Raising the SDP dimension exponent

Raise the exponent $2/7$ in the dimension-dependent lower bounds for objective-independent semidefinite extended formulations approximating the separable set.

Background

The paper proves lower bounds whose dimension-dependent exponent is limited by any fixed value below $2/7$. The authors explicitly ask whether this exponent can be increased, making this an unresolved problem concerning the quantitative semidefinite extension complexity of separability.

References

On the SDP side, the main quantitative questions are whether the $a{-1/3}$ dependence can be improved and whether the exponent $2/7$ can be raised.

— Semidefinite extension complexity of the separable set, with applications to approximate disentanglers  (2609.09033 - Gharibian et al., 8 Sep 2026) in Section 1, Introduction, paragraph “Open questions”

In particular, can these methods yield size lower bounds exponential in a power of $d$ at sufficiently small constant additive error, even for formulations that do not arise from quantum channels?

— Semidefinite extension complexity of the separable set, with applications to approximate disentanglers  (2609.09033 - Gharibian et al., 8 Sep 2026) in Section 1, Introduction, paragraph “Open questions”