The common information of two dependent random variables

The problem of finding a meaningful measure of the "common information" or "common randomness' of two discrete dependent random variables
X,Y
is studied. The quantity
C(X; Y)
is defined as the minimum possible value of
I(X, Y; W)
where the minimum is taken over all distributions defining an auxiliary random variable
W \in \mathcal{W}
, a finite set, such that
X, Y
are conditionally independent given
W
. The main result of the paper is contained in two theorems which show that
C(X; Y)
is i) the minimum
R_0
such that a sequence of independent copies of
(X,Y)
can be efficiently encoded into three binary streams
W_0, W_1,W_2
with rates
R_0,R_1,R_2
, respectively,
[\sum R_i = H(X, Y)]
and
X
recovered from
(W_0, W_1)
, and
Y
recovered from
(W_0, W_2)
, i.e.,
W_0
is the common stream; ii) the minimum binary rate
R
of the common input to independent processors that generate an approximation to
X,Y
.