  The scalar additive Gaussian noise channel has the "single crossing point"
property between the minimum-mean square error (MMSE) in the estimation of the
input given the channel output, assuming a Gaussian input to the channel, and
the MMSE assuming an arbitrary input. This paper extends the result to the
parallel MIMO additive Gaussian channel in three phases: i) The channel matrix
is the identity matrix, and we limit the Gaussian input to a vector of Gaussian
i.i.d. elements. The "single crossing point" property is with respect to the
snr (as in the scalar case). ii) The channel matrix is arbitrary, the Gaussian
input is limited to an independent Gaussian input. A "single crossing point"
property is derived for each diagonal element of the MMSE matrix. iii) The
Gaussian input is allowed to be an arbitrary Gaussian random vector. A "single
crossing point" property is derived for each eigenvalue of the MMSE matrix.
  These three extensions are then translated to new information theoretic
properties on the mutual information, using the fundamental relationship
between estimation theory and information theory. The results of the last phase
are also translated to a new property of Fisher's information. Finally, the
applicability of all three extensions on information theoretic problems is
demonstrated through: a proof of a special case of Shannon's vector EPI, a
converse proof of the capacity region of the parallel degraded MIMO broadcast
channel (BC) under per-antenna power constrains and under covariance
constraints, and a converse proof of the capacity region of the compound
parallel degraded MIMO BC under covariance constraint.
