XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

The question in the title was originally posed by Zyczkowski, Horodecki, Sanpera and Lewenstein in 1998. To make it more precise, we mention that the state space of a composite quantum system is a disjoint union of separable and entangled states and if we endow the state space with some probability measure, then we may ask about the probability of separable states that is called separability probability. Separability probability in rebit-rebit and qubit-qubit quantum systems with respect to the Hilbert-Schmidt measure was extensively studied during the last decades. However, no exact mathematical proofs were presented for separability probability values conjectured by Slater. We show that the statistical manifold of 2nx2n density matrices is diffeomorphic to the Cartesian product of the space of nxn density matrices, the (-I,I) operator interval of the nxn selfadjoint matrices and the unit ball of nxn matrices with respect to the canonical operator norm. Using this decomposition and the Peres-Horodecki criterion for separability we parametrize the space of 4x4 (qubit-qubit) and 6x6 (qubit-qutrit system) separable states and its surface. We present explicit integral formulas for separability probability in these systems. Introducing the concept of the separability function, we give mathematical proof for the rebit-rebit separability probability. The decomposition presented in this talk leads to a better understanding of the geometry of entangled quantum states.

Click here to access the slides.