  We characterize the Schatten class $S^p$ of the commutator of Riesz
transforms $[b,R_j]$ in $\mathbb R^n$ ($j=1,\ldots, n$) in the two weight
setting for $n< p<\infty$, by introducing the condition that the symbol $b$
being in Besov spaces associated with the given two weights. At the critical
index $p=n$, the commutator $[b,R_j]$ belongs to Schatten class $S^{n}$ if and
only if $b$ is a constant, and to the weak Schatten class $S^{n,\infty}$ if and
only if $b$ is in an oscillation sequence space associated with the given two
weights. As a direct application, we have the Schatten class estimate for A.
Connes' quantised derivative in the two weight setting.
