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On the state space structure of tripartite quantum systems (2104.06938v2)

Published 14 Apr 2021 in quant-ph

Abstract: State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say $\mathcal{B}{int}(ABC)$] is a strict subset of the set of states having positive partial transposition (PPT) across the three bipartite cuts [say $\mathcal{P}{int}(ABC)$] for all the tripartite Hilbert spaces $\mathbb{C}_A{d_1}\otimes\mathbb{C}_B{d_2}\otimes\mathbb{C}_C{d_3}$ with $\min{d_1,d_2,d_3}\ge2$. The claim is proved by constructing state belonging to the set $\mathcal{P}{int}(ABC)$ but not belonging to $\mathcal{B}{int}(ABC)$. For $(\mathbb{C}{d}){\otimes3}$ with $d\ge3$, the construction follows from specific type of multipartite unextendible product bases. However, such a construction is not possible for $(\mathbb{C}{2}){\otimes3}$ since for any $n$ the bipartite system $\mathbb{C}2\otimes\mathbb{C}n$ cannot have any unextendible product bases [Phys. Rev. Lett. 82, 5385 (1999)]. For the $3$-qubit system we, therefore, come up with a different construction.

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