Partitewise entanglement

It is known that ${\ensuremath{\rho}}^{AB}$ as a bipartite reduced state of the three-qubit Greenberger-Horne-Zeilinger (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, $n\ensuremath{\geqslant}3$, and call it partitewise entanglement, which includes pairwise entanglement proposed by Dong et al. [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. [Note: If a positive function $h$ satisfies $h({\ensuremath{\rho}}^{A})=E(|\ensuremath{\psi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}{|}^{AB})$ for some bipartite entanglement measure $E, h$ is called the reduced function (of $E$).] 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, from which we find that the maximal partitewise entanglement extension is its purification. Lastly, the relation between this extensibility and the partitewise entanglement is discussed.