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
Reports do not certify correctness or automatically change the problem's status. Progress policy.
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.