  An analytical solution for the posterior estimate in Bayesian tomography of
the unknown quantum state of an arbitrary quantum system (with a
finite-dimensional Hilbert space) is found. First, we derive the Bayesian
estimate for a pure quantum state measured by a set of arbitrary rank-1 POVMs
under the uninformative (i.e. the unitary invariant or Haar) prior. The
expression for the estimate involves the matrix permanents of the Gram matrices
with repeated rows and columns, with the matrix elements being the scalar
products of vectors giving the measurement outcomes. Second, an unknown mixed
state is treated by the Hilbert-Schmidt purification. In this case, under the
uninformative prior for the combined pure state, the posterior estimate of the
mixed state of the system is expressed through the matrix $\alpha$-permanents
of the Gram matrices of scalar products of vectors giving the measurement
outcomes. In the mixed case, there is also a free integer parameter -- the
Schmidt number -- which can be used to optimise the Bayesian reconstruction
(for instance, in case of Schimdt number being equal to 1, the mixed state
estimates reduces to the pure state estimate). We also discuss the perspectives
of approximate numerical computation and asymptotic analytical evaluation of
the Bayesian estimate using the derived formula.
