Full local-unitary optimization of nonlocal magic measures

Determine the full local-unitary optimizers of the nonlocal stabilizer Rényi entropy, stabilizer extent, generalized robustness, and mana, in order to establish how far the sorted computational-basis geometry extends beyond the stabilizer fidelity.

Background

The paper exactly solves the local-unitary optimization defining nonlocal magic for the stabilizer fidelity, showing that an ordered computational-basis representative of the Schmidt decomposition is globally optimal. It also derives bounds relating the resulting nonlocal min-relative entropy of nonstabilizerness to the nonlocal stabilizer Rényi entropy, stabilizer extent, and generalized robustness.

The corresponding optimizing local unitaries for those other measures, as well as for mana, are not determined. Solving these optimization problems would reveal whether the spectral and sorted-basis structure identified for stabilizer fidelity persists for broader classes of nonlocal magic measures.

References

Ref. further conjectures that the descending reference state Schmidt coefficients results in the smallest value of Eq.~eq:nlref and leads through the conjecture of Ref. to $M_2{\rm NL}=M_2(|\tilde{\psi}\rangle)$. We adopt this as our estimate of the non-local magic.

— Non-stabilizerness and entanglement in $(2+1)$-dimensional SU(2) lattice gauge theory using tensor networks  (2609.28634 - Jha et al., 23 Sep 2026) in Appendix, Section \ref{app:subsec_lowerbound_and_nlmagic}, subsection “Lower bound and computation of non-local magic”

Several questions remain open. Finding and proving the full local-unitary optimizers of the nonlocal SRE, stabilizer extent, generalized robustness and mana would determine how far the sorted computational basis geometry persists beyond the stabilizer fidelity.

— Exact quantification of nonlocal magic  (2608.28563 - Sierant, 28 Aug 2026) in Discussion

A natural next step is to determine whether analogous closed-form expressions in terms of the entanglement spectrum can be obtained for other magic monotones, in particular the stabilizer Rényi entropies (SREs), and whether the minimizers of SRE-based non-local magic coincide with those identified here for the stabilizer fidelity.

— Non-local Magic: closed-form solution and equivalence with magic of purification  (2609.04119 - Viscardi et al., 3 Sep 2026) in Section Conclusions and perspectives