Generalized wave-particle-entanglement triality

¹ School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University - Shanghai 200240, China
² School of Computer Science and Engineering, Sun Yat-sen University - Guangzhou 510006, China
³ School of Information Science and Technology, Southwest Jiaotong University - Chengdu 610031, China
⁴ Hefei National Laboratory, University of Science and Technology of China - Hefei 230026, China
⁵ School of Mathematical Sciences, Capital Normal University - Beijing 100048, China
⁶ Max-Planck-Institute for Mathematics in the Sciences - 04103 Leipzig, Germany
⁷ Shanghai Seres Information Technology Co., Ltd - Shanghai 200040, China
⁸ Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology Shenzhen 518055, China

Received: 19 September 2025
Accepted: 8 April 2026

Abstract

Photon's dual nature, manifesting as both wave-like and particle-like behavior, is a phenomenon known as wave-particle duality and remains one of the most perplexing mysteries in quantum mechanics. In this work, we explore the relationships among the wave behavior, particle behavior, and entanglement of quantum states. We show the strong complementary relationships for bipartite isolated systems in terms of the generalized quantum entropic measures. This provides a measure-independent result going beyond all the existing complementary results based on specific measures. We further extend the results for non-isolated systems.