Research proof index

V_rho^H * V_rho simplifies to sandwiching the traceRight by the inverse square root.

V_rho is an isometry.

Core inequality: W†W ≤ I. This is the key step, following from the isometry argument: V_rho ⊗ I_C and I_A ⊗ V_sigma are isometries, their cross product has norm ≤ 1, and the result can be related to W_mat through the MES computation (Eq. 6 in Lin-Kim-Hsieh).

An asymmetric weak-PFR estimate. Let $A,B$ be nonempty finite subsets of a rank-$n$ free $\mathbb Z$-module $G$. There are a subgroup $N\le G$, cosets $x,y\in G/N$, and nonempty fibers $A_x=\{a\in A:a+N=x\}$ and $B_y=\{b\in B:b+N=y\}$ such that $n\log2\le\log|G/N|+40d[U_A;U_B]$ and $\log|A|+\log|B|-\log|A_x|-\log|B_y|\le34\bigl(d[U_A;U_B]-d[U_{A_x};U_{B_y}]\bigr).$