In this paper, we primarily focus on analyzing the stability property of phase retrieval by examining the bi-Lipschitz property of the map $\Phi_{\boldsymbol{A}}(\boldsymbol{x})=|\boldsymbol{A}\boldsymbol{x}|\in \mathbb{R}_+^m$, where $\boldsymbol{x}\in \mathbb{H}^d$ and $\boldsymbol{A}\in \mathbb{H}^{m\times d}$ is the measurement matrix for $\mathbb{H}\in\{\mathbb{R},\mathbb{C}\}$. We define the condition number $\beta_{\boldsymbol{A}}=\frac{U_{\boldsymbol{A}}}{L_{\boldsymbol{A}}}$, where $L_{\boldsymbol{A}}$ and $U_{\boldsymbol{A}}$ represent the optimal lower and upper Lipschitz constants, respectively. We establish the first universal lower bound on $\beta_{\boldsymbol{A}}$ by demonstrating that for any ${\boldsymbol{A}}\in\mathbb{H}^{m\times d}$, \begin{equation*} \beta_{\boldsymbol{A}}\geq \beta_0^{\mathbb{H}}=\begin{cases} \sqrt{\frac{\pi}{\pi-2}}\,\,\approx\,\, 1.659 & \text{if $\mathbb{H}=\mathbb{R}$,}\\ \sqrt{\frac{4}{4-\pi}}\,\,\approx\,\, 2.159 & \text{if $\mathbb{H}=\mathbb{C}$.} \end{cases} \end{equation*} We prove that the condition number of a standard Gaussian matrix in $\mathbb{H}^{m\times d}$ asymptotically matches the lower bound $\beta_0^{\mathbb{H}}$ for both real and complex cases. This result indicates that the constant lower bound $\beta_0^{\mathbb{H}}$ is asymptotically tight, holding true for both the real and complex scenarios. As an application of this result, we utilize it to investigate the performance of quadratic models for phase retrieval. Lastly, we establish that for any odd integer $m\geq 3$, the harmonic frame $\boldsymbol{A}\in \mathbb{R}^{m\times 2}$ possesses the minimum condition number among all $\boldsymbol{A}\in \mathbb{R}^{m\times 2}$. We are confident that these findings carry substantial implications for enhancing our understanding of phase retrieval.

翻译：本文主要研究相位恢复的稳定性性质，通过分析映射 $\Phi_{\boldsymbol{A}}(\boldsymbol{x})=|\boldsymbol{A}\boldsymbol{x}|\in \mathbb{R}_+^m$ 的双Lipschitz性质来展开，其中 $\boldsymbol{x}\in \mathbb{H}^d$，$\boldsymbol{A}\in \mathbb{H}^{m\times d}$ 为测量矩阵，$\mathbb{H}\in\{\mathbb{R},\mathbb{C}\}$。我们定义条件数 $\beta_{\boldsymbol{A}}=\frac{U_{\boldsymbol{A}}}{L_{\boldsymbol{A}}}$，其中 $L_{\boldsymbol{A}}$ 和 $U_{\boldsymbol{A}}$ 分别表示最优下、上Lipschitz常数。我们建立了 $\beta_{\boldsymbol{A}}$ 的首个通用下界，证明对任意 ${\boldsymbol{A}}\in\mathbb{H}^{m\times d}$，\begin{equation*} \beta_{\boldsymbol{A}}\geq \beta_0^{\mathbb{H}}=\begin{cases} \sqrt{\frac{\pi}{\pi-2}}\,\,\approx\,\, 1.659 & \text{若 $\mathbb{H}=\mathbb{R}$，}\\ \sqrt{\frac{4}{4-\pi}}\,\,\approx\,\, 2.159 & \text{若 $\mathbb{H}=\mathbb{C}$。} \end{cases} \end{equation*} 我们证明 $\mathbb{H}^{m\times d}$ 中标准高斯矩阵的条件数在实数和复数情形下渐近匹配下界 $\beta_0^{\mathbb{H}}$。这一结果表明常数下界 $\beta_0^{\mathbb{H}}$ 是渐近紧的，对实数和复数情形均成立。作为该结果的应用，我们将其用于研究相位恢复中二次模型的性能。最后，我们证明对任意奇数 $m\geq 3$，调和框架 $\boldsymbol{A}\in \mathbb{R}^{m\times 2}$ 在所有 $\boldsymbol{A}\in \mathbb{R}^{m\times 2}$ 中具有最小条件数。我们相信这些发现对促进相位恢复的理解具有重要意义。