# Why are slice states called so?

The "slice" states are defined by equation 50 in this (freely available) article. I wonder why they call them so!

## 2 Answers

I will hazard a guess that there is a typo present. The given set of states have $$T_1=\begin{pmatrix}p&0\\0&0\end{pmatrix},\quad T_2=\begin{pmatrix}ac&ad\\bc&bd\end{pmatrix}=\begin{pmatrix}a\\b\end{pmatrix}\begin{pmatrix}c&d\end{pmatrix}.$$ The authors define slice states as having $$a=0$$ or $$b=0$$ with $$bd\neq 0$$, which makes only a slice of T_2 be nonzero: either the top row or the bottom row. But they have the requirement $$bd\neq 0$$, so the bottom row is never zero, so that slice interpretation falls away. I would suggest the requirement $$a=0$$ or $$c=0$$ with $$bd\neq 0$$ to define slice states. Then $$T_2$$ would either have a single vertical slice nonzero (when $$c=0$$) or a single horizontal slice nonzero (when $$a=0$$). This works because in (50) the two options have $$a=0$$ or $$c=0$$ (neither option would be true if $$b=0$$). This also helps motivate their subsequent Case 3, which generalizes the slice states to when both $$a$$ and $$c$$ are zero. Notably, the slice must always include $$bd\neq 0$$ and $$ac=0$$ so that it differs from $$T_1$$.

There may be a more geometrical answer explaining to what the slices refer, but I strongly suspect that the typo I mentioned is necessary to correct.

My guess would simply be that they're visualising the 4-dimensional parameter space, and they're thinking that fixing one parameter, say c=0, is like looking at a (hyper)-plane through that parameter space: a slice.