Entangled states: notation

Here we discuss entangled states in more detail, setting up nation that will help us understand the information paradox

Existence of entangled states :   Entangled states are very common in quantum theory. Here are some examples:

(a) Vacuum fluctuations produce particle pairs which are typically in an entangled state, as discussed above.

(b) The hydrogen atom is composed of a proton and an electron. The proton has a spin which can be in two possible states; these states are typically called spin 'up' and spin 'down'. Similarly, the electron has a spin which can also be 'up' or 'down'.

The lowest energy state for the hydrogen atom is one where the two spins are in an entangled state: if the proton spin is 'up', then the electron spin is 'down', and if the proton spin is 'down', then the electron spin is 'up'.

(c) Suppose we start with two objects, and assume that their state is not entangled. Now let the bodies interact. For most kinds of interactions, the resulting state ends up being an entangled state. Since most things around us (like the molecules of air) keep interacting with each other, entanglement is very ubiquitious.

Implications for the black hole problem:   Recall that vacuum fluctuations near the horizon had lead to the creation of particle pairs