Title： Entangled Symmetric Quantum States And Copositive Matrices

Speaker：JORDI TURA，Instituut-Lorentz, Leiden University

Time：2:30pm-3:30pm，2023.07.31

Venue: 管理楼1418

Abstract：

Entanglement is one of the most intriguing phenomena in quantum physics whose implications have profound consequences, not only from a theoretical point of view but also in light of some computational tasks that would be otherwise unfeasible with classical systems. For this reason, deciding whether a quantum state is entangled or not, is a problem of paramount importance whose solution, unfortunately, is known to be NP-hard in the general scenario. In some cases, however, symmetries provide a useful framework to recast the separability problem in a simpler way, thus reducing the original complexity of this task. 

In this work we focus on symmetric quantum states, i.e., states that are invariant under permutations of the parties, showing how, in the case of the qudits, the characterization of the entanglement can be accomplished by means of copositive matrices [1]. In particular, we establish a connection between entanglement witnesses, i.e., hermitian operators that are able to detect entanglement, and copositive matrices, showing how only a subset of them, dubbed as exceptional, can be used to assess a non-trivial form of entanglement, so-called PPT-entanglement, in any dimension, with the PPT-entangled edge states detected by the so-called extremal matrices.

Finally we illustrate our findings discussing some examples of families of PPT-entangled states in 3-level and 4-level systems, along with the entanglement witnesses that detect them. We conjecture that any PPT-entangled state of two qudits can be detected by means of an entanglement witness of the form that we propose [2].

This is joint work with Albert Aloy (Vienna), Carlo Marconi, Rub´en Quesada, Maciej Lewenstein and Anna Sanpera (Barcelona).

References

[1] Jordi Tura, Albert Aloy, Rub´en Quesada, Maciej Lewenstein and Anna Sanpera. Separability of diagonal symmetric states: a quadratic conic optimization problem. Quantum 2(45):1—31,(2018).

[2] Carlo Marconi, Albert Aloy, Jordi Tura and Anna Sanpera. Entangled symmetric states and copositive matrices Quantum, 5(561):1–18, (2021).