Analytical Landscape of Maximal Magic for Two-Qutrit States and Beyond

2026-07-08

AI Summary

This work investigates the trade-off between entanglement and magic (non-stabilizerness) in high-dimensional bipartite quantum systems, focusing on the extremal boundary of magic for fixed entanglement. By leveraging Schmidt decomposition, stabilizer Rényi entropy, Weyl–Heisenberg covariance analysis, numerical optimization, and mutually unbiased bases theory, the authors derive—for the first time—a tight analytical expression for the maximal magic state in two-qutrit systems. They identify 18 distinct maxima, which fall into three equivalence classes, and generalize these findings to propose a universal formula applicable to any prime-dimensional bipartite system. The maximal magic values are found to be ln(81/17) ≈ 1.561 for two qutrits and ln(625/49) ≈ 2.546 for two ququints, with the formula validated across qubit, qutrit, and ququint cases.

Abstract

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.

Problem

• magic
• entanglement
• qutrit
• Pareto frontier
• nonstabilizerness

Innovation

• magic
• entanglement
• Pareto frontier
• stabilizer Rényi entropy
• Weyl-Heisenberg covariance