The condition number in phase retrieval from intensity measurements

Abstract This paper investigates the fundamental stability properties of the phase retrieval problem, which are critical for ensuring robust signal reconstruction across a wide range of applications. We present a rigorous analysis of the condition number $\beta _{\varPsi _{\boldsymbol A}}^{\ell _{p}}$ associated with the nonlinear mapping $\varPsi _{{\boldsymbol A}}(\boldsymbol{x}) = \bigl (\lvert \langle{\boldsymbol a}_{j}, {\boldsymbol x} \rangle \rvert ^{2} \bigr )_{1 \le j \le m}$, evaluated with respect to the $\ell _{p}$ norm. Here, ${\boldsymbol a}_{j}\in{\mathbb R}^{n}$ or ${\mathbb C}^{n}$ denote the sensing vectors. A smaller condition number corresponds to enhanced stability against noise and perturbations. Our main contribution is the derivation of the first explicit and universal lower bounds on this condition number, which hold for all sensing vectors and thereby characterize intrinsic limitations on stability. Specifically, we establish that for the $\ell _{2}$ norm, the condition number $\beta _{\varPsi _{\boldsymbol A}}^{\ell _{2}}$ admits the following lower bounds: $$\begin{align*} \beta^{\ell_{2}}_{\varPsi_{\boldsymbol A}} \geq \begin{cases} \sqrt{3}, & \text{in the real case}, \\ 2, & \text{in the complex case}. \end{cases} \end{align*}$$ Similarly, with respect to the $\ell _{1}$ norm, the condition number $\beta _{\varPsi _{\boldsymbol A}}^{\ell _{1}}$ asymptotically satisfies $$\begin{align*} \beta^{\ell_{1}}_{\varPsi_{\boldsymbol A}} \geq \begin{cases} \pi/2, & \text{in the real case}, \\ 2, & \text{in the complex case}. \end{cases} \end{align*}$$ Importantly, we demonstrate that these fundamental bounds are asymptotically tight or exactly attained in significant instances. For example, in the real setting with $m \times 2$ sensing matrices, the harmonic frame achieves the optimal lower bound of $\sqrt{3}$ in the $\ell _{2}$ norm for all $m\geq 3$, thereby establishing itself as a provably optimal sensing matrix. By delineating these fundamental stability limits and identifying optimal measurement structures, our work provides profound insights into the intrinsic nature of phase retrieval and establishes a theoretical foundation for the development of more robust and efficient signal recovery algorithms.