We derive the strong subadditivity of von Neumann entropy with a strict lower bound, dependent on distribution quantum correlation in system. investigate structure states saturating bounded and examine its consequences for data processing inequality. The achieves bound associated locally inaccessible information.

We show that every algorithm for testing $n$-variate Boolean functions monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds this problem were designed nonadaptive algorithms and, as a result, the best bound general (possibly adaptive) testers was only $\Omega(\log n)$. Combined with of tester Khot, Minzer, and Safra (FOCS 2015), our shows adaptivity can r...

Answering a key point left open in recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ \|b\|_{\text{BMO}_{\gamma}(\mathbb{R}^2)}\lesssim \|[b,H_{\gamma}]\|_{L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)}, where $H_{\gamma}$ is parabolic Hilbert transform.