This paper presents a localized treatment of quantum Rényi entropy. Specifically, based on the localized characteristics of Local Quantum Bernoulli Noises (LQBNs), a new definition of quantum Rényi entropy, that is, quantum Rényi entropy with localization characteristics, is given through the local density operator constructed by local conservative operators \(l_{k}^{\circ } \) . We also verify that this new definition possesses properties such as unitary invariance, additivity, monotonicity, and weak subadditivity. Furthermore, through the monotonicity of the local quantum Rényi entropy, we derive the local quantum Rényi minimum entropy and the local quantum Rényi maximum entropy. The local quantum Rényi entropy can be used to study quantum entanglement and coherence. For instance, in the contexts of quantum phase transitions and quantum state transmission, the local quantum Rényi entropy can provide important insights into the flow of information and interactions within quantum systems.
