We have two places, call them Here and There where a particle could be. When the particle isn’t spatially superposed we write its state as $ |1\rangle_{Here}\otimes|0\rangle_{There}$ zero being the vacuum mode.
My question is why when the particle is superposed, for example in the state $$\frac{1}{\sqrt{2}}(|Here\rangle + |There\rangle)$$ if we want to write its state including the vacuum modes we write it like this $$\frac{1}{\sqrt{2}}(|1\rangle_{Here}\otimes|0\rangle_{There} + |0\rangle_{Here}\otimes|1\rangle_{There})$$ which looks like entangled form. And we don’t write it in some separable form in analogy with the case in first equation: $$\frac{|1\rangle_{Here}+|1\rangle_{There}}{\sqrt{2}}\otimes\frac{|0\rangle_{Here}-|0\rangle_{There}}{\sqrt{2}}.$$ Or even $$|1\rangle_{\frac{H+T}{\sqrt{2}}}\otimes|0\rangle_{\frac{H-T}{\sqrt{2}}}.$$
Superpositional +/- states are as legitimate basis as localized ones, but vacuum modes notation seems to suggest there is difference - one state is separable, the other - entangled. I was thinking that this happens because the creation and annihilation operators are defined for local points like Here, but then again, why not define them for $\frac{H+T}{\sqrt{2}}$.