We investigate the Bell inequalities derived from the graph states with violations detectable even with the presence of noises, which generalizes the idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501 (2008)]. Firstly we constru
ct a family of valid Bell inequalities tolerating arbitrary $t$-qubit errors involving $3(t+1)$ qubits, e.g., 6 qubits suffice to tolerate single qubit errors. Secondly we construct also a single-error-tolerating Bell inequality with a violation that increases exponentially with the number of qubits. Exhaustive computer search for optimal error-tolerating Bell inequalities based on graph states on no more than 10 qubits shows that our constructions are optimal for single- and double-error tolerance.
Let $D$ be an unbounded domain in $RR^d$ with $dgeq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $overline D$ is transient. Next assume that RBM $X$ on $overline D$ is transient
and let $Y$ be its time change by Revuz measure ${bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $overline D$. We further show that if there is some $r>0$ so that $Dsetminus overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.
