Partitewise Entanglement

TL;DR

This work introduces partitewise entanglement (PWE) as a framework to quantify entanglement shared by a fixed subset of parties within an n‑partite system, capturing cases where reduced states may be separable yet still participate in global n‑partite entanglement. It develops three families of k‑PWEMs—GEM-based, minimal-bipartition, and distance-based—each built from reduced-state information and LOCC monotonicity, enabling a resource-theoretic treatment of PWE. The paper also defines and analyzes partitewise entanglement extensibility (PWEE), presenting an extensibility measure E_ext derived from genuine entanglement in purifications and showing maximal extensibility occurs at certain highly entangled extensions. Together, these results reveal how entanglement can be shared and extended in multipartite systems, providing a structured approach to quantify and compare PWE across different states and partitions with potential implications for quantum networks and information processing.

Abstract

It is known that $ρ^{AB}$ as a bipartite reduced state of the 3-qubit GHZ state is separable, but part $A$ and part $B$ indeed ``share tripartite entanglement'' in the GHZ state. Namely, whether a state can ``share'' more entanglement is dependent on the global system it lives in. Here we explore such kind of entanglement in any $n$-partite system with arbitrary dimensions, $n\geqslant3$, and call it partitewise entanglement (PWE) which includes pairwise entanglement (PE) proposed in [Phys. Rev. A 110, 032420(2024)] as a special case. We propose three classes of the partitewise entanglement measures which are based on the genuine entanglement measure, the minimal bipartition, and the minimal distance from the partitewise separable states, respectively. The former two methods are far-ranging since all of them are defined by the reduced function. Consequently, we establish the framework of the resource theory of the partitewise entanglement. In addition, we investigate the partitewise entanglement extensibility and give a measure of such extensibility, and from which we find that the maximal partitewise entanglement extension is its purification. At last, the relation between this extensibility and the partitewise entanglement is discussed.

Figures (4)

• Figure 1: The PE of (a) $|\Psi_1\rangle^{ABC}$, and (b) $|\Psi_2\rangle^{ABC}$. We take $h_C(\rho)=\sqrt{2(1-{\rm Tr}\rho^2)}$, and $C_g^{(3)}(|\psi\rangle^{ABC})=\frac{1}{2}\delta(|\psi\rangle^{ABC})[h_C(\rho^A)+h_C(\rho^B)+h_C(\rho^C)]$. From the plot, we see that $\check{C}^{AB}$, $\check{C}_s^{AB}$, and $\check{C}_{\min}^{AB}$ are inequivalent to each other, where $\check{C}^{AB}(|\psi\rangle^{ABC})=C(\rho^{AB})+C_g^{(3)}(|\psi\rangle^{ABC})$, $\check{C}_s^{AB}(\rho^{AB})=C_g^{(3)}(|\Phi\rangle^{ABC})$, and $\check{C}_{\min}^{AB}(|\psi\rangle^{ABC})=\min\{C(A|BC), C(B|AC)\}$. Case (b) also reveals that the PE entanglement can decrease while the entanglement increases.
• Figure 2: The PE extensibility of $\rho^{AB}=p|\phi\rangle\langle\phi|+(1-p)\frac{I_4}{4}$ with $|\phi\rangle=\sqrt{t}|00\rangle+\sqrt{1-t}|11\rangle$, $0\leqslant p, t\leqslant 1$. Here, $h=h_C$, $C_{\text{ext}}(\rho^{AB})=C_g^{(3)}(|\Phi\rangle^{ABC})$. For any fixed $p<0.5$, $E(AB)\nearrow$ when $E_{\text{ext}}\nearrow$, but for any fixed $0<t<1$, $E(AB)\searrow$ when $E_{\text{ext}}\nearrow$. In addition, $E_{\text{ext}}\nearrow$ whenever $S_L^{AB,A,B}\nearrow$, where $S_L(\rho)=1-{\rm Tr}\rho^2$ denotes the purity of $\rho$.
• Figure 3: The PE extensibility of (a) $\rho^{AB}=p|\Phi^+\rangle\langle\Phi^+|+(1-p)\sigma^{AB}$, $\sigma^{AB}=|\varphi\rangle\langle\varphi|^{A}\otimes|\varphi\rangle\langle\varphi|^{B}$, $|\varphi\rangle^{A}=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, $|\varphi\rangle^{B}=|0\rangle$, and (b) $\rho^{AB}=p|\Phi^+\rangle\langle\Phi^+|+(1-p)\frac{I_4}{4}$. Here, $h=h_C$, $C_{\text{ext}}(\rho^{AB})=C_g^{(3)}(|\Phi\rangle^{ABC})$, $|\Phi^+\rangle=\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$, $p_0=\frac{1}{3}$. In case (a), $E_{\text{ext}}$ is monotonically decreasing convergent to 1 whenever $E(AB)$ is monotonically increasing convergent to 1 for $p>0.681$. But for $p<0.681$, both of them are monotonically increasing. At the same time, $S_L:=\frac{1}{3}(S_L(AB)+S_L(A)+S_L(B))$ is increasing whenever $p<0.7$ and decreasing whenever $p>0.7$. For case (b), the relation is more clear: $\rho^{AB}$ is entangled when $p>\frac{1}{3}$ and $E(AB)$ is monotonically increasing for $p>\frac{1}{3}$, while $S_L$ is monotonically decreasing. In both cases, the shape of line for $E_{\text{ext}}$ and that for $S_L$ keep almost the same variation tendency.
• Figure 4: The pairwise entanglement, entanglement, and the pairwise entanglement extensibility of $|\Psi_3\rangle^{ABC}=\sqrt{p}|\text{GHZ}\rangle+\sqrt{1-p}|W\rangle$ under $\check{C}^{AB}$ and $\mathcal{C}_{A'B'}^2$. $p_0=0.4$.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 2
• Definition 7
• Theorem 1