}}, \qquad \mathcal T_{d,M}(X)_{v,w} =\sum_i\sum_{p,q\in\{0,\ldots,d-1\}\times\mathcal A}\sum_{r,t\in\mathcal A} \mathbf1_{\{r_i=v,\ t_i=w,\ r_j=t_j\ (j\ne i)\}} (H_i)_{q,p}X_{p_0,q_0}\beta_M((p_1,r),(q_1,t)).$$

Let

$$\delta_{d,M} =\sup_{\|X\|_1\le1} \|(\operatorname{id}_d\otimes(\operatorname{id}-\mathcal T_{d,M}))(X)\|_1, \qquad \|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X},$$

where the supremum ranges over all complex matrices $X$ on $\mathbb C^d\otimes\mathbb C^d$. Define $K(\mathcal E)$ to be the extended-real supremum of all real $r\ge0$ for which there is a sequence of codes with positive use counts $u_n\to\infty$, natural key-bit counts $k_n$ satisfying $k_n/u_n\to r$, and errors tending to zero. Each code starts from a separable density matrix on $(\mathbb C^{a_n+1}\otimes\mathbb C^d)\otimes\mathbb C^{b_n+1}$, uses $\mathcal E$ exactly $u_n$ times by sending the distinguished input register from Alice to Bob, and allows finite local protocols between uses and at termination. Those protocols are generated by identities and complete local Kraus instruments with recursively chosen branch protocols, allowing changes of dimension. Its error is half the trace norm between its output and a density-matrix private target on $(\mathbb C^{2^{k_n}}\otimes\mathbb C^{s_n+1})^{\otimes2}$, whose entries are zero unless the two parties’ key indices agree on both row and column, and otherwise equal $2^{-k_n}(U_i\omega U_j^\dagger)_{(x,y),(x',y')}$ for a shield density matrix $\omega$ and shield unitaries $U_i$. Finally put

$$c(t)=(1+t)\log_2(1+t)-t\log_2t,\qquad B_{d,M}(\mathcal E) =M\mathcal R(J(\mathcal E)) +\frac{2d(d-1)}M\log_2d +c\!\left(\frac{d(d-1)}M\right),$$

with scalar logarithms totalized by $\log_2 0=0$. Then, for every natural $M\ge2$, all five assertions hold:

$$\begin{gathered} \mathfrak A(\mathcal E\circ\mathcal T_{d,M})\le\mathcal R(J_M(\mathcal E)),\\ \mathcal R(J_M(\mathcal E))\le M\mathcal R(J(\mathcal E)),\\ \mathfrak A(\mathcal E) \le\mathcal R(J_M(\mathcal E)) +\delta_{d,M}\log_2d+c(\delta_{d,M}/2),\\ K(\mathcal E) \le\mathcal R(J_M(\mathcal E)) +\delta_{d,M}\log_2d+c(\delta_{d,M}/2) \le B_{d,M}(\mathcal E). \end{gathered}$$

The second inequality is a real inequality; the others use the extended real line, embedding every displayed real bound there. In addition,

$$K(\mathcal E)\le\inf_{\substack{M\in\mathbb N\\M\ge2}}B_{d,M}(\mathcal E),$$

where this last infimum is also in the extended real line.