Positive maps and Entanglement Witnesses in different dimensions

• URL: http://arxiv.org/abs/2308.07019v5
• Date: Fri, 04 Oct 2024 20:14:56 GMT
• Title: Positive maps and Entanglement Witnesses in different dimensions
• Authors: Vahid Jannesary, Vahid Karimipour,
• Abstract summary: We derive a closed-form criterion for detecting entanglement in general density matrices based on Entanglement Witnesses. We prove that non-unital EWs, corresponding to non-unital maps, are not more powerful than unital EWs.
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• Abstract: We present a continuous, multiparameter family of positive maps between spaces of differing dimensions. This framework facilitates the construction of Entanglement Witnesses (EWs) specifically designed for systems in $d_1\times d_2$ dimensions. We derive a simple, closed-form criterion for detecting entanglement in general density matrices based on these witnesses. To demonstrate the effectiveness of this criterion, we apply it to a range of Positive Partial Transpose (PPT) entangled states, revealing that the parameter regions where these states exhibit entanglement are larger than previously reported. Furthermore, we prove that non-unital EWs, corresponding to non-unital maps, are not more powerful than unital EWs, thus supporting the focus on unital positive maps in recent studies. Our method complements existing approaches to separability criteria for density matrices in different dimensions.

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