Entanglement, loss, and quantumness: When balanced beam splitters are best

Quantum optics routinely uses beam splitters to generate entanglement, including in pioneering experiments conducted by Hanbury-Brown and Twiss and Hong, Ou, and Mandel. The quantum interference at beam splitters lies at the heart of what makes boson sampling hard to emulate by classical computers and is a vital component of quantum computation with light. Yet, despite overwhelming positive evidence, the conjecture that beam splitters with equal reflection and transmission probabilities generate the most entanglement for any state interfered with the vacuum has remained unproven for almost two decades [Asb\'oth et al., Phys. Rev. Lett. 94, 173602 (2005)]. We prove this conjecture for ubiquitous entanglement monotones including mixed-state generalizations of entanglement entropy and purity by uncovering monotonicity and convexity with respect to photon loss for these monotones. At the same time, we highlight an infinite class of lesser-used monotones for which the conjecture fails. Because beam splitters are so fundamental, our results yield numerous corollaries for quantum optics, including proof of a recent conjecture for the evolution of a measure of quantumness through loss and a more efficient computational strategy for optimizing entanglement generation over linear optics. These results justify the value of seeking mathematical rigour behind commonly accepted facts and the danger of trusting them unconditionally.