# 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).

Lin Chen

Seung-Hyeok Kye

Kyung Hoon Han

Lin Chen is an associate professor in the School of Mathematics and Systems Science, Beihang University, Beijing. He was previously a postdoctoral researcher in the National University of Singapore, University of Waterloo, and Singapore University of Technology and Design. He is interested in the theoretical and mathematical problems of entanglement theory, tensor rank and mutually unbiased bases.

Kyung Hoon Han is an assistant professor in the Department of Mathematics, the University of Suwon. He is the corresponding author of this work.

Seung-Hyeok Kye is a professor in the Department of Mathematics, Seoul National University.