Quantitative midpoint-versus-endpoint entanglement inequality

Determine whether the stronger quantitative conjecture comparing the minimum eigenvalue of the partially transposed Choi matrix of the sequential qubit channel with an SDP bound for the corresponding parallel channel holds for all qubit channels.

Background

The paper proves midpoint optimality only in a qualitative feasibility sense: if the sequential composition of two qubit channels is not entanglement breaking, then their parallel action is not entanglement annihilating. A stronger conjecture from the cited source seeks a quantitative ordering between the two source placements rather than merely an implication about whether some entanglement survives.

Specifically, the unresolved conjecture compares the minimum eigenvalue of the partially transposed Choi matrix associated with the sequential channel to an SDP-derived bound associated with the parallel channel. The paper explicitly states that its theorem does not establish this pointwise quantitative relationship for negativity, concurrence, fidelity, or other entanglement measures.