On the state space structure of tripartite quantum systems

TL;DR

This paper investigates the structure of tripartite quantum state spaces, demonstrating that PPT states form a larger set than separable states across all bipartitions, with specific constructions for different Hilbert space dimensions.

Contribution

The paper proves that PPT states strictly contain all fully separable states in tripartite systems and provides explicit constructions for these states in various dimensions.

Findings

PPT states form a strict superset of fully separable states in tripartite systems.

Explicit constructions of PPT states outside the set of separable states for different dimensions.

Different methods are used for systems with dimensions greater than or equal to 3 and for 3-qubit systems.

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.