Operator Schmidt Rank and the Geometry of Schmidt Profiles

Abstract

The main result of the paper states that if $0\ne v\in {\mathbb{C}}^m\otimes {\mathbb{C}}^n$, $d=\min \{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf{U})$ denotes the operator Schmidt rank of $\mathbf{U}$. Furthermore, if $\mathbf{U}\in {\mathbb{M}}_m\otimes {\mathbb{M}}_n$ is unitary and $OSR(\mathbf{U})\le k$, then, for every unit vector $x\in {\mathbb{C}}^m\otimes {\mathbb{C}}^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.