  Explicit constructions of polar codes and polar lattices for both lossless
and lossy Gray-Wyner problems are studied. Polar codes are employed to extract
Wyner's common information of doubly symmetric binary source; polar lattices
are then extended to extract that of a pair of Gaussian sources or multiple
Gaussian sources. With regard to the discrete sources, the entire best-known
region of the lossless Gray-Wyner problem are achieved by specifying the test
channels to construct polar codes without time-sharing. As a result, we are
able to give an interpretation that the Wyner's common information remains the
same to the lossy case when the distortion is small [1]. Finally, the entire
best-known lossy Gray-Wyner region for discrete sources can also be achieved
using polar codes. With regard to the Gaussian sources, the best-known lossy
Gray-Wyner region for bivariate Gaussian sources with a specific covariance
matrix [1] can be achieved by using polar lattices. Moreover, we prove that
extracting Wyner's common information of a pair of Gaussian sources is
equivalent to implementing the lossy compression for a single Gaussian source,
which implies that the common information can be extracted by a polar lattice
for quantization. Furthermore, we extend this result to the case of multiple
Gaussian sources.
