Invariance of separability probability over reduced states in 4x4 bipartite systems

Abstract: 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.