  Finding the optimal attainable precisions in quantum multiparameter metrology
is a non trivial problem. One approach to tackling this problem involves the
computation of bounds which impose limits on how accurately we can estimate
certain physical quantities. One such bound is the Holevo Cramer Rao bound on
the trace of the mean squared error matrix. The Holevo bound is an
asymptotically achievable bound when one allows for any measurement strategy,
including collective measurements on many copies of the probe. In this work we
introduce a tighter bound for estimating multiple parameters simultaneously
when performing separable measurements on finite copies of the probe. This
makes it more relevant in terms of experimental accessibility. We show that
this bound can be efficiently computed by casting it as a semidefinite program.
We illustrate our bound with several examples of collective measurements on
finite copies of the probe. These results have implications for the necessary
requirements to saturate the Holevo bound.
