  In this paper we will establish a relation between geometric uncertainty
relation and the determinant of the quantum covariance matrix for mixed quantum
states. We will show that determinant of the covariance matrix represents the
squared metric area of a parallelogram. In this setting the geometric
uncertainty relation compares a metric area to a symplectic area. Moreover, we
will in details investigate relation between J-holomorphic maps and geometric
uncertainty relation for mixed quantum states. We will argue that determinant
of the quantum covariance matrix is equal to the harmonic energy of a
holomorphic map that minimize the areas.
