  We study quantum tomography from a continuous measurement record obtained by
measuring expectation values of a set of Hermitian operators obtained from
unitary evolution of an initial observable. For this purpose, we consider the
application of a random unitary, diagonal in a fixed basis at each time step
and quantify the information gain in tomography using Fisher information of the
measurement record and the Shannon entropy associated with the eigenvalues of
covariance matrix of the estimation.
  Surprisingly, very high fidelity of reconstruction is obtained using random
unitaries diagonal in a fixed basis even though the measurement record is not
informationally complete. We then compare this with the information generated
and fidelities obtained by application of a different Haar random unitary at
each time step.
  We give an upper bound on the maximal information that can be obtained in
tomography and show that a covariance matrix taken from the Wishart-Laguerre
ensemble of random matrices and the associated Marchenko-Pastur distribution
saturates this bound. We find that physically, this corresponds to an
application of a different Haar random unitary at each time step. We show that
repeated application of random diagonal unitaries gives a covariance matrix in
tomographic estimation that corresponds to a new ensemble of random matrices.
We analytically and numerically estimate eigenvalues of this ensemble and show
the information gain to be bounded from below by the Porter-Thomas
distribution.
