This property seems to be used in the proof of Theorem 3 of these notes. In particular, see (3.4) and (3.5).
Suppose that we have a complex finite dimensional Hilbert space $\mathcal{H}$ and consider vectors $\vert\psi\rangle \in \mathcal{H}$. These are quantum states so are subject to normalization condition $\langle\psi\vert\psi\rangle = 1$ and can be expressed as a linear combination of orthogonal basis states $\{\vert\psi_i\rangle\}$. For a set of operators $\{E_i\}$, let $E$ be obtained by some linear combination of $E_i$. Consider the condition (see (3.4) of the notes)
$$\langle\psi\vert E^\dagger E\vert\psi\rangle = C(E)$$
with $C(E)$ being some function that is independent of $\vert\psi\rangle$. This is equivalent to (see (3.5) of the notes)
$$\langle\psi_i\vert E^\dagger_a E_b\vert\psi_j\rangle = C_{ab}\delta_{ij}$$
with $C_{ab}$ being the coefficients of a Hermitian matrix.
Why is this equivalence true? In particular, how does a Hermitian matrix and a Kronecker-delta function emerge from the first condition?