Dive into the first lecture of a series on entanglement in quantum field theory and quantum gravity, delivered by Tom Hartman at the Kavli Asian Winter School on Strings, Particles and Cosmology. Explore fundamental concepts such as Euclidean path integrals, quantum states defined by cut path integrals, and operators defined by path integrals with two cuts. Examine quantum entropy, coarse-grained entropy, and their applications to Unruh radiation and thermofield double states. Investigate entanglement entropy in 2D conformal field theory using the replica method. Engage with multiple Q&A sessions throughout the lecture, providing opportunities for deeper understanding and clarification of complex topics in theoretical physics.