Achievability of the Rains bound under completely PPT-preserving channels
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Problem
Is the Rains bound asymptotically achievable by completely positive-partial-transpose-preserving (completely PPT-preserving) channels for every full-rank two-qubit Bell-diagonal state, and what channel family achieves it? Consider
where \(p_i>0\) and \(p_I+p_X+p_Y+p_Z=1\), and \(\lvert\Phi^\pm\rangle:=(\lvert00\rangle\pm\lvert11\rangle)/\sqrt2\) and \(\lvert\Psi^\pm\rangle:=(\lvert01\rangle\pm\lvert10\rangle)/\sqrt2\).
A channel \(\Lambda_n:A^nB^n\to A'_nB'_n\) is completely PPT-preserving if
where \(\Gamma\) is partial transposition on the indicated subsystem. Equation (2) requires PPT preservation also with arbitrary local ancillas. Write \(D(\rho\|\tau)=\operatorname{Tr}\rho(\log_2\rho-\log_2\tau)\) with the standard support convention, and define
For the state in Eq. (1), achievability of the Rains bound in Eq. (3) means a sequence satisfying Eq. (2) and
where \(M_n\) is a positive integer and \(\Phi_M:=M^{-1}\sum_{i,j=1}^M|ii\rangle\langle jj|\). Equation (4) imposes no efficiency requirement on the construction.
Source
Rains introduced the distillation framework and converse bound for completely PPT-preserving channels; this Bell-diagonal achievability question is an implicit specialization of that work [Rai99], [Rai01]. The operation class is explicitly the one in Definition 6 of Regula, Fang, Wang, and Gu [RFWG19]. This corrects the earlier, weaker requirement of preserving PPT input states without ancillary extensions.
Progress
For the completely PPT-preserving operation class, Rains’ converse is
\begin{equation} D_{\mathrm{cPPT}}(\rho_{\mathbf p})\leq R(\rho_{\mathbf p}), \tag{5} \end{equation}where \(D_{\mathrm{cPPT}}\) is the supremum of asymptotically achievable distillation rates. Equation (5) gives the upper bound whose attainability is asked in Eq. (4) [Rai99], [Rai01].
For this state family, the Rains bound is additive. Let \(p=\max_i p_i\) and \(h_2(x)=-x\log_2x-(1-x)\log_2(1-x)\). If \(p\leq1/2\), the state is PPT and the bound is zero. If \(p>1/2\), let \(\Phi\) be its largest Bell component, and let \(\sigma\) have Bell probabilities \(q_{\max}=1/2\) and \(q_i=p_i/[2(1-p)]\) otherwise. This normalized state is PPT. Its relative-entropy supporting functional is \(A=\rho_{\mathbf p}\sigma^{-1}=2(1-p)I+2(2p-1)\Phi\). The eigenvalues of \(A^{\Gamma_B}\) are \(1\) and \(3-4p\), hence \(\|A^{\Gamma_B}\|_\infty=1\) and \(\operatorname{Tr}A\tau\leq1=\operatorname{Tr}A\sigma\) for every Rains-feasible \(\tau\). This is the convex first-order optimality condition for \(\sigma\). The same argument uses \(A^{\otimes n}\) for every tensor power, since partial transpose and the operator norm are multiplicative. Consequently,
\begin{equation} R(\rho_{\mathbf p}^{\otimes n}) =E_{R,\mathrm{PPT}}(\rho_{\mathbf p}^{\otimes n}) =\begin{cases}0,&p\leq1/2,\\ n[1-h_2(p)],&p>1/2, \end{cases} \tag{6} \end{equation}where \(E_{R,\mathrm{PPT}}\) minimizes \(D\) over normalized PPT states. Equation (6) evaluates the converse but does not construct a completely PPT-preserving distillation protocol.
The weaker class of channels preserving PPT states without ancillary extensions is \(\mathrm{PPTP}_+\) in Definition 7 of Regula, Fang, Wang, and Gu. Their Corollary 10, Eq. (33), gives its distillation rate as \(E_{R,\mathrm{PPT}}^\infty\), so Eq. (6) solves achievability for that weaker class [RFWG19]. The generalized quantum Stein lemma used in this result has a complete proof in Lami’s Theorem 1 [Lam25]. For \(p>1/2\) and \(0<r<1-h_2(p)\), it supplies effects \(0\leq T_n\leq I\) with \(\operatorname{Tr}T_n\rho_{\mathbf p}^{\otimes n}\to1\) and \(\sup_{\omega\in\mathrm{PPT}}\operatorname{Tr}T_n\omega\leq1/M_n\), where \(M_n=\lfloor2^{nr}\rfloor\geq2\) for sufficiently large \(n\). An optimization-defined channel family is
\begin{equation} \Lambda_n(X)=\operatorname{Tr}(T_nX)\Phi_{M_n} +\operatorname{Tr}[(I-T_n)X] \frac{I-\Phi_{M_n}}{M_n^2-1}. \tag{7} \end{equation}Each PPT input to Eq. (7) yields an isotropic state with maximally entangled weight at most \(1/M_n\), so the map is PPT-state-preserving. Letting \(r\) approach the value in Eq. (6) gives that asymptotic rate. This does not establish the complete PPT condition in Eq. (2).
Comment
The open question uses completely PPT-preserving channels as in Rains’ framework. Its former PPT-state-preserving wording described a strictly larger class and is corrected explicitly here. The weaker-class result in Eq. (7) does not settle the intended stronger operation class. The zero-rate case \(p\leq1/2\) is immediate; the remaining question concerns entangled full-rank Bell-diagonal states.