Analysis of optical quantum state preparation using photon detectors in the finite-temporal-resolution regime

Quantum state preparation is important for quantum information processing. In particular, in optical quantum computing with continuous variables, non-Gaussian states are needed for universal operation and error correction. Optical non-Gaussian states are usually generated by heralding schemes using photon detectors. In previous experiments, the temporal resolution of the photon detectors was sufficiently high relative to the time width of the quantum state, so that the conventional theory of non-Gaussian state preparation treated the detector's temporal resolution as negligible. However, when using various photon detectors including photon-number-resolving detectors, the temporal resolution is non-negligible. In this paper, we extend the conventional theory of quantum state preparation using photon detectors to the finite temporal resolution regime, analyze the cases of single-photon and two-photon preparation as examples, and find that the generated states are characterized by the dimensionless parameter $B$, defined as the product of the temporal resolution of the detectors $\Delta t$ and the bandwidth of the light source $\Delta f$. Based on the results, $B\sim0.1$ is required to keep the purity and fidelity of the generated quantum states high.