Certification of high stellar ranks of quantum states of light with a pair of click detectors

Abstract

Stellar rank of quantum states of light quantifies the amount of non-Gaussian resources required for their generation. One popular and practical approach to certification of stellar rank is based on measurement of click statistics with an array of binary detectors that can only distinguish the presence and absence of photons. Specifically, it was shown that measurements with an array of m+1 detectors allows one to certify stellar rank of approximate Fock state |m>, even when the state is subjected to losses or certain noise. Here we address the question how many click detectors are in principle required to certify high stellar ranks. We show that two click detectors arranged in a Hanbury Brown-Twiss setup suffice. Interestingly, detection of higher stellar rank is greatly facilitated by making the total detection efficiency of the detectors sufficiently low but well calibrated. Losses affect the response of the detection scheme in a way that can be exploited to certify stellar rank of high Fock states. We explicitly construct the corresponding stellar rank witnesses and discuss dependence of the stellar rank thresholds on parameters of the considered setup. Our results reveal that it is possible to certify high stellar ranks even with a minimalistic scheme that provides only very coarse grained information about the photon number statistics of the characterized state.

Figures (5)

• Figure 1: Proposed measurement scheme (a). The input signal is attenuated by attenuator Att and split at a beam splitter BS to be detected by two binary click detectors D$_1$ and D$_2$. Example of tunable attenuator for input linearly polarized light (b). The attenuator is formed by a sequence of half-wave plate HWP and polarizing beam splitter PBS.
• Figure 2: Probabilities $R_1(n)$ of click of detector D$_1$ and simultaneous no-click of D$_2$ for input Fock states $|n\rangle$ are plotted for balanced detection scheme with $T=0.5$ and four different total detection efficiencies $\eta=1$ (a), $\eta=0.6$ (b), $\eta=0.4$ (c) and $\eta=0.2$ (d).
• Figure 3: Probability $R_1(n)$ of click of detector D$_1$ and simultaneous no-click of D$_2$ for input Fock state $|n\rangle$ is plotted as function of $T$ for $\eta=2/3$ and $n=1$ (blue solid line), $n=2$ (red dashed line), and $n=3$ (green dot-dashed line).
• Figure 4: Witnesses of stellar rank based on the pair of probabilities $R_1$ and $R_2$ are plotted for balanced setup with $T=0.5$, and total detection efficiency $\eta=1$ (a) and $\eta=0.5$ (b). The color of each area indicates the minimum stellar rank that is certified when the point $[R_2,R_1]$ lies in that area. Points in the light gray area are compatible with Gaussian states and their mixtures. For $\eta=1$, only stellar rank $1$ can be certified (the light blue area), while higher stellar ranks become certifiable for $\eta=0.5$. The black lines indicate the boundary of all physically achievable probability pairs $[R_2,R_1]$. See text for additional details.
• Figure 5: The same as Fig. \ref{['figW']}, but $T=0.75$. Witnesses of stellar rank based on the pair of probabilities $R_1$ and $R_2$ are plotted for total detection efficiency $\eta=1$ (a) and $\eta=0.5$ (b). The color of each area indicates the minimum stellar rank that is certified when the point $[R_2,R_1]$ lies in that area. The black lines indicate the boundary of all physically achievable probability pairs $[R_2,R_1]$.