LCU projected-block correctness

Let $w_j:=\mathrm{\mid }{\beta }_j\mathrm{\mid }{\alpha }_j\mathrm{/}\mathrm{\Lambda }$. The amplitude list is normalized because ${\sum }_j{w}_j=1$. Identity-padding a component ancilla preserves its unitary and contributes factor $⟨e_0,e_0⟩=1$ to its all-zero block, so by [F2] its compression remains $A_j\mathrm{/}{\alpha }_j$. SELECT is unitary because it is block diagonal on orthogonal index subspaces, with each block a unit-modulus phase times a unitary (and identity on unused labels); hence $W=P^{†}\mathrm{S}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{C}\mathrm{T}P$ is unitary. Projecting SELECT onto the common component-ancilla zero state replaces each $U_j$ by $A_j\mathrm{/}{\alpha }_j$, so the full projected block of $W$ is the coefficient-register matrix element of $P^{†}({\sum }_j\mathrm{\mid }j⟩⟨j\mathrm{\mid }\otimes ({\beta }_j\mathrm{/}\mathrm{\mid }{\beta }_j\mathrm{\mid })A_j\mathrm{/}{\alpha }_j)P$.

The amplitudes in [F1] are real and nonnegative, so the $(0,j)$ and $(j,0)$ entries of $P^{†}$ and $P$ in this matrix element both equal $\sqrt{w_j}$. Expanding gives ${\sum }_j{w}_j({\beta }_j\mathrm{/}\mathrm{\mid }{\beta }_j\mathrm{\mid })A_j\mathrm{/}{\alpha }_j={\sum }_j{\beta }_j{A}_j\mathrm{/}\mathrm{\Lambda }$, as claimed.

For every unit vector $x$, [F3] and [F4] give $\mathrm{\parallel }{\sum }_j{\beta }_j{A}_{j}x\mathrm{\parallel }\le {\sum }_j\mathrm{\mid }{\beta }_j\mathrm{\mid }\mathrm{\parallel }{A}_{j}x\mathrm{\parallel }\le {\sum }_j\mathrm{\mid }{\beta }_j\mathrm{\mid }{\alpha }_j=\mathrm{\Lambda }$. Taking the maximum over unit $x$ yields $\mathrm{\parallel }{\sum }_j{\beta }_j{A}_j{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\Lambda }$. Since $\mathrm{\Lambda }>0$, the definition of block encoding applies with zero error.