Supplement V.B, Eq.(183): finite-block private-rate bound, corrected dimension

Let $d,e,M$ be natural numbers with $d,e\ne0$, $d\ge2$, and $M\ge2$, and let $\mathcal E$ map complex $d\times d$ matrices to complex $e\times e$ matrices with a finite Kraus representation $\mathcal E(X)=\sum_jK_jXK_j^\dagger$, $\sum_jK_j^\dagger K_j=I$. Throughout, a density matrix is positive semidefinite with trace $1$, a separable density matrix is a finite sum $\sum_jw_j\rho_j\otimes\sigma_j$ of product density matrices with $w_j\ge0$ and $\sum_jw_j=1$, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. Consider any code with a natural number $k$ of key bits, a positive natural number $u$ of channel uses, natural numbers $a,b,s$, a separable density matrix $\rho_0$ on $(\mathbb C^{a+1}\otimes\mathbb C^d)\otimes\mathbb C^{b+1}$, and an evolution $\mathcal P$ to $(\mathbb C^{2^k}\otimes\mathbb C^{s+1})\otimes(\mathbb C^{2^k}\otimes\mathbb C^{s+1})$. The evolution uses exactly $u$ copies of $\mathcal E$: each use applies $\mathcal E$ to Alice’s distinguished $d$-dimensional register and transfers its output to Bob, followed by a finite local protocol that prepares the memory layout for the remaining uses, with a finite local protocol after the last use. Such local protocols are generated recursively by identity maps and complete local Kraus instruments, followed by a branch-dependent protocol and summation over branches; the local dimensions may change. The code also specifies a density-matrix target $\gamma$ that, for some shield density matrix $\omega$ and shield unitaries $U_i$, has entries

Write $\operatorname{err}=\|\mathcal P(\rho_0)-\gamma\|_1/2$. Define the following map $\mathcal T_{d,M}$, which fixes the precise port-based simulation used in the bound. With $\mathcal A=\{0,\ldots,d-1\}^M$, let

where $f(S)$ is functional calculus and the scalar inverse at zero is zero, and put $H_i=RF_iR+M^{-1}(I-RSR)$. Let

For a matrix $X$ on $\mathbb C^d$, define

For a matrix map $\mathcal F$, define

where $X$ ranges over all complex matrices on $\mathbb C^d\otimes\mathbb C^d$, without positivity or Hermiticity restrictions, and identity amplification applies the map separately to each reference-indexed block. Let $\epsilon\ge0$ satisfy $2\operatorname{err}\le\epsilon$. Let $Q$ be any real-valued functional on bipartite matrices for every pair of finite nonempty index types with decidable equality, and let $g,h:\mathbb R\to\mathbb R$ and $\mathcal D\subseteq\mathbb R$ satisfy all of the following: $Q$ is nonnegative on density matrices and does not increase under the finite local protocols just described, including changes of local dimension; $g(t),h(t)\to0$ as $t\downarrow0$; $\mathcal D\subseteq[0,\infty)$; $Q(\rho\otimes\sigma)\le Q(\rho)+Q(\sigma)$ for all bipartite density matrices, with corresponding parties grouped together; every private-state matrix of the displayed form with $2^\ell$ key values and equal shield dimensions $r+1$ satisfies $Q(\gamma)\ge\ell$; and, for all density matrices $\rho,\sigma$ on the same finite nonempty bipartite space $A\otimes B$ and every $t\in\mathcal D$,

If $u\delta_{d,M}+\epsilon\in\mathcal D$, then, writing $J(\mathcal E)=(\operatorname{id}_d\otimes\mathcal E)(|\Phi_d\rangle\langle\Phi_d|)$ for $\Phi_d=d^{-1/2}\sum_i|i,i\rangle$,

Key size $k=0$, shield parameter $s=0$, zero code error, and $\epsilon=0$ are included; membership in $\mathcal D$ is required at the actual argument appearing above.