Exact directional steerability thresholds for full-rank mixed noise

Determine the exact directional steerability thresholds, for both projective measurements and arbitrary positive operator-valued measures where applicable, along Bell-state mixtures with generic full-rank mixed two-qubit X noise and with full-rank product noise.

Background

The paper derives exact steerability thresholds for several rank-deficient cases, including arbitrary pure two-qubit noise, two diagonal mixed X families with one missing population, and product noise with at least one pure local factor. In these cases, entanglement and steerability coincide in both directions, and the projective-measurement and POVM thresholds agree.

For generic full-rank mixed X noise, the paper cannot determine the full projective-measurement steerability boundary and instead uses finite-setting semidefinite programs to produce upper bounds. For full-rank product noise, the correlation tensor determines optimized CHSH and CJWR witness thresholds but not necessarily the exact directional steerability thresholds; states with identical correlation tensors can have different directional steerability thresholds. The unresolved problem is therefore to characterize or calculate these exact directional boundaries using information beyond the correlation tensor.