Analytical Landscape of Maximal Magic for Two Qutrit States and Beyond

Jul 16, 2026

Upload by Quinn

Achieving quantum advantage requires two non-classical resources that hinder efficient classical simulation: entanglement and magic (nonstabilizerness). The work analyzes their interplay by deriving Pareto frontiers of extreme magic at fixed entanglement for two qutrits (d=3) and two ququints (d=5). For qutrits, the two-dimensional Pareto surfaces arise from two entanglement parameters. Lower and upper frontiers yield compact formulas for minimal magic and a tightened maximal stabilizer Rényi entropy bound, with discrete maximally magical states identified.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages34

Reading Time1 h 26 min

Abstract

Introduction

Conventions and Notations for Two-Qutrit States

Two-Qutrit State Parametrization

Measures of Entanglement

Measures of Magic

Optimization Setup

Plotting Conventions

Results for Two-Qutrit States

The Lower Pareto Frontier: States with Minimal Magic

The Upper Pareto Frontier: States with Maximal Magic

Validity Range of the Analytical Formulas for Maximal Magic

Generalization Beyond Qutrits

A System of Two Ququints

Conclusions

Acknowledgments

Numerical optimization

References

Analytical Landscape of Maximal Magic for Two-Qutrit States and Beyond Marco Knipfer, Alexander Roman, Katia Matcheva, and Konstantin T. Matchev Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL USA (Dated: July 9, 2026) arXiv :2607 .07197v1 [ quant-ph] 8 Jul 2026 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 ln2 . For the upper frontier, we determine the maximal stabilizer R´enyi entropy to be M2 = ln(81/17) ≈ 1.561, which tightens the previous theoretical bound of ln5 ≈ 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 turnout 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 M2 = ln(625/49) ≈ 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[d4 /(2d2 − 1)], which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints. CONTENTS I. Introduction 3 II. Conventions and Notations for Two-Qutrit States 6 A. Two-Qutrit State Parametrization 6 B. Measures of Entanglement 7 C. Measures of Magic 9 D. Optimization Setup 10 E. Plotting Conventions 11 III. Results for Two-Qutrit States 13 A. The Lower Pareto Frontier: States with Minimal Magic 13 B. The Upper Pareto Frontier: States with Maximal Magic 15 1. The first maximum: G3 = 1/3 17 2. The second maximum: G = 0 19 3. The third maximum: G3 = 1/9 20 C. Validity Range of the Analytical Formulas for Maximal Magic 23 IV. Generalization Beyond Qutrits 26 A. A System of Two Ququints 27 V. Conclusions 28 Acknowledgments 30 A. Numerical optimization 30 References 32 I. INTRODUCTION The difficulty of simulating a quantum system on classical hardware is governed by various quantum resources, including entanglement and magic [1] . States with low entanglement admit an efficient classical description in terms of tensor networks, irrespective of their magic content. Conversely, by the Gottesman–Knill theorem [2–4], states with low magic can be simulated efficiently with stabilizer techniques, irrespective of their entanglement. This observation led Bravyi and Kitaev to promote magic to a quantum resource in its own right, which quantifies the overhead of classical simulation [5] . Genuine quantum advantage therefore requires states that are simultaneously rich in both resources. Rather than attempting to survey the large and rapidly growing literature on entanglement and magic as quantum resources, we refer the reader to the recent comprehensive review [6] . While each resource is by now well understood in isolation, their interplay is much less explored [7–12] . The two resources are not independent: part of the magic of a state resides in the local degrees of freedom, where it can be freely created or removed by local unitary operations, while the remainder, the non-local magic, is inseparably linked to entanglement [6] . As a

FAQs

Why do entanglement and magic both matter for quantum advantage?

Classical simulation efficiency is limited by having both resources simultaneously. Low entanglement can be efficiently described by tensor networks, while low magic can be efficiently simulated by stabilizer techniques, so genuine advantage needs strong presence of both.

What distinguishes the two-qutrit entanglement–magic Pareto landscape from two qubits?

For two qutrits, the Schmidt spectrum has two independent entanglement parameters, producing two-dimensional Pareto surfaces rather than the single continuous curve found for two qubits’ minimal-magic frontier.

What are the main results for maximal magic in the two-qutrit and two-ququint cases?

For two qutrits, the maximal stabilizer Rényi entropy is M2 = ln(81/17) ≈ 1.561, with eighteen permutation-equivalent maxima grouped into three families. For two ququints (d=5), numerical analysis yields six permutation-inequivalent maxima with peak value M2 = ln(625/49) ≈ 2.546 and motivates a prime-d conjectured formula.

Jul 16, 2026

Upload by Quinn