Maximum heralding probabilities of non-classical state generation from two-mode Gaussian state via photon counting measurements

TL;DR

The paper tackles the problem of maximizing the heralding probability $P_n$ for generating non-Gaussian states by measuring $n$ photons in one mode of a two-mode pure Gaussian state. It formulates a core two-mode state and uses the Bargmann representation to derive the conditioned signal state $|\psi_n\rangle_a=(\hat{a}^\u0307 + s_0 \hat{a} + \delta_0)^n|0\rangle$ with $P_n= (\lambda^{2n}/n!) |Z|^2 \langle\psi_n|\psi_n\rangle$, optimizing over the free parameter $\lambda$. The analysis yields analytic results in special cases (e.g., $\delta_0=0$ or $s_0=0$) and shows polynomial scaling of $P_n$ with $n$ (such as $P_n\sim n^{-1}$ or $P_n\sim n^{-\gamma}$), with a distinct $n^{-3/4}$ scaling for photon-added coherent states. These findings imply that high-stellar-rank non-Gaussian states can be generated with realistic squeezing and repetition rates, and the framework readily extends to multimode conditional-state schemes.

Abstract

Highly non-classical states of light - such as the approximate Gottesman-Kitaev-Preskill states or cat-like states - can be generated from experimentally accessible Gaussian states via photon counting measurements on selected modes, conditioned on specific outcomes of these heralding events. A simplest yet important example of this approach involves performing photon number measurements on one mode of a two-mode entangled Gaussian state. The heralding probability of this scheme is a key figure of merit, as it determines the generation rate of the targeted non-classical state. In this work we show that the maximum heralding probability for the two-mode setting can be calculated analytically, and we investigate its dependence on the number of detected photons n. Our results show that the number of required experimental trials scales only polynomially with n. Generation of highly complex optical quantum states with high stellar rank is thus practically feasible in this setting, given access to sufficiently strong squeezing.

Figures (4)

• Figure 1: Conditional state preparation via Gaussian boson sampling Su2019Hanamura2025. Mode c of pure two-mode Gaussian state $|G\rangle$ is measured with photon number resolving detector. Detection of $n$ photons heralds preparation of state $|\psi_n\rangle$ in mode a.
• Figure 2: Dependence of the heralding probability $P_n$ on detected number of photons $n$ is ploted for three different values of control parameter $s_0$, and $\delta_0=0$.
• Figure 3: Dependence of the heralding probability $P_n$ on the control parameter $s_0$ is plotted for four different values of $n$, and $\delta_0=0$.
• Figure 4: The maximum heralding probability $P_n$ is plotted as a function of $n$ for $\delta_0=1$ (a), $\delta_0=i$ (b), and $\delta_0=e^{i\pi/4}$ (c). In each case, results for three different values of the oher control parameter $s_0$ are plotted, $s_0=0.5$ (blue circles), $s_0=1$ (red triangles), and $s_0=1.5$ (green squares).