I'd like to understand the math of how something specific in the environment (like photons) can be said to be entangled with a system (like dust particles). Im reading a newbie friendly intro to superposition and entanglement "Home of the Wave Function"... I'd like to visualize entanglement using purely Schroedinger wave equations. I read that in entanglement (is what follows right)? "We have already discussed the linear superposition of different quantum waves for the same system. But supposed we have two or more systems. Two or more quantum systems can share the same quantum wave. When this happens, we say that the systems are entangled with each other. The separate systems no longer have quantum waves of their own. The process "quantum connects" the two systems no matter how far apart they get from each other, provided that they are protected against collitions with other particles. Getting to Hilbert space. I know the Schroedinger Wave is a vector in Hilbert space.. but in the following passage where vanesh mentioned: "In the case that a quantum state of a multi-system is NOT one of those very special states which are "factorisable" (psi(p,q) = f(p) g(q)), then we say that the quantum state is an ENTANGLED state of the subsystems." What is meant by factorisable and what does this mean psi(p,q) = f(p) g(q)? what is f and g? I know vectors and components axis in Hilbert space... What is the equivalent of factorisable in pure Schroedinger wave equation? Can you give an actual example of how an actual specific thing in the environment can be entangled with a system in decoherence? like photon with dust? How would this form entanglement where it is not factorisable?