On the state space structure of tripartite quantum 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 $\min \{d_1,d_2,d_3\}\ge 2$. The claim is proved by constructing state belonging to the set ${\mathcal{P}}^{int}(ABC)$ but not belonging to ${\mathcal{B}}^{int}(ABC)$. For $({\mathbb{C}}^d{)}^{\otimes 3}$ with $d\ge 3$, the construction follows from specific type of multipartite unextendible product bases. However, such a construction is not possible for $({\mathbb{C}}^2{)}^{\otimes 3}$ 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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References in corpus (38)

• Quantum entanglement
• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Entanglement detection
• Separability criterion and inseparable mixed states with positive partial transposition
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Quantum Nonlocality without Entanglement
• Unextendible Product Bases and Bound Entanglement
• General Entanglement Breaking Channels
• Bound entanglement can be activated
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Secure key from bound entanglement
• Taming multiparticle entanglement
• Entanglement detection via mutually unbiased bases
• Strong Quantum Nonlocality without Entanglement
• General paradigm for distilling classical key from quantum states
• Irreversibility for all bound entangled states
• All entangled states are useful for information processing
• Separability properties of tripartite states with UxUxU-symmetry
• Irreversibility in asymptotic manipulations of entanglement
• Construction of quantum states with bound entanglement
• Low dimensional bound entanglement with one-way distillable cryptographic key
• Genuinely Nonlocal Product Bases: Classification and Entanglement Assisted Discrimination
• Genuine multiparticle entanglement of permutationally invariant states
• Activating NPPT distillation with an infinitesimal amount of bound entanglement
• Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition
• Quantum metrology: Heisenberg limit with bound entanglement
• Separability of diagonal symmetric states: a quadratic conic optimization problem
• Family of bound entangled states on the boundary of Peres set
• Multi-particle entanglement manipulation under positive partial transpose preserving operations
• Extreme points of the set of density matrices with positive partial transpose
• Low rank extremal PPT states and unextendible product bases
• Numerical studies of entangled PPT states in composite quantum systems
• Nonlocality Without Entanglement: Quantum Theory and Beyond
• Searching for extremal PPT entangled states
• The four postulates of quantum mechanics are three
• Construction of three qubit genuine entanglement with bi-partite positive partial transposes
• Mutually Unbiased Bases and Bound Entanglement

Cited by in corpus (2)

• Strongly nonlocal unextendible product bases do exist
• Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$