# Progress on the separability problem of Quantum Mechanics: A Lab Talk

In their recent paper in Journal of Physics A: Mathematical and Theoretical researchers from China and South Korea investigate a Separability criterion for three-qubit states with a four dimensional norm.  Kyung Hoon Han spoke to JPhys+ to explain more about their progress on the separability problem.

In quantum physics, it is known that the quantum system is characterized by a quantum state, which is either entangled or separable. The problem distinguishing entanglement from separability is NP-hard and has received a lot of attentions from the community of mathematics, physics and computer science in the past decades.

Multi-qubit entanglement is a fundamental resource in quantum computation and information theory. The separability of two-qubit states can be analytically determined by the famous Peres-Horodecki criterion proposed in 1996. It goes back to the Woronowicz duality argument in the seventies and Størmer’s result on positive maps in the sixties. However the separability of three-qubit states turns out to be mathematically hard, and so far there is no analytical criterion. In this work, we give a necessary and sufficient condition for the separablity of three-qubit states when all entries of the density matrix are zero except for diagonal and anti-diagonals, so called X-states. Many important three qubit states like Greenberger-Horne-Zeilinger diagonal states are in this class. Figure 1: Two thick curves, circle and rectangle, represent the boundary of the separability region of the state. From J. Phys. A: Math. Theor.50, 345303 (2017). Copyright IOP Publishing.

The condition is expressed in terms of a norm arising from anti-diagonal entries. We compute this norm in several cases to get operational criterion. A certain three parameter family of X-states gives rise to an interpolation between l1 and l norms. The next target of our work is to get the corresponding results for general multi-qubit cases.

This paper was originally reported in J. Phys. A: Math. Theor.50, 345303 (2017).