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Maximally entangled real states and SLOCC invariants: the 3-qutrit case

Hamza Jaffali, Frédéric Holweck, Luke Oeding

TL;DR

The paper investigates entanglement in real $3$-qutrit systems by analyzing the absolute values of SLOCC invariants, notably the hyperdeterminant $\Delta_{333}$ and the fundamental invariants $I_6,I_9,I_{12}$. Using both numerical optimization and symbolic methods, it identifies real states that maximize $|\Delta_{333}|$, finds the global maximum $\sqrt{3}/(2^{19}3^{14})$, and proves that the Aharonov state $\ket{\mathcal{A}}$ simultaneously maximizes all three fundamental invariants. It provides a detailed enumeration of real semiclassical maximizers, demonstrates that the Aharonov state is a critical point for the invariants, and develops practical, Lie-algebra–based techniques to evaluate invariants on any $3$-qutrit state. The work combines algebraic geometry, computational invariant theory, and random-state statistics to illuminate the landscape of entanglement in $3$-qutrits and suggests avenues for quantum-information applications and extensions to higher dimensions.

Abstract

The absolute values of polynomial SLOCC invariants (which always vanish on separable states) can be seen as measures of entanglement. We study the case of real 3-qutrit systems and discover a new set of maximally entangled states (from the point of view of maximizing the hyperdeterminant). We also study the basic fundamental invariants and find real 3-qutrit states that maximize their absolute values. It is notable that the Aharonov state is a simultaneous maximizer for all 3 fundamental invariants. We also study the evaluation of these invariants on random real 3-qutrit systems and analyze their behavior using histograms and level-set plots. Finally, we show how to evaluate these invariants on any 3-qutrit state using basic matrix operations.

Maximally entangled real states and SLOCC invariants: the 3-qutrit case

TL;DR

The paper investigates entanglement in real -qutrit systems by analyzing the absolute values of SLOCC invariants, notably the hyperdeterminant and the fundamental invariants . Using both numerical optimization and symbolic methods, it identifies real states that maximize , finds the global maximum , and proves that the Aharonov state simultaneously maximizes all three fundamental invariants. It provides a detailed enumeration of real semiclassical maximizers, demonstrates that the Aharonov state is a critical point for the invariants, and develops practical, Lie-algebra–based techniques to evaluate invariants on any -qutrit state. The work combines algebraic geometry, computational invariant theory, and random-state statistics to illuminate the landscape of entanglement in -qutrits and suggests avenues for quantum-information applications and extensions to higher dimensions.

Abstract

The absolute values of polynomial SLOCC invariants (which always vanish on separable states) can be seen as measures of entanglement. We study the case of real 3-qutrit systems and discover a new set of maximally entangled states (from the point of view of maximizing the hyperdeterminant). We also study the basic fundamental invariants and find real 3-qutrit states that maximize their absolute values. It is notable that the Aharonov state is a simultaneous maximizer for all 3 fundamental invariants. We also study the evaluation of these invariants on random real 3-qutrit systems and analyze their behavior using histograms and level-set plots. Finally, we show how to evaluate these invariants on any 3-qutrit state using basic matrix operations.
Paper Structure (17 sections, 2 theorems, 38 equations, 6 figures)

This paper contains 17 sections, 2 theorems, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Previous Works
  3. The Hyperdeterminant
  4. Classification of SLOCC Orbits
  5. Orbit Classification Over
  6. Real Semi-simple Elements
  7. Expressing the Hyperdeterminant
  8. New Maximally Entangled 3-qutrit States
  9. Maximizing the Fundamental Invariants
  10. Evaluating Invariants on Random States
  11. Random States and
  12. Random States and
  13. Random States and
  14. Random states and
  15. Combination of
  16. ...and 2 more sections

Key Result

Theorem 3.1

The global maximum of the absolute value of the hyperdeterminant $|\Delta_{333}|$, when restricted to real states is $\frac{\sqrt{3}}{2^{19}\times 3^{14}}$. The global max is reached at 12 semi-simple points $a\ket{v_1} + b\ket {v_2} + c\ket{v_3}$ with the following values and their permutations:

Figures (6)

  • Figure 1:
  • Figure 2: A plot of the level sets of the invariants $I_6$ (top left, (A)), $I_9$ (top right, (B)), $I_{12}$ (bottom left, (C)), $\Delta_{333}$ (bottom right, (D)) on the sphere of semi-simple states. The solid curves represent the zero-sets of the invariants, the small circles indicate multiple roots. The line colors of Yellow, Blue, Green, Black respectively correspond to the level sets $I_6=0$, $I_9=0$, $I_{12}=0$, $\Delta_{333} = 0$. The other point values are according to the color scheme above.
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3