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Constructing \(2 \times 2 \times 4\) and \(4 \times 4\) unextendible product bases and positive-partial-transpose entangled states. (English) Zbl 1458.81009

Summary: The 4-qubit unextendible product bases (UPBs) has been recently studied by N. Johnston [J. Phys. A, Math. Theor. 47, No. 42, Article ID 424034, 19 p. (2014; Zbl 1304.81038)]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in \(\mathcal{H} = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^4\), three UPBs of size 9 in \(\mathcal{K} = \mathbb{C}^4 \otimes \mathbb{C}^4\), and no UPB of size 7 in \(\mathcal{H}\) and \(\mathcal{K}\). Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in \(\mathcal{H}\) and \(\mathcal{K}\), respectively. We analytically derive a geometric measure of a special PPT entangled states.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
81P42 Entanglement measures, concurrencies, separability criteria
81P55 Special bases (entangled, mutual unbiased, etc.)
81V70 Many-body theory; quantum Hall effect

Citations:

Zbl 1304.81038
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References:

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