Induced measures in the space of mixed quantum states

We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several, averages with respect to this measure and show that the mean entanglement of N x N pure states behaves as ln N - 1/2.