Symmetry Classification of Non-Reciprocal Responses in Multiterminal Ring Devices

We present a symmetry-based framework to classify the non-reciprocal responses of multiterminal ring quantum devices. The device is modeled as a ring of $n$ vertices, where a binary variable $e_k\in\{+1,-1\}$ on each bond encodes the preferred direction of signal flow between terminals. Non-reciprocity corresponds to a preferred current configuration on the ring, and the symmetry group of the device partitions all $2^n$ configurations into equivalence classes(orbits) characterized by a topological winding number $W$. Using the minimal non-trivial case $n=3$, we establish two results independent of microscopic details. First, lifting the degeneracy within an orbit generates non-reciprocal responses. For $n=3$ this requires simultaneous breaking of both time-reversal $T$ and spatial inversion $I$. Breaking either alone is insufficient. Second, the residual geometry symmetry after $T$ and $I$ are broken determines which responses are observable. For an isosceles triangular geometry, only two types of response are allowed: uniform circulation (all bonds carrying current in the same direction) and semi-circulation with the reversed bond on the geometrically distinct base. Semi-circulation with the reversed bond on either equal leg is symmetry-forbidden. Both predictions are validated using a minimal toy model of three quantum dots coupled to superconducting baths, which demonstrates a reactive quantum circulator response.