Partitewise Entanglement

It is known that $ρ^{AB}$ as a bipartite reduced state of the 3-qubit GHZ state is separable, but part $A$ and part $B$ are indeed entangled with each other without tracing out part $C$. Namely, whether a state containing entanglement is always dependent on the global system it lives in. We explore here such entanglement in any $n$-partite system with arbitrary dimensions, $n\geqslant3$, and call it partitewise entanglement (PWE) which contains pairwise entanglement (PE) proposed in [Phys. Rev. A 110, 032420(2024)] as a special case. In particular, we investigate the partitewise entanglement extensibility and give a measure of such extensibility, and from which we find that the reduced state of a genuine entangled state could be any mixed state without pure reduced states, and that the maximal partitewise entanglement extension is its purification. We then propose three classes of the partitewise entanglement measures which are based on the genuine entanglement measure, minimal bipartition measure, 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 functions. Consequently, we establish the framework of the resource theory of the partitewise entanglement and the genuine partitewise entanglement for which the free states are the states with pure reduced states.