Achievability of the Rains bound under PPT-preserving channels
- Field
- Topics
Problem
Consider distilling the bipartite Bell-diagonal state
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 \(\lvert\Psi^\pm\rangle:=(\lvert01\rangle\pm\lvert10\rangle)/\sqrt2\). Is the Rains bound of the state in Eq. (1) achievable by a positive-partial-transpose-state-preserving (PPT-state-preserving, or PPT-preserving) quantum channel? If so, what is the constructive quantum channel? A channel is PPT-state-preserving if every PPT input state is mapped to a PPT output state.
Source
Rains introduced the PPT-preserving distillation framework and its semidefinite-programming bound; the present Bell-diagonal achievability question is an implicit specialization of that work [Rai99], [Rai01].
Progress
Rains showed that the Rains bound \(R(\rho_{\mathbf p})\) upper-bounds the entanglement distillable from \(\rho_{\mathbf p}\) by PPT-preserving operations:
\begin{equation} D_{\mathrm{PPT}}(\rho_{\mathbf p})\le R(\rho_{\mathbf p}). \tag{2} \end{equation}Equation (2) is the relevant converse bound for the operational class in this problem [Rai99], [Rai01].
Comment
The problem asks whether the upper bound in Eq. (2) is achievable for the Bell-diagonal family in Eq. (1), and, if so, for an explicit construction of a PPT-preserving channel attaining it.