Sum of Entanglement and Subsystem Coherence Is Invariant under Quantum Reference Frame Transformations

Recent work on quantum reference frames (QRFs) has demonstrated that superposition and entanglement are properties that change under QRF transformations. Given their utility in quantum information processing, it is important to understand how a mere change of perspective can produce or reduce these resources. Here we prove the existence of a QRF invariant which can be decomposed such that it captures the trade-off between entanglement and subsystem coherence: we demonstrate the invariance of the sum of entanglement and subsystem coherence for two pairs of resource quantifiers. Moreover, we find a weaker trade-off that holds for any possible pair of measures. Finally, we discuss the implications of this interplay for violations of Bell's inequalities, clarifying that for any choice of QRF, there is a quantum resource responsible for the violation. These findings contribute to a better understanding of the quantum information theoretic aspects of QRFs, offering a foundation for future exploration in both quantum theory and quantum gravity.