Lemma: Four-effect certificate

Let $R\in \mathcal{C}$, and let $X_0,X_1,Y_0,Y_1⪰0$. Put

Then

No POVM normalization of the four local matrices is required.

Proof. Choose (eq:C), and write

First make the $A_i$ commute by a change of representation. For $H=A_0+A_1$, let $S$ equal $H^{-1/2}$ on $\mathrm{ran}H$ and the identity on $\ker H$. Replace

The output is unchanged. The new pullbacks sum to the support projection of $H$, so they commute. The PPT condition persists because

Suppress the new notation, set $C=L\otimes M$, and define

Applying Lemma lem:block to $C$, and then to the CP map $C(T_r\otimes T_s)$, gives

For the second estimate use the input vectors $\overline{v},\overline{u}$; transposition exchanges the two diagonal inputs and leaves the desired off-diagonal output.

Let $P$ be the common spectral projection where $A_0\ge A_1$, and split

The pullback cross terms vanish, and

Equation (eq:two-block) bounds the four terms in the expansion of $C(|\chi ⟩⟨\chi |)$, respectively, by

The triangle inequality proves (eq:four). No inverse of an output state or effect was used.