Curator's Take
This article delivers the first analytic description of how “magic” – the non‑stabilizer resource that fuels quantum speed‑up – is limited by entanglement in two‑qutrit and two‑ququint systems, tightening long‑standing bounds on maximal magic and revealing a rich two‑dimensional Pareto frontier. By pinpointing exact extremal states (which turn out to be Weyl–Heisenberg covariant fiducials for mutually unbiased bases) the work provides concrete targets for hardware designers seeking high‑magic resources while keeping entanglement under control, and it sharpens simulation benchmarks that rely on magic‑state quantifiers. In the broader landscape of quantum resource theory, these results bridge the gap between abstract magic‑state distillation protocols and realistic multi‑level qudit platforms, offering a clearer roadmap for exploiting both entanglement and magic in near‑term devices.
— Mark Eatherly
Summary
Achieving a genuine quantum advantage relies on two distinct non-classical resources that restrict efficient classical simulation: entanglement and magic (nonstabilizerness). We investigate the interplay between these resources by characterizing the Pareto frontiers of extreme magic at fixed entanglement for systems of two qutrits ($d=3$) and two ququints ($d=5$). Unlike the case of two qubits, the Schmidt spectrum for two qutrits features two independent entanglement parameters, resulting in two-dimensional Pareto surfaces. For the lower frontier, we recast the minimal magic as a compact function of concurrence and negativity, with a maximal value of $\ln 2$. For the upper frontier, we determine the maximal stabilizer Rényi entropy to be $M_2 = \ln(81/17) \approx 1.561$, which tightens the previous theoretical bound of $\ln 5\approx 1.609$ and improves on earlier numerical estimates. The maximum magic is achieved at eighteen distinct maxima categorized into three families of six permutation-equivalent spectra. We provide analytical expressions for the maximal magic in the neighborhood of each maximum and for the corresponding maximally magical states which turn out to be Weyl-Heisenberg-covariant fiducial states for mutually unbiased bases. Finally, numerical analysis of two ququints ($d=5$) reveals six permutation-inequivalent maxima with a peak magic value of $M_2 = \ln(625/49) \approx 2.546$. Based on these findings, we conjecture that the maximal magic for a bipartite system of two qudits with prime dimension $d$ is given by $\ln [ d^4 / (2d^2 - 1) ]$, which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints.