Lemma 2 (PBT stretching), Eqs.(15)–(18)

For all natural numbers $d,e,o,n,M$, equipped with $d,e,o\ne0$, with $d\ge2$ and $M\ge2$, fix an adaptive protocol $P$ consisting of a positive memory dimension, an initial positive semidefinite trace-one matrix on memory and the $d$-dimensional input, a channel from memory and the $e$-dimensional output back to memory and input for every natural-number step index, and two terminal channels to dimension $o$, one for zero uses and one for positive uses. Every channel here means a map with finitely many Kraus operators $K_j$, acting by $\sum_jK_jXK_j^\dagger$, with $\sum_jK_j^\dagger K_j=I$. Write $\mathcal O_P(\mathcal E)$ for its output: for $n=0$ apply the zero-use terminal channel to the initial matrix; for $n>0$, apply $\mathcal E$ to the input register $n$ times, use the prescribed intervening channels after its first $n-1$ applications, and then apply the positive-use terminal channel. Define $\Phi_d=d^{-1}\sum_{x,y}|x,x\rangle\langle y,y|$, $J_{\mathcal E}=(\operatorname{id}_d\otimes\mathcal E)(\Phi_d)$, and the port map $\mathcal P_{d,M}$ as follows: its signal matrices are $S_i[(x,a),(y,b)]=d^{-1}\mathbf1_{\{x=a_i,\ y=b_i,\ a_j=b_j\ \forall j\ne i\}}$; with $S=\sum_iS_i$, $R=f(S)$, $f(t)=(\sqrt t)^{-1}$, $f(0)=0$, its effects are $Q_i=RS_iR+M^{-1}(I-RSR)$; on $X\otimes\Phi_d^{\otimes M}$, contract the input and first halves with $Q_i$, trace out all second halves except $i$, and sum over $i$. Put $\widetilde{\mathcal E}_M=\mathcal E\circ\mathcal P_{d,M}$, $\epsilon_{d,M}=\Delta_d(\operatorname{id}-\mathcal P_{d,M})$, and $\Delta_d(\mathcal F)=\sup_{\|X\|_1\le1}\|(\operatorname{id}_d\otimes\mathcal F)(X)\|_1$, where the supremum ranges over all matrices on two input registers and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. There exists a single channel $\mathcal L$, from $nM$ registers of dimension $de$ to dimension $o$, such that for every channel $\mathcal E:d\to e$,

In addition, for every such $\mathcal E$, all natural numbers $a,b,c,f$, every separable density matrix $\rho$ on $(\mathbb C^{a+1}\otimes\mathbb C^d)\otimes\mathbb C^{b+1}$, and every evolution $\mathcal V$ to dimensions $c+1,f+1$ having the specified $n$-use two-way form, there is a channel $\mathcal L'$ that is finite LOCC and satisfies

Here separability means a finite convex combination of tensor products of positive semidefinite trace-one local matrices; $J_{\mathcal E}^{[nM]}$ is the same product of Choi entries with all $d$-registers grouped on one side and all $e$-registers on the other. The specified two-way form has a finite LOCC terminal map at zero uses and, at each positive use, sends the designated input register through $\mathcal E$ to the other party, applies a finite LOCC map to registers of the next prescribed dimensions, and continues recursively; finite LOCC itself is generated by identities and local complete finite Kraus measurements with outcome-dependent continuations. The natural number $n$ may be zero, in which case the tensor product has zero factors, is the scalar matrix $1$, and both displayed error bounds are zero.