Table 1 A comparison of different methods for the verification of high-dimensional entanglement

Our witness

• Measure in at least two coordinated local (arbitrary) orthonormal bases

• Only need to know the (minimum and maximum) absolute values of the overlaps between the local measurement bases

Reference ¹⁰’s witness

• Precise control over the absolute values & the complex phases of the overlaps between different local measurement bases and within each basis

• Measure in the computational basis + at least one coordinated “tilted” basis

SDP witness⁵⁵

• Treat bases bias as imperfect implementations of measurements in MUBs

• Memory issue/long computational runtime for large dimensions

• Efficient only for small dimensions in which case it is possible to obtain tighter bounds compared to our witness

Entropic uncertainty relationship^25,27

[eq. (17.135) in ref. ⁵⁶]

• Need to know the absolute values of the bases overlaps of only one party and the classical entropies corresponding to the two parties’ measurements

• Can lower bound the distillable entanglement instead of the Schmidt number

Generalized Bell

inequalities^57,58

• To witness Schmidt number without any measurement assumptions, require non-trivial optimization over all possible local measurements and all states with Schmidt number ≤ k for every local dimension, which is computationally costly

• Certify lower Schmidt numbers than other methods can in general

• Standard approaches involve finding the largest eigenvalue corresponding to eigenvectors with Schmidt rank ≤ k of the Bell operator⁵⁹ associated with restricted measurement settings^58,60 to keep optimization problems numerically tractable (more specifically, ref. ⁵⁸ uses physical arguments to restrict the maximization of the Bell operator’s expectation value to restricted sets of states with different maximum Schmidt numbers—referred to therein as entanglement dimensions—of which the union is believed to contain the experimental states)  →  also require measurement assumptions and are computationally feasible only for small dimensions

Correlation-matrix norms from randomized measurements⁶¹

• Independent of the relative reference frame between the two parties as the matrix \(p\)-norms for all even \(p\in {\mathbb{N}}\) of the correlation matrix remain unchanged under any local unitary transformation of a bipartite state⁶¹.

• Require sampling local unitaries randomly from (Haar measure or) t-designs, where exact sampling is highly inefficient⁶².

• While approximate sampling from \({\mathrm{t}}\)-designs can be efficient⁶³, it is unclear how approximate sampling can affect the Schmidt -number witness in ref. ⁶¹.

• Analytic bounds of the 2- and 4-norms of the correlation matrices corresponding to states with Schmidt number ≤ k are known only for k = 2

• For certifying Schmidt number ≥ 3, require numerical optimizations which can be computationally costly for large dimensions and the bounds can be loose since there are no tighter known constraints on the singular values of the correlation matrix other than the purity bound \(\text{Tr}(\rho^2)<1\) (see Appendix C of ref. ⁶¹.)