Theorem: Complete common-charge phase instrument

The maps (eq:phaseinstrument) form a complete destructive instrument. For every $\gamma \in \mathbb{R}$, let

Then

on all operators and with arbitrary untouched references.

On (eq:phaseinput),

For fixed $b_1$, comparison of $b_2=b_1+d$ with $b_2=b_1$ has

It is positive unless $d/{\sigma }_{\mathrm{p}}\in 2\pi \mathbb{Z}$.

Proof. The two computational even-parity projectors and the two odd Bell projectors sum to the identity. Each Kraus bra in the failure sum acquires only a scalar phase under $G_{\gamma }$; each ${\chi }_{\pm }$ acquires the common phase $e^{i\gamma }$. Their selective maps are therefore unchanged. Tensoring the operator identities with a reference identity proves the complete version.

The odd-parity amplitudes are $(e^{i\beta }\pm e^{i\alpha })/(2\sqrt{2})$. The even-parity total probability is $1/2$. These give the laws and their exact TV distance.