entropy1 said:
But the states (the four arrows) must represent something right? If the spins (let's assume) of the entangled pair are not defined before measurement, this entangled state does not represent those spins, right? Yet.
Well, it's a bit vaguely formulated, but I guess what's meant is that you consider two spin-1/2 particles.
Each particle's spin states are described in a 2D Hilbert space, spanned by the eigenvectors of one spin component (usually one chooses ##\hat{s}_z##) with eigenvalues ##+1/2## and ##-1/2##. The corresponding eigenstates are often denoted by ##|\uparrow \rangle## and ##|\downarrow \rangle##, which seems to be the notation used in your book. Each spin state can be written as a superposition of these eigenstates,
$$|\psi \rangle=\psi_{\uparrow} |\uparrow \rangle + \psi_{\downarrow} |\downarrow \rangle.$$
Now take two indepdendent particles. Then their spins are described by superpositions of product states ##|\psi_1 \rangle \otimes |\psi_2 \rangle \equiv |\psi_1,\psi_2 \rangle##. Now the general two-spin state can be written in terms of the four basis vectors
$$|\uparrow ,\uparrow \rangle, \quad |\uparrow,\downarrow \rangle, \quad |\downarrow,\uparrow \rangle, \quad |\downarrow,\downarrow \rangle,$$
i.e., the two-spin Hilbert space is four-dimensional. The most general two-spin state thus is given by
$$|\Psi \rangle = \sum_{a,b \in \{\uparrow,\downarrow\}} \Psi_{ab} |a,b \rangle.$$
I don't know, who has invented thus ##\uparrow##-##\downarrow## notation instead of simply using ##|\sigma_z \rangle## with ##\sigma_z \in \{-1/2,1/2 \}##, which is a much more intuitive notation, but that doesn't really matter.
