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A multi-reservoir quantum transport relation. Here $I_i$ is conventional current injected from reservoir $i$, $N_i$ counts outgoing spin-resolved modes and $T_{ij}$ counts transmission from reservoir $j$ into $i$. Ideal voltage probes draw zero net current and equilibrate incident carriers. On a chiral integer Hall edge with $N$ modes, $I_i=(N{e}^2/h)(V_i-V_{i-1})$ for the stated circulation order.

A conducting scanning-probe tip acts as a movable electrostatic gate. Mapping conductance changes as the tip perturbs a quantum point contact or surrounding electron gas probes quantum transport. The image is a response to the tip, not automatically a noninvasive map of the unperturbed local current density.