A new entanglement measure based on the total concurrence

TL;DR

The paper proposes a new entanglement measure, the $\mathcal{C}^{t}_q$-concurrence, derived from the $q$-concurrence and its complementary dual, and normalized via a convex-roof extension. It proves the measure is a bona fide entanglement monotone by establishing the concavity of the associated entropy functional and LOCC nonincreasability, and it provides analytical lower bounds via PPT and realignment criteria. Closed-form expressions are obtained for isotropic and Werner states, with explicit tightness results in several low-dimensional cases. The work further analyzes monogamy relations in multipartite systems and introduces the $\mathcal{C}^{t}_\alpha$ variant, outlining future directions for assisted and polygamy properties. Overall, the framework yields a refined, parameterized characterization of bipartite entanglement with operational links to standard criteria and well-known state families.

Abstract

Quantum entanglement is a crucial resource in quantum information processing, advancing quantum technologies. The greater the uncertainty in subsystems' pure states, the stronger the quantum entanglement between them. From the dual form of $q$-concurrence ($q\geq 2$) we introduce the total concurrence. A bona fide measure of quantum entanglement is introduced, the $\mathcal{C}^{t}_q$-concurrence ($q \geq 2$), which is based on the total concurrence. Analytical lower bounds for the $\mathcal{C}^{t}_q$-concurrence are derived. In addition, an analytical expression is derived for the $\mathcal{C}^{t}_q$-concurrence in the cases of isotropic and Werner states. Furthermore, the monogamy relations that the $\mathcal{C}^{t}_q$-concurrence satisfies for qubit systems are examined. Additionally, based on the parameterized $α$-concurrence and its complementary dual, the $\mathcal{C}^{t}_α$-concurrence $(0\leqα\leq\frac{1}{2})$ is also proposed.

Figures (12)

• Figure 1: A comparison between the $2$-concurrence $C_2(Y)$ and the total concurrence $C^{t}_2$ as a function of the probability distribution $\{p_i\}$ is presented. The results indicate that the total concurrence $C^{t}_2(Y)$ (light blue) consistently exceeds the $2$-concurrence $C_2(Y)$ (blue), highlighting its ability to capture a greater degree of uncertainty by incorporating both the direct and complementary probability distributions.
• Figure 2: The plot illustrates the first (solid black line) and second (dashed green line) derivatives of $\zeta\left(\mathcal{F},3,3\right)$ with respect to $\mathcal{F}$. The solid black line represents the first derivative, while the dashed green line shows the second derivative of the function.
• Figure 3: The plot shows $\mathcal{C}^{t}_3\left(\varrho_\mathcal{F}\right)$ versus $\mathcal{F}$ for $d=3$. The solid black line represents the exact value given by equation (\ref{['c333']}), while the dashed blue line represents the lower bound given by equation (\ref{['exc33']}).
• Figure 4: The plot of $\mathcal{C}^{t}_4\left(\varrho_\mathcal{F}\right)$ versus $\mathcal{F}$ for $d=2$ is as follows: the solid black line represents the expression in (\ref{['example5']}), while the dashed blue line corresponds to the lower bound given by (\ref{['example42']}).
• Figure 5: The first (solid black line) and second (dashed green line) derivatives of $\zeta\left(\mathcal{F},4,3\right)$ with respect to $\mathcal{F}$ are shown in the plot. The solid black line represents the first derivative, while the dashed green line represents the second derivative.

Theorems & Definitions (8)

• Lemma 1
• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Corollary 2