A new class of distances between pure quantum states

An $n$-dimensional pure quantum state can be regarded as a norm-one vector $\textbf{x} \in \mathbb{C}^n$ defined up to a global phase factor ($\textbf{x} \sim \rm{e}^{\rm{i}\varphi}\textbf{x}$). In my talk I will show that, for any $n \times n$ distance matrix $(D_{ij})$ and any $p \geq 2$, the map $$d_p(\textbf{x},\textbf{y})=\left(\sum_{i\lt j}D_{ij}^p|x_iy_j - x_jy_i|^2 \right)^{1/p}$$ s a bona fide distance on the space of $n$-dimensional pure states. This is a far-reaching generalization of the standard distance on this space given by the wedge product $$\|\textbf{x} \wedge \textbf{y}\| = \sqrt{\sum_{i\lt j}|x_iy_j - x_jy_i|^2}.$$ will also discuss how this result carries over to the $n \rightarrow +\infty$ case, offering a way to lift the metric structure from some underlying metric space $(X,D)$ to the suitably defined `space of wave functions', and --- time permitted --- touch upon the mixed states case. The talk builds on and significantly extends the results of R.Bistroń, M.Eckstein, S.Friedland, TM, K.Życzkowski, A new class of distances on complex projective spaces, Linear Algebra Appl. 721, 577--611, (2024).