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We study quantum tomography from a continuous measurement record obtained bymeasuring expectation values of a set of Hermitian operators obtained fromunitary evolution of an initial observable. For this purpose, we consider theapplication of a random unitary, diagonal in a fixed basis at each time stepand quantify the information gain in tomography using Fisher information of themeasurement record and the Shannon entropy associated with the eigenvalues ofcovariance matrix of the estimation.  Surprisingly, very high fidelity of reconstruction is obtained using randomunitaries diagonal in a fixed basis even though the measurement record is notinformationally complete. We then compare this with the information generatedand fidelities obtained by application of a different Haar random unitary ateach time step.  We give an upper bound on the maximal information that can be obtained intomography and show that a covariance matrix taken from the Wishart-Laguerreensemble of random matrices and the associated Marchenko-Pastur distributionsaturates this bound. We find that physically, this corresponds to anapplication of a different Haar random unitary at each time step. We show thatrepeated application of random diagonal unitaries gives a covariance matrix intomographic estimation that corresponds to a new ensemble of random matrices.We analytically and numerically estimate eigenvalues of this ensemble and showthe information gain to be bounded from below by the Porter-Thomasdistribution.

Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory