  Let ${\cal H}_1,$ ${\cal H}_2$ be finite dimensional complex Hilbert spaces
describing the states of two finite level quantum systems. Suppose $\rho_i$ is
a state in ${\cal H}_i, i=1,2.$ Let ${\cal C} (\rho_1, \rho_2)$ be the convex
set of all states $\rho$ in ${\cal H} = {\cal H}_1 \otimes {\cal H}_2$ whose
marginal states in ${\cal H}_1$ and ${\cal H}_2$ are $\rho_1$ and $\rho_2$
respectively. Here we present a necessary and sufficient criterion for a $\rho$
in ${\cal C} (\rho_1, \rho_2)$ to be an extreme point. Such a condition
implies, in particular, that for a state $\rho$ to be an extreme point of
${\cal C} (\rho_1, \rho_2)$ it is necessary that the rank of $\rho$ does not
exceed $(d_1^2 + d_2^2 - 1)^{{1/2}},$ where $d_i = \dim {\cal H}_i, i=1,2.$
When ${\cal H}_1$ and ${\cal H}_2$ coincide with the 1-qubit Hilbert space
$\mathbb{C}^2$ with its standard orthonormal basis $\{|0 >, |1> \}$ and $\rho_1
= \rho_2 = {1/2} I$ it turns out that a state $\rho \in {\cal C} ({1/2}I,
{1/2}I)$ is extremal if and only if $\rho$ is of the form $|\Omega>< \Omega|$
where $| \Omega > = \frac{1}{\sqrt{2}} (|0> | \psi_0 > + |1 > | \psi_1 >),$
$\{| \psi_0 >, | \psi_1> \}$ being an arbitrary orthonormal basis of
$\mathbb{C}^2.$ In particular, the extremal states are the maximally entangled
states.
