Symmetric inseparability and entanglement of charge coherence in number conserving mixed states

We explore sufficient conditions for inseparability in mixed states with a globally conserved charge. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state can not be separated into charge conserving components. As a witness of symmetric inseparability we study the entanglement of charge coherence (ECC), $\Delta S_m$, defined as the entropy change due to a subsystem's charge measurement. Whenever $\Delta S_m > 0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state is either separable or has vanishing negativity. We demonstrate that the ECC is not only a witness of symmetric inseparability, but also an entanglement monotone in systems with a conserved charge. Finally, we study the scaling of $\Delta S_m$ in thermal 1D systems combining high temperature expansion and conformal field theory.