Topological structure of projective Hilbert spaces associated with phase retrieval vectors

Projective Hilbert spaces as the underlying spaces of this paper are obtained by identifying two vectors of a Hilbert space $\mathcal{H}$ which have the same phase and denoted by $\hat{\mathcal{H}}$. For a family $\Phi$ of vectors of $\mathcal{H}$ we introduce a topology $\tau_{\Phi}$ on $\hat{\mathcal{H}}$ and provide a topology-based approach for analyzing $\hat{\mathcal{H}}$. This leads to a new classification of phase retrieval property. We prove that $(\hat{\mathcal{H}}, \tau_{\Phi})$ is $\sigma$-compact, as well as it is Hausdorff if and only if $\Phi$ does phase retrieval. In particular, if $\Phi$ is phase retrieval, then we prove that $(\hat{\mathcal{H}}, \tau_{\Phi})$ is metrizable and $\hat{\mathcal{H}}$ is paracompact by a direct limit topology. Also, we make a comparison between $\tau_{\Phi}$ and some known topologies including the quotient topology, the weak topology and the direct-limit topology. Furthermore, we establish a metric $d_{\Phi}$ on $\hat{\mathcal{H}}$ and show that $d_{\Phi}$ is weaker than the Bures-Wasserstein distance on $\hat{\mathcal{H}}$. As a result, in the finite dimensional case, we prove that $\tau_{\Phi}$ coincides with the weak topology and $\tau_{d_{\Phi}}$ on $\hat{\mathcal{H}}$ if and only if $\Phi$ is phase retrieval.