  We prove a tight and close-to-optimal lower bound on the effectiveness of
local quantum measurements (without classical communication) at discriminating
any two bipartite quantum states. Our result implies, for example, that any two
orthogonal quantum states of a $n_A\times n_B$ bipartite quantum system can be
discriminated via local measurements with an error probability no larger than
$\frac12 \left(1 - \frac{1}{c \min\{n_A, n_B\}} \right)$, where $1\leq c\leq
2\sqrt2$ is a universal constant, and our bound scales provably optimally with
the local dimensions $n_A,n_B$. Mathematically, this is achieved by showing
that the distinguishability norm $\|\cdot\|_{LO}$ associated with local
measurements satisfies that $\|\cdot\|_1\leq 2\sqrt2 \min\{n_A,n_B\}
\|\cdot\|_{LO}$, where $\|\cdot\|_1$ is the trace norm.
