We propose a theoretical scheme to improve the resolution and precision of phase measurement with parity detection by using a nonclassical state generated by applying a number-conserving generalized superposition of products (GSP) operation, ${\left(sa{a}^{†}+t{a}^{†}a\right)}^m$ with $s^2+t^2=1$, on a two-mode squeezed vacuum (TMSV) state as the input of Mach-Zehnder interferometer. Then, the statistical properties of the proposed GSP-TMSV are investigated via average photon number (APN), antibunching effect, and two-mode squeezing. Particularly, both higher order $m$ GSP operation and lower proportion parameter $s$ are beneficial for presenting larger total APN, which leads to the improvement of quantum Fisher information (QFI). In addition, we also compare the phase measurement precisions with and without photon losses between our scheme and the previous photon subtraction or photon addition schemes. It is found that our scheme, especially for the case of $s=0$, even in the presence of photon losses, has the best performance via the enhanced phase resolution, sensitivity, and QFI when compared to those previous schemes. Interestingly, without losses, the standard quantum-noise limit (SQL) is always broken through for our scheme and the phase uncertainty associated with the state of our scheme is closer to the corresponding Heisenberg limit (HL) than the TMSV in the larger total APN region, especially for the case of two-side GSP operation $\left(m,n\right)\in \left(1,1\right)$. However, in the presence of photon losses, the HL cannot be beaten, but the SQL can be broken through, particularly for the large total APN regimes.

中文翻译：

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使用数字保留操作改善相位估计

我们提出一种理论方案，以通过使用通过应用乘数保全的广义乘积（GSP）运算生成的非经典状态，通过奇偶校验检测来提高相位测量的分辨率和精度， ${\left(s\mathrm{一种}{\mathrm{一种}}^{†}+Ť{\mathrm{一种}}^{†}\mathrm{一种}\right)}^{米}$ 和 $s^{\mathrm{2个}}+{Ť}^{\mathrm{2个}}=\mathrm{1个}$在两模压缩真空（TMSV）状态下，作为Mach-Zehnder干涉仪的输入。然后，通过平均光子数（APN），抗聚束效应和双模压缩研究了提出的GSP-TMSV的统计特性。特别是，两者都更高$米$ GSP操作和较低比例参数 $s$有助于呈现更大的总APN，从而改善了量子Fisher信息（QFI）。此外，我们还比较了我们的方案与先前的光子减法或光子加和方案之间在有无光子损失的情况下的相位测量精度。发现我们的方案，特别是对于$s=0$与以前的方案相比，即使存在光子损耗，也可以通过增强的相位分辨率，灵敏度和QFI来获得最佳性能。有趣的是，在没有损失的情况下，我们的方案始终会突破标准的量子噪声限值（SQL），并且与方案状态相关的相位不确定性比较大的APN中的TMSV更接近相应的海森堡限值（HL）。区域，尤其是对于两侧的GSP操作$\left(米，ñ\right)\in \left(\mathrm{1个}，\mathrm{1个}\right)$。但是，在存在光子损失的情况下，无法克服HL，但是可以突破SQL，尤其是对于较大的总APN机制而言。