Some examples of magic states (these can all be found here): the $ |T\rangle $ state for implementing the $ T $ gate is $$ T | + \rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{i\pi/4}|1\rangle) $$
A $ |\text{CS}\rangle $ state for implementing the $ CS $ gate is $$ CS | + \rangle^{\otimes 2} = \frac{1}{2} (|00\rangle + |01\rangle + |10\rangle +i |11\rangle) $$
A $ |\text{CCZ}\rangle $ state for implementing the $ CCZ $ gate is $$ CCZ | + \rangle^{\otimes 3} \\ = \frac{1}{\sqrt{8}}(|000\rangle + |001\rangle + |010\rangle + |011\rangle + |100\rangle + |101\rangle + |110\rangle - |111\rangle ) $$
A $|\text{Toffoli}\rangle $ state for implementing the Toffoli (also known as $ CCX $) gate is $$ CCX | + \rangle^{\otimes 2} | 0 \rangle = \frac{1}{2} (|000\rangle + |100\rangle + |010\rangle + |111\rangle) $$ It is well known that a magic state is always a non-stabilizer state. And indeed this is true for the above examples.
But is it true that every non-stabilizer state, for example the $ W $ state, is a magic state?
