Transitivity of entanglement

Gelo Noel Tabia^2,1,3*, Yu-Chun Yin^1,3, Chung-Yun Hsieh⁴, Yeong-Cherng Liang^1,3

¹Department of Physics, National Cheng Kung University, Tainan 701, Taiwan
²Center for Quantum Technology, National Tsing Hua University, Hsinchu 300, Taiwan
³Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
⁴ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels , Barcelona, Spain

* Presenter:Gelo Noel Tabia, email:gelo.tabia@gmail.com

Two entangled states σ and τ are said to exhibit entanglement transitivity if for all tripartite states with two marginal states equal to σ and τ, the remaining marginal state is also entangled. Using the Peres-Horodecki criterion, the transitivity of any pair of entangled qubit states can be tested via a semidefinite program. From the quantum channel-state duality, we can also define entanglement transitivity in terms of the degradability properties of quantum operations associated with the Stinespring representation. When σ and τ are the same, the existence of a joint state reduces to the problem of symmetric extendibility. From several examples derived from symmetric extensions, we prove that a two-qubit state has a pure unique symmetric extension if and only if it is the Choi state of a non-entanglement-breaking, self-complementary quantum operation. In this case, entanglement transitivity follows if all the bipartite marginal states are entangled.

Keywords: entanglement, transitivity, Stinespring dilation, symmetric extension