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

Abstract

The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest 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 Prebit−rebit=2964 and Pqubit−qubit=833 conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of (D1CC∗D2) depends on D=D1+D2 (on single qubit subsystems), moreover it depends only on the Bloch radii (r) of D and it is constant in r. Using the Peres–Horodecki criterion for separability we give a mathematical proof for the 2964 probability and we present an integral formula for the complex case which hopefully will help to prove the 833 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)=x is also studied in detail. We show that even in this setting Milz and Strunz’s conjecture holds true and we give an integral formula for separability probability according to this measure.