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arXiv:2112.12643

New examples of entangled states on $\mathbb{C}^3 \otimes \mathbb{C}^3$

Anita Buckley

Published Dec 23, 2021• 0 citations

Abstract

We build apon our previous work, the Buckley-\vSivic method for simultaneous construction of families of positive maps on $3 \times 3$ self-adjoint matrices by prescribing a set of complex zeros to the associated forms. Positive maps that are not completely positive can be used to prove (witness) that certain mixed states are entangled. We obtain entanglement witnesses that are indecomposable and belong to extreme rays of the cone of positive maps. Consequently our semidefinite program returns new examples of entangled states whose entanglement cannot be certified by the transposition map nor by other well-known positive maps. The constructed states as well as the method of their construction offer some valuable insights for quantum information theory, in particular into the geometry of positive cones.

DOI: 10.48550/arxiv.2112.12643

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New examples of entangled states on

\mathbb{C}^3 \otimes \mathbb{C}^3

We build apon our previous work, the Buckley-\vSivic method for simultaneous construction of families of positive maps on

3 \times 3

self-adjoint matrices by prescribing a set of complex zeros to the associated forms. Positive maps that are not completely positive can be used to prove (witness) that certain mixed states are entangled. We obtain entanglement witnesses that are indecomposable and belong to extreme rays of the cone of positive maps. Consequently our semidefinite program returns new examples of entangled states whose entanglement cannot be certified by the transposition map nor by other well-known positive maps. The constructed states as well as the method of their construction offer some valuable insights for quantum information theory, in particular into the geometry of positive cones.

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.

We give an affirmative answer to the question posed by Yang et al. on the existence of indecomposable 2-positive linear maps from

\mathcal{M}_3(\mathbb{C})

to

\mathcal{M}_4(\mathbb{C})

. In this study, we construct an explicit class of linear maps in

\mathcal{B}(\mathcal{M}_3(\mathbb{C}), \mathcal{M}_4(\mathbb{C}))

that are positive, yet fail to be completely positive or completely copositive. Furthermore, we demonstrate that the associated Choi matrices of these maps serve as entanglement witnesses.

We define an entanglement witness in a composite quantum system as an observable having a nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are nonnegative biquadratic forms or positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\ifmmode\times\else\texttimes\fi{}3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of

\mathcal{D}

, the set of density matrices, fills out

\mathcal{D}

maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.