Quantum correlations and Basis-Independent Coherence Distribution in Two Gravitational Cat States

We study the distribution of quantum correlations and basis-independent coherence in a pair of massive particles confined in a double-well potential and coupled through their mutual Newtonian gravitational interaction. Non-classical correlations are characterized using Bures distance of entanglement and quantum discord, while coherence is quantified through the square root of the quantum Jensen--Shannon divergence (QJSD) from the maximally mixed state, yielding a measure that is invariant under arbitrary unitary transformations and is therefore genuinely basis-independent. The total coherence $C_T$ decomposes into two operationally distinct contributions: the collective coherence $C_C$, which captures quantum correlations between the two subsystems, and the localized coherence $C_L$, which captures the intrinsic quantum coherence of each individual subsystem. We analyze how temperature $T$, the gravitational coupling $\Delta$, and the single-particle energy scale $w$ govern the redistribution of coherence between its collective and localized components. Our results show that $C_L$ is more robust against thermal fluctuations than $C_C$, and that increasing $\Delta$ preferentially enhances collective coherence by strengthening gravitationally induced inter-particle correlations.