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The Pareto Frontiers of Magic and Entanglement: The Case of Two Qubits

Alexander Roman, Marco Knipfer, Jogi Suda Neto, Konstantin T. Matchev, Katia Matcheva, Sergei Gleyzer

Abstract

Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, $M_2$, and entanglement by the concurrence $Δ$. We find that the Pareto frontier of maximal magic $M_2^{(max)}(Δ)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(Δ)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.

The Pareto Frontiers of Magic and Entanglement: The Case of Two Qubits

Abstract

Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, , and entanglement by the concurrence . We find that the Pareto frontier of maximal magic is composed of three separate segments, while the boundary of minimal magic is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.

Paper Structure

This paper contains 12 sections, 34 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Conventions and Notation
  3. Parametrization of two-qubit states
  4. Measures of entanglement
  5. Measures of magic
  6. Lower Pareto Frontier: Minimally Magical Entangled States
  7. Upper Pareto Frontier: Maximally Magical Entangled States
  8. The middle branch GFE
  9. The left branch IHG
  10. The right branch ED
  11. Summary and Discussion
  12. Catalog of the States on the IHG Boundary

Figures (8)

  • Figure 1: 2D histogram of the $|\psi\rangle$ number density in the $(\Delta, M_2)$ plane. The distribution was generated by sampling 50 million $|\psi\rangle$ states according to the Haar measure and binned into a $1000 \times 1000$ grid. Points A through I mark interesting features discussed in the text. Colored lines represent the analytic predictions from eqs. (\ref{['eq:M2min_delta']}-\ref{['eq:f_functions']}) derived in this paper. Dashed lines are extensions of those lines into the bulk.
  • Figure 2: The 16 individual matrix element values (\ref{['eq:MABs']}) contributing to the sum in (\ref{['eq:M2def']}) for 16 randomly chosen states in the bulk (left panel) or on the lower Pareto boundary ABC (right panel).
  • Figure 3: The 18 patterns of expectation values $\langle\psi\vert P_1\otimes P_2\vert \psi \rangle$ for the states on the lower Pareto boundary ABC (excluding points A and C). White cells identify the expectation values which are identically zero. The patterns are labelled with python's zero-based indexing convention, starting from 0.
  • Figure 4: The 9 patterns for any state on the GFE boundary, including point F.
  • Figure 5: The six patterns for the states at point H on the IHG boundary.
  • ...and 3 more figures