Equalities and inequalities from entanglement, loss, and beam splitters

Abstract Quantum optics connects fundamental phenomena such as entanglement and superposition with practical applications like metrology and communication. Throughout, there is an interplay between information theoretic concepts such as entropy and physical considerations such as quantum system design, noise, and loss. Therefore, a result at the heart of these fields can have numerous ramifications in development of applications and advancing our understanding of quantum physics. Our recent proof for the entanglement properties of states interfering with the vacuum on a beam splitter led to monotonicity and convexity properties for quantum states undergoing photon loss [Lupu-Gladstein et al., PRA 112 , 033722 (2025)] by breathing life into a decades-old conjecture. In this work, we extend these fundamental properties to measures of similarity between states, provide inequalities for creation and annihilation operators beyond the Cauchy–Bunyakovsky–Schwarz inequality, prove a conjecture [Hertz et al., PRA 110 , 012408 (2024)] dictating that nonclassicality through the quadrature coherence scale is uncertifiable beyond a loss of 50%, place constraints on quasiprobability distributions of all physical states, and explore entanglement generation optimized over all input states and linear optical networks. These ideas can now be used throughout quantum optics.