Optimal trajectory detection with an array of sensor qubits

An array of qubits can be used as a detector for identifying the trajectory of an incoming particle. The interaction between the particle and the detector qubits changes the state of the array, with this change depending on the position of the interaction. This problem can be reframed as one of state discrimination, with the underlying symmetry that, for a fixed trajectory length, the states to be discriminated are shifted versions of one another. This problem has been studied in the strong interaction case where a single measurement can perfectly distinguish between different trajectories. We are looking at the weaker interaction case where there can be errors, but we seek to minimize the error probability.

Our goal is to maximize the detection capability. Optimal detection depends on many factors, including the initial state of the detector array. Previously, it was shown that the optimal state of the detector array for a one-qubit-long trajectory is the symmetric superposition state, i.e., the Dicke state. Here, we extend this idea to more realistic scenarios, where the particle interacts with multiple qubits along its trajectory. We find that the Dicke states are not always the optimal ones. We provide analytical expressions for the discrimination probability using minimum-error and pretty-good measurements, by exploiting the fact that the states to be discriminated are related to each other by the shift operator. Numerical simulations were performed to confirm our results.