  The geometric separability probability of composite quantum systems is
extensively studied in the last decades. One of most simple but strikingly
difficult problem is to compute the separability probability of qubit-qubit and
rebit-rebit quantum states with respect to the Hilbert-Schmidt measure. A lot
of numerical simulations confirm the P(rebit-rebit)=29/64 and
P(qubit-qubit)=8/33 conjectured probabilities. Milz and Strunz studied the
separability probability with respect to given subsystems. They conjectured
that the separability probability of qubit-qubit (and qubit-qutrit) depends on
sum of single qubit subsystems (D), moreover it depends just on the Bloch radii
(r) of D and it is constant in r. Using the Peres-Horodecki criterion for
separability we give mathematical proof for the P(rebit-rebit)=29/64
probability and we present an integral formula for the complex case which
hopefully will help to prove the P(qubit-qubit)=8/33 probability too. We prove
Milz and Strunz's conjecture for rebit-rebit and qubit-qubit states. The case,
when the state space is endowed with the volume form generated by the operator
monotone function f(x)=sqrt(x) is studied too in detail. We show, that even in
this setting the Milz and Strunz's conjecture holds and we give an integral
formula for separability probability according to this measure.
