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

Abstract: 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 $Δ$, 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 $Δ$ preferentially enhances collective coherence by strengthening gravitationally induced inter-particle correlations.

• The paper introduces a basis-independent QJSD coherence decomposition that separates total coherence into collective correlations and localized single-particle superpositions in a thermal two-qubit gravcat model.
• Collective coherence is more sensitive to temperature and increases with gravitational coupling, while localized coherence is more thermally robust and becomes dominant as the single-particle energy scale grows.
• The results show that coherence can persist without entanglement or quantum discord, including at zero local energy splitting, while strong thermal entanglement requires balanced gravitational coupling and local energy scales.

Overview and motivation

The paper studies how quantum coherence is distributed between collective and localized degrees of freedom in a two-particle "gravitational cat" (gravcat) system: two massive particles confined in a one-dimensional double-well potential and coupled only through their mutual Newtonian gravitational interaction (2608.13493). The work builds on the proposal by Bose et al. and Marletto–Vedral that gravitationally induced entanglement between massive particles constitutes evidence for the quantum nature of gravity, and on the thermal gravcat model of Anastopoulos–Hu and Rojas–Lobo. Its contribution is to replace the basis-dependent $\ell_1$-norm of coherence used in earlier analyses with the basis-independent coherence measure of Radhakrishnan et al., based on the square root of the quantum Jensen–Shannon divergence (QJSD) from the maximally mixed state, and to decompose the total coherence $C_T$ into a collective component $C_C$ (correlations between the two particles) and a localized component $C_L$ (single-particle superpositions). The central question is how temperature $T$, gravitational coupling $\Delta$, and single-particle energy scale $w$ govern this redistribution.

Model and formalism

The two-level truncation of each double-well yields an effective two-qubit Hamiltonian

$$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$

where $\Delta = (\alpha/2)(1/d - 1/d')$ with $\alpha = Gm^2$ encodes the anisotropy of the Newtonian potential energy across the four spatial superposition branches. The thermal Gibbs state is X-shaped in the computational basis, with all matrix elements real and $C_T$0 by exchange symmetry. Because the X-form holds, the concurrence, quantum discord (QD), and all QJSD-based coherence quantities admit closed-form expressions without numerical optimization — a significant technical simplification that the paper exploits throughout.

The basis-independent coherence is defined as $C_T$1, where $C_T$2 is the square root of the QJSD. Since $C_T$3 is invariant under all unitaries, the measure is genuinely basis-independent, in contrast to resource-theoretic coherence measures that presuppose a preferred incoherent basis. The collective coherence $C_T$4 is the distance to the closest product state $C_T$5 (diagonal, hence incoherent), and the localized coherence is $C_T$6, depending on $C_T$7 only through the marginal occupation probability $C_T$8.

Thermal entanglement and discord

The Bures-distance entanglement (normalized so that $C_T$9 for maximally entangled states) shows three regimes. At $C_C$0, $C_C$1 grows with $C_C$2 and reaches $C_C$3 for $C_C$4 at $C_C$5, while remaining nearly zero for $C_C$6; for $C_C$7 the ground state approaches a Bell-like state. The threshold temperature for entanglement survival increases with $C_C$8: entanglement persists up to $C_C$9 for $C_L$0 but vanishes at $C_L$1 for $C_L$2. Notably, the dependence on $C_L$3 at fixed $C_L$4 is non-monotonic: $C_L$5 reaches a maximum at a $C_L$6 value that depends on $C_L$7, then decreases. Similarly, at fixed $C_L$8, $C_L$9 exhibits a pronounced maximum near $T$0 at low temperature, and is suppressed both for $T$1 (small gap, enhanced thermal mixing) and $T$2 (separable ground state). Strong thermal entanglement therefore requires a balance between the local splitting and the gravitational coupling.

Quantum discord behaves similarly but decays smoothly and monotonically with temperature, without the sharp finite-temperature vanishing seen for entanglement. A structurally important result is that at $T$3 all quantum correlations vanish despite $T$4: the commutator of the local and interaction terms, $T$5, vanishes, so the non-commutativity that generates correlations is absent. This identifies the competition between non-commuting local and interaction terms — rather than the coupling strength alone — as the source of nontrivial correlations.

Coherence distribution

The decomposition yields a clear physical picture. Total coherence $T$6 decreases monotonically with $T$7 for all $T$8, and at $T$9 takes the value $\Delta$0 for all $\Delta$1 considered at $\Delta$2 (except $\Delta$3, where $\Delta$4). The key findings are:

• Localized coherence is more thermally robust than collective coherence. $\Delta$5 vanishes around $\Delta$6 for $\Delta$7, while larger $\Delta$8 sustains $\Delta$9 to higher temperatures; $w$0 follows a similar decay pattern for all $w$1 and outlasts $w$2. The paper states that $w$3 vanishes at a threshold temperature distinct from, and lower than, that of $w$4, which the authors interpret as evidence that the two components carry independent physical information not accessible to global measures alone.
• Increasing $w$5 preferentially enhances collective coherence. $w$6 grows monotonically with $w$7 and saturates; $w$8 rises from zero at $w$9 and saturates, with a steeper initial slope for smaller $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$0; $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$1 decreases gradually with $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$2 for fixed $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$3. For the representative case $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$4 at $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$5, a crossover occurs around $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$6, beyond which collective coherence exceeds localized coherence as the dominant contribution to $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$7.
• Increasing $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$8 shifts the balance toward localized coherence. At fixed $$\mathcal{H} = \frac{w}{2}(\sigma_z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma_z) - \Delta\,(\sigma_x \otimes \sigma_x),$$9, $\Delta = (\alpha/2)(1/d - 1/d')$0 increases with $\Delta = (\alpha/2)(1/d - 1/d')$1 at all temperatures, while $\Delta = (\alpha/2)(1/d - 1/d')$2 at $\Delta = (\alpha/2)(1/d - 1/d')$3 is highest for the smallest $\Delta = (\alpha/2)(1/d - 1/d')$4. The local energy gap protects single-particle superpositions against thermal fluctuations, whereas a small $\Delta = (\alpha/2)(1/d - 1/d')$5 relative to $\Delta = (\alpha/2)(1/d - 1/d')$6 facilitates gravitationally induced global superpositions.

A comparison across measures at $\Delta = (\alpha/2)(1/d - 1/d')$7 shows that entanglement ($\Delta = (\alpha/2)(1/d - 1/d')$8) requires $\Delta = (\alpha/2)(1/d - 1/d')$9, whereas collective coherence and QD are generated as soon as $\alpha = Gm^2$0; entanglement thus demands a stronger gravitational coupling to overcome thermal mixing. Conversely, at $\alpha = Gm^2$1 the state remains coherent ($\alpha = Gm^2$2, entirely collective) while both entanglement and QD vanish, demonstrating that coherence and quantum correlations are genuinely distinct resources in this system.

Limitations and open questions

The analysis is confined to the two-level, weak-coupling regime justified by the WKB approximation, and to thermal equilibrium states; no dynamical evolution or decoherence channel is treated, so the reported thresholds are equilibrium quantities rather than operationally timed survival times. The metric character of $\alpha = Gm^2$3 for mixed states is conjectured and numerically supported rather than proven, which the paper inherits from the underlying literature. The claim that the distinct vanishing temperatures of $\alpha = Gm^2$4 and $\alpha = Gm^2$5 reflect "independent physical information" is asserted from the observed separation of thresholds; no operational task distinguishing the two components is provided. The crossover value $\alpha = Gm^2$6 and the maximum near $\alpha = Gm^2$7 are established numerically for specific parameter choices ($\alpha = Gm^2$8, $\alpha = Gm^2$9), and their dependence on the mass, separation $C_T$00, and superposition size $C_T$01 is not explored. The authors themselves leave open the effect of correlated dephasing channels on the $C_T$02/$C_T$03 redistribution and the extension to multipartite gravcat systems.

Conclusion

This paper applies the basis-independent QJSD coherence framework to the thermal gravcat model and provides a decomposition of total coherence into gravitationally induced collective and single-particle localized parts. Its main quantitative results are the higher thermal robustness of localized coherence relative to collective coherence, the selective amplification of collective coherence by the gravitational coupling with a crossover near $C_T$04 at representative parameters, and the demonstration that coherence persists in regimes (e.g., $C_T$05, $C_T$06) where entanglement and discord vanish identically. The work supplies a coherence-theoretic counterpart to the known fragility hierarchy of gravcat resources and identifies parameter regimes in which the gravitationally generated component of coherence dominates.