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Two-parameter bipartite entanglement measure

Chen-Ming Bai, Yu Luo

TL;DR

We introduce a two-parameter bipartite entanglement measure, the unified $C_{q,s}$-concurrence, to unify and extend existing concurrences. For pure states it is defined by $C_{q,s}(|\psi\rangle)=\varepsilon_{q,s}(1-(\mathrm{Tr}\rho_A^q)^s)$ and extended to mixed states via convex roof; the measure recovers standard concurrence and $q$-concurrence in appropriate limits. We derive analytical lower bounds using PPT and realignment criteria and obtain explicit expressions for isotropic states via a convex envelope $\xi(F,q,s,d)$ and for Werner states via $\xi(\rho_w)=1-\big[(\tfrac{1+G}{2})^q+(\tfrac{1-G}{2})^q\big]^s$, with $G=2\sqrt{w(1-w)}$. The framework also establishes monogamy relations for $q\ge2$, $0\le s\le1$, $1\le qs\le3$ in qubit systems and entanglement polygon inequalities for multipartite qudit systems, providing utility for analyzing entanglement distribution in quantum networks.

Abstract

Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified $(q,s)$-concurrence. Both the standard entanglement concurrence and the recently proposed $q$-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified $(q,s)$-concurrence in the cases of isotropic and Werner states under the constraint $q>1$ and $qs\geq 1$. Furthermore, we explore the monogamy properties of the unified $(q,s)$-concurrence for $q\geq 2$, $0\leq s\leq 1$ and $1\leq qs\leq 3$, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified $(q,s)$-concurrence with $q\geq 1$ and $qs\geq 1$, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.

Two-parameter bipartite entanglement measure

TL;DR

We introduce a two-parameter bipartite entanglement measure, the unified -concurrence, to unify and extend existing concurrences. For pure states it is defined by and extended to mixed states via convex roof; the measure recovers standard concurrence and -concurrence in appropriate limits. We derive analytical lower bounds using PPT and realignment criteria and obtain explicit expressions for isotropic states via a convex envelope and for Werner states via , with . The framework also establishes monogamy relations for , , in qubit systems and entanglement polygon inequalities for multipartite qudit systems, providing utility for analyzing entanglement distribution in quantum networks.

Abstract

Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified -concurrence. Both the standard entanglement concurrence and the recently proposed -concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified -concurrence in the cases of isotropic and Werner states under the constraint and . Furthermore, we explore the monogamy properties of the unified -concurrence for , and , in qubit systems. In addition, we derive an entanglement polygon inequality for the unified -concurrence with and , which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
Paper Structure (10 sections, 9 theorems, 136 equations, 6 figures)

This paper contains 10 sections, 9 theorems, 136 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Parametrized entanglement measures
  3. Establishing lower bounds for the unified $(q,s)$-concurrence
  4. Entanglement inequalities based on the unified $(q,s)$-concurrence in multipartite systems
  5. Monogamy of the unified $(q,s)$-concurrence in multipartite systems
  6. Entanglement polygon inequalities for the unified $(q,s)$-concurrence
  7. Conclusion
  8. Proof of the Lemma \ref{['le:concave']}
  9. Analytical Expression of $\xi(\rho_F)$ for Isotropic States
  10. Analytical Expression of $\xi(\rho_w)$ for Werner States

Key Result

Lemma 2

The function $F_{q,s}(\rho)=\epsilon_{q,s}(1-({\operatorname{Tr}}\rho^q)^s)$ is concave, where (1) Concavity. For any probability distribution $\{p_i\}$ and corresponding density matrices $\rho_i$, the following inequality holds where the equality occurs if and only if all $\rho_i$ are identical for all $p_i > 0$. (2) Subadditivity. For a general bipartite state $\rho_{AB}$, $q\geq 1$ and $qs\g

Figures (6)

  • Figure 1: First and second derivatives of $\xi(F, 2,2, 3)$ with respect to $F$.
  • Figure 2: Entanglement measures for isotropic states: the exact value from Eq.(\ref{['eq:c22function']}) (blue solid); the lower bound from Eq.(\ref{['eq:c22bound']}) (red dashed); and the $C_2$-concurrence from Eq.(\ref{['eq:c2function']}) (green dashed).
  • Figure 3: First and second derivatives of $\xi(\rho_w)$ with respect to $w$.
  • Figure 4: Entanglement measures for Werner states: the exact value from Eq.(\ref{['eq:c32function']}) (blue solid); the lower bound from Eq.(\ref{['eq:c32bound']}) (red dashed); and the $C_3^t$-concurrence from Eq. (\ref{['eq:c3tfunction']}) (green solid).
  • Figure 5: The residual entanglement $\tau_{\hat{C}_{q,s}}=K-K_1 -K_2$ of the normalized unified $(q,s)$-concurrence in Eq.(\ref{['eq:rentanglement1']}).
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Example 6
  • Lemma 7
  • ...and 10 more