Bound Entanglement of Bell Diagonal Pairs of Qutrits and Ququarts: A Comparison

• URL: http://arxiv.org/abs/2209.15267v1
• Date: Fri, 30 Sep 2022 06:58:27 GMT
• Title: Bound Entanglement of Bell Diagonal Pairs of Qutrits and Ququarts: A Comparison
• Authors: Christopher Popp and Beatrix C. Hiesmayr
• Abstract summary: We classify Bell diagonal bipartite qudits with positive partial transposition (PPT) as entangled or separable. We estimate the volumes of separable and free and bound entangled states.
• Score: 0.06091702876917279
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We classify Bell diagonal bipartite qudits with positive partial transposition (PPT) as entangled or separable and compare their properties for different dimensions. The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. Leveraging a geometrical representation of states in Euclidean space, we analyze a family of bipartite Bell diagonal qudits that can be separable, free entangled or bound entangled. Extending and applying analytical and numerical methods that almost completely solve the separability problem for Bell diagonal qutrits ($d=3$), we successfully classify more than $75\%$ of representative Bell diagonal PPT states for $d=4$. Via those representative states we are able to estimate the volumes of separable and free and bound entangled states. We find that at least $75.7\%$ of all PPT states are separable, while only $1.7\%$ are found to be bound entangled and for $22.6\%$ it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities and relate those to differences of the Euclidean geometry for qutrits ($d=3$) and ququarts ($d=4$). Finally, using a detailed visual analysis of the set of separable Bell diagonal states, a conjecture relating the group structure of Bell diagonal states of the analyzed family to necessary and sufficient mixing conditions for separable states is motivated.

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