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

Explicit form of the Bayesian posterior estimate of a quantum state under the uninformative prior