Determine optimal bounds for approximate disentanglers

Determine the optimal upper and lower bounds on the number of input qubits required by an $(\epsilon,\delta)$-disentangler with output dimension $D$, particularly in the regime $\epsilon+\delta>2/3$ and, more generally, for fixed $\epsilon+\delta<1$.

Background

An (ϵ,δ)(\epsilon,\delta)-disentangler is a quantum channel whose outputs are approximately separable and whose image approximately contains every separable state. The paper proves exponential lower bounds in the number of output qubits, with a near-optimal Ωϵ,δ(D)\Omega_{\epsilon,\delta}(\sqrt D) lower bound when ϵ+δ<2/3\epsilon+\delta<2/3.

For larger error sums, the paper obtains weaker polynomial-in-DD lower bounds and explicitly notes that these bounds are probably not tight. It identifies determining the optimal upper and lower bounds as an unresolved problem.

References

This scaling might be the correct form for small $\eta$, but our bound is almost certainly not tight up to constants. The proof could be improved to achieve a slightly improved bound (in some regimes) of $D{1/\left(\lceil 1/\eta\rceil - 1\right)}$ by modifying the construction to use antisymmetric subspace projectors on different sized registers; however, this is also probably not tight. We leave the task of finding the optimal upper and lower bounds on disentanglers as interesting future work.

— A quantum oracle separation between QMA(2) and QMA  (2609.02865 - Bostanci et al., 2 Sep 2026) in Section 5, paragraph beginning “To summarize the results in this section”

Determining the optimal dependence as $a$ approaches one remains open.

— 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”

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”