Optical states with higher stellar rank

TL;DR

This work addresses certifying quantum non-Gaussian traveling-light states with higher stellar rank using measurement-based heralding and hierarchical witnesses (m-PQnG). It develops a mathematical model of a two-mode squeezed vacuum circuit with lossy channels and Kraus losses, derives the conditional state $\hat{\varrho}_{(m)}$ and heralding probability $P_{(m)}$, and uses Monte Carlo simulations to quantify tolerable loss for certifiable $m$-PQnG states ($m=4,5$). The study compares true photon-number-resolving detectors with cascaded avalanche photodiodes (CAP) and shows that high-rank certifiable states demand near-ideal detectors or large CAP arrays, establishing practical guidelines for experiments. The results demonstrate robustness of the hierarchical, stellar-rank-based certification against loss and provide a path toward realizing higher-photon-number non-Gaussian resources for optical quantum information processing.

Abstract

Quantum non-Gaussian states of traveling light fields are crucial components of quantum information processing protocols; however, their preparation is experimentally challenging. In this paper, we discuss the minimal requirements imposed on the quantum efficiency of photon number resolving detectors and the quality of the squeezing operation in an experimental realization of certifiable quantum non-Gaussian states of individual photonic states with three, four, and five photons.

Figures (4)

• Figure 1: Schematic illustration of the non-Gaussian state preparation circuit with a two mode squeezed vacuum state serving as a source of entangled quantum states. One of the modes is measured, thus projecting the other mode onto the resolved Fock state. The effective loss, characterized by the intensity transmittances $\zeta_{1}$ and $\zeta_{2}$, aggregate the various sources of loss present in generation of the entangled state, propagation and mode matching of its output modes, and detector efficiencies.
• Figure 2: Illustration of the certification procedure of genuine $4$-PQnG states for the simulated statistical ensembles. (a) The blue line represents the threshold curve $F_{4} (x)$. States with $y_{4} > F_{4}(x_{4})$ are genuine $4$-PQnG states. Four example experimental states are depicted in the figure. The bullet points represent the expectation values obtained by simulating the experiment. The dashed boxes represent their uncertainty and span three standard deviations in both axial directions. The pictured boxes are exaggerated in size for the legibility of the illustration. States marked with red bullets failed the certification as they lie either under the threshold curve or their respective uncertainty boxes intersect the curve. States marked with blue bullets are certifiably genuine $4$-PQnG states according to the hierarchical criteria; their boxes are well above the curve and do not intersect the threshold curve. The black cross above the threshold curve marks a $4$-PQnG state corresponding to $2.75\%$ loss in characterization and $20\%$ loss in heralding. Its certification is visualised in (b) where the solid box represents the actual three sigma uncertainties in both computed quantities. (c) The photon number distribution of the prepared quantum state. It is contaminated with higher-ordered contributions due to the loss incurred in heralding and lower-ordered components caused by characterization loss.
• Figure 3: The tolerable loss in preparation of certifiable genuine $4$-PQnG (top) and $5$-PQnG (bottom) states. Individual tiles represent the best attainable probability of success. Values for each tile are obtained by maximizing the probability of successfully preparing a certifiable state over the initial squeezing rate ${0 \leq r \leq 10\ \mathrm{dB}}$. White colored tiles correspond to statistically insignificant cases with probabilities of success below $10^{-5}$. Different heralding detectors were used in the analysis. The results obtained for a true PNR detector are presented in the leftmost column, while the remaining columns represent CAP detectors with ${n = 20, 15}$, and $10$ constituent avalanche detectors.
• Figure 4: Thresholds of the tolerable loss in preparation of certifiable genuine $3$ (black), $4$ (red), and $5$-PQnG (blue) states. Their values are determined using the same methodology as the results visualised in Figure \ref{['f-res-45']}where the minimal viable probability of success is limited to $10^{-5}$. Solid lines represent heralding with true PNR detectors, whereas the dashed and dotted lines correspond to CAP detectors comprising $20$ and $10$ photodiodes.