Extending Heisenberg's measurement--disturbance relation to the twin-slit case

Heisenberg's position-measurement--momentum-disturbance relation is derivable from the uncertainty relation $\sigma(q)\sigma(p) \geq \hbar/2$ only for the case when the particle is initially in a momentum eigenstate. Here I derive a new measurement--disturbance relation which applies when the particle is prepared in a twin-slit superposition and the measurement can determine at which slit the particle is present. The relation is $d \times \Delta p \geq 2\hbar/\pi$, where $d$ is the slit separation and $\Delta p=D_{M}(P_{f},P_{i})$ is the Monge distance between the initial $P_{i}(p)$ and final $P_{f}(p)$ momentum distributions.