On the state space structure of tripartite quantum systems

• URL: http://arxiv.org/abs/2104.06938v2
• Date: Tue, 20 Apr 2021 15:41:22 GMT
• Title: On the state space structure of tripartite quantum systems
• Authors: Hari Krishnan S V, Ashish Ranjan, and Manik Banik
• Abstract summary: It has been shown that the set of states separable across all the three bipartitions [say $mathcalBint(ABC)$] is a strict subset of the set of states having positive partial transposition (PPT) across the three bipartite cuts [say $mathcalPint(ABC)$] The claim is proved by constructing state belonging to the set $mathPint(ABC)$ but not belonging to $mathcalBint(ABC)$.
• Score: 0.22741525908374005
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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