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

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2 Answers 2

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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.

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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.

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