New bound on $S_{1}\times S_{2}$-setting Bell locality of a nonseparable Werner state

In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers $S_{1},S_{2}\geq1$ of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension $d\leq\min\{S_{1},S_{2}\}$ to satisfy all Bell inequalities under every $S_{1}\times S_{2}$-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous $-$ that is, to be $S_{1}\times S_{2}$-setting Bell local, for short. For a variety of $S_{1},S_{2}\geq1$ values, this new general locality condition is beyond Werner's and Barrett's locality conditions for a nonseparable Werner state. We also prove explicitly in the operator terms the optimization result by Terhal et. el. [Phys. Rev. Lett. \textbf{90,} 157903 (2003)] via semi-programming that every nonseparable Werner state with a dimension $d>\min\{S_{1},S_{2}\}$ is $S_{1}\times S_{2}$ -setting Bell local. The new results of the present article are important both for Bell nonlocality theory and for quantum applications based on Bell nonlocality.