ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Linear Algebra and Quantum Computation

Tue 16:00–16:30, Auditorium Jean Monnet
Positivity of multi-linear maps arising from entanglement witnesses in multi-partite systems
Seung-Hyeok Kye (Seoul National University)

A non positive semi-definite Hermitian matrix $W$ is an entanglement witness if $\langle W,\varrho\rangle := {\text{\rm Tr}} (\varrho W^t)\ge 0$ for every separable state $\varrho$, where $W^t$ denotes the transpose of $W$. In the bi-partite system, it is nothing but the Choi matrix of a positive linear map between two systems which is not completely positive.

In the multi-partite systems, there are several notions of separability and the corresponding entanglement witnesses can be interpreted as various kinds of multi-linear maps. We study positivity of these multi-linear maps characterizing various witness properties. In the tri-partite system, we introduce positivity of bi-linear maps motivated by operator system theory which extends the notion of $s$-positivity of linear maps.

In the general multi-partite system, the notion of genuine entanglement is the most important resource in quantum information theory. From the corresponding witnesses we extract various linear maps which must be simultaneously positive. We apply these results to the so called X-shaped matrices to characterize multi-qubit X-shaped genuine entanglement in terms of the entries. We also characterize various kinds of separability of X-states as well as PPT properties.

  1. S.-H. Kye, Three-qubit entanglement witnesses with the full spanning properties, J. Phys. A: Math. Theor. 48 (2015), 235303, arXiv:1501.00768.
  2. K. H. Han and S.-H. Kye, Various notions of positivity for bi-linear maps and applications to tri-partite entanglement, J. Math. Phys. 57 (2016), 015205. arXiv:1503.05995
  3. K. H. Han and S.-H. Kye, Construction of multi-qubit optimal genuine entanglement witnesses, J. Phys. A: Math. Theor., to appear, arXiv:1510.03620