Induced measures in the space of mixed quantum states

We analyse several product measures in the space of mixed quantum states. I n 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 tak es place with respect to the first subsystem). For K = N the induced measur e is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian mat rices 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.