Szegedy row-preparation map and discriminant

For a finite row-stochastic kernel $(X,P)$, let ${\mathbb{C}}^X$ and ${\mathbb{C}}^{X\times X}$ carry the standard orthonormal label and ordered-product bases. If $X\mathrm{\ne }\mathrm{\varnothing }$, these coordinate spaces are implemented as two ordered registers; if $X=\mathrm{\varnothing }$, they are zero spaces and every map below is the unique zero-dimensional map. Define the row-preparation linear map $T:{\mathbb{C}}^X\to {\mathbb{C}}^{X\times X}$ and the swap $S$ on basis vectors by

$T\mathrm{\mid }x⟩=\mathrm{\mid }x⟩\otimes {\sum }_{y\in X}\sqrt{P_{xy}}\mathrm{\mid }y⟩,S\mathrm{\mid }x,y⟩=\mathrm{\mid }y,x⟩\mathrm{.}$

Every scalar square root is the unique nonnegative real root. Define the discriminant by $D:=T^{\ast }ST$, using the adjoint characterized by $⟨Tv,w⟩=⟨v,T^{\ast }w⟩$ for the root inner product, which is linear in its first argument. The row-preparation map is an isometry; its complete verification is The Szegedy discriminant is self-adjoint and contractive ↗. This is the general row-stochastic construction and assumes no reversibility.