On Optimal Quantum Data Hiding and Maximal Separable Ball

Quantum data hiding asks how much distinguishing power can be lost when global measurements are restricted to local measurements and classical communication. In this work, we establish sharp results and improved bounds for several natural classes of restricted measurements. For bipartite systems on $\mathbb C^n\otimes\mathbb C^m$, we prove that the optimal data-hiding ratios against separable and LOCC measurements are both $\min\{n,m\}$. This result follows from a stronger result that, for every $2\le p\le\infty$, the largest centered Schatten $p$-ball whose associated binary measurements are implementable by finite-round LOCC has radius $\min\{n,m\}^{2/p-1}$. This strengthens the classic separable-ball theorems, while also providing an explicit finite-round LOCC implementation. For Alice-first one-way LOCC with Alice's local dimension equal to $n$, we prove that the optimal ratio is $(1+o(1))n$, with the upper bound obtained from a Gaussian rank-one POVM. For local operations without communication, we improve the universal upper bound to $(\pi\sqrt3/4+o(1))\min\{n,m\}$.