This lecture, "Tensor Norms for Quantum Entanglement (Lecture 1)" delivered by Ion Nechita at the International Centre for Theoretical Sciences, explores fundamental concepts in quantum information theory as part of the "Quantum Trajectories" program. Dive into the mathematical foundations of quantum entanglement through the lens of tensor norms during this 1-hour 17-minute presentation. The lecture contributes to a broader program addressing central theoretical questions in quantum-based technologies, including quantum trajectories, measurement-induced phase transitions, and quantum information and computation. Organized by Michel Bauer, Cedric Bernardin, Raphael Chetrite, and Abhishek Dhar, this session represents the collaborative efforts of theoretical physicists and mathematicians working toward the "second quantum revolution" in which quantum encryption, communication, and computing become practical realities.