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Classification of three-qubit genuine entangled states using concurrence fill

Shruti Aggarwal

TL;DR

This paper addresses the challenge of reliably detecting and classifying genuine three-qubit entanglement, where standard measures can fail, by employing concurrence fill, a geometric measure based on the area of a concurrence triangle. It reformulates concurrence fill for pure three-qubit states in terms of the three-tangle and partial tangles, enabling an operational criterion to distinguish GHZ and W classes. The authors then analyze a rank-2 mixture of generalized GHZ and W states, deriving explicit expressions for the concurrence fill of eigenstates, a lower bound, and an upper bound for zero-tangle mixed states using convex-roof ideas. These results provide a practical diagnostic for GME and extend geometric entanglement characterization to mixed three-qubit states with quantitative bounds.

Abstract

Despite the successful experimental generation and verification of genuine multipartite entanglement, several existing entanglement measures remain insufficient to reliably capture its presence. In this study, we overcome this challenge by utilizing a geometric measure known as concurrence fill, which quantifies genuine multipartite entanglement of pure states using the area associated with the underlying concurrence triangle. Firstly, the concurrence fill for three-qubit pure states is reformulated using three tangle and partial tangles. This yields a representation that is more tractable and operational. Using this formulation, we derive a criterion to classify GHZ and W class of states. Next, we analyse the rank-2 mixture of generalized GHZ and W class of states for which three-tangle is known to vanish over a zero polytope. Furthermore, we derive the concurrence fill for the eigenstates of the corresponding mixture and obtain its upper bound for mixed states with zero tangle.

Classification of three-qubit genuine entangled states using concurrence fill

TL;DR

This paper addresses the challenge of reliably detecting and classifying genuine three-qubit entanglement, where standard measures can fail, by employing concurrence fill, a geometric measure based on the area of a concurrence triangle. It reformulates concurrence fill for pure three-qubit states in terms of the three-tangle and partial tangles, enabling an operational criterion to distinguish GHZ and W classes. The authors then analyze a rank-2 mixture of generalized GHZ and W states, deriving explicit expressions for the concurrence fill of eigenstates, a lower bound, and an upper bound for zero-tangle mixed states using convex-roof ideas. These results provide a practical diagnostic for GME and extend geometric entanglement characterization to mixed three-qubit states with quantitative bounds.

Abstract

Despite the successful experimental generation and verification of genuine multipartite entanglement, several existing entanglement measures remain insufficient to reliably capture its presence. In this study, we overcome this challenge by utilizing a geometric measure known as concurrence fill, which quantifies genuine multipartite entanglement of pure states using the area associated with the underlying concurrence triangle. Firstly, the concurrence fill for three-qubit pure states is reformulated using three tangle and partial tangles. This yields a representation that is more tractable and operational. Using this formulation, we derive a criterion to classify GHZ and W class of states. Next, we analyse the rank-2 mixture of generalized GHZ and W class of states for which three-tangle is known to vanish over a zero polytope. Furthermore, we derive the concurrence fill for the eigenstates of the corresponding mixture and obtain its upper bound for mixed states with zero tangle.

Paper Structure

This paper contains 14 sections, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement Measures
  3. Concurrence
  4. Tangle
  5. Partial Tangle
  6. Concurrence fill
  7. Classification of GHZ and W-class of states
  8. Concurrence Fill for a Rank-two Mixture of states
  9. Concurrence fill for eigenstates of $\rho(p)$
  10. Concurrence fill for mixed states
  11. Conclusion
  12. Acknowledgement
  13. DATA AVAILABILITY STATEMENT
  14. Appendix

Figures (3)

  • Figure 1: The above figure shows the concurrence triangle for the three-qubits A, B, and C. The edges of the triangle represents the squared concurrences between bipartitions.
  • Figure 4: The solid curves represent the concurrence fill of the state $|\psi_{p,\phi}\rangle$ given in (\ref{['psi']}) for $a=b=\frac{1}{\sqrt{2}}$; $c=d=f=\frac{1}{\sqrt{3}}$ and various values of $\phi$ with $0 \leq p \leq 1$.
  • Figure 5: The blue curve represents the upper bound of the concurrence fill given in (\ref{['Cub']}) for the zero-tangle states in the generalized mixture $\rho(p)$ with $0 \leq p \leq p_0$.

Theorems & Definitions (5)

  • proof
  • proof
  • Example 1
  • proof
  • Example 2