Proposition: A finite equal-slice residual separation

For input ${\rho }_{+}=|+⟩⟨+|$, the histories $+-$ and $-+$:

1. have equal probability $4/25$;

2. select the same terminal marking $T=T_0+2\tau$;

3. have conditional residual states of trace distance $1/\sqrt{2}$.

Nevertheless the complete geometric history law is identical for ${\rho }_{+}$ and $I_2/2$.

Proof. The two middle effects give (i), and their terminal $q_2=0$ gives (ii). Their normalized residuals are

For equatorial qubit pure states with these relative phases, their overlap has modulus $1/\sqrt{2}$; their trace distance is $\sqrt{1-1/2}=1/\sqrt{2}$. Equivalently, subtract their $2\times 2$ matrices. The final assertion follows from their equal input populations.