Searching for extremal PPT entangled states

Abstract

We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states. First we investigate qubit--ququart states and show that the only candidates for extremal PPT entangled states (PPTES) have ranks of the state and its partial transposition (5,5) or (5,6) (equivalently (6,5)). Then, examples of extremal states of (5,5) type and the so--called edge states of type (5,6) are provided. We also make an attempt to explore the set of PPT states with ranks (5,6). Finally, we discuss what are the possible configurations of ranks of density matrices and their respective partial transposition in general three-qubit and four-qubit symmetric states for which there may exist extremal entangled PPT states. For instance in the first case we show that the only possibilities are (4,4,4) and (4,4,5).

12 pages, 2 figures, revised version due to the partial overlap with results of arXiv:0704.3348, some new results on extremality in multi-qubit systems added, contribution to the special issue of Optics Communications in memory of Krzysztof Wodkiewicz

References in corpus (10)

• A complete family of separability criteria
• Entanglement and permutational symmetry
• Detecting multipartite entanglement
• Geometry of quantum systems: density states and entanglement
• Characterizing multiparticle entanglement in symmetric N-qubit states via negativity of covariance matrices
• Separability criteria and entanglement witnesses for symmetric quantum states
• A (5,5) and (6,6) PPT edge state
• Extreme points of the set of density matrices with positive partial transpose
• Separability of entangled qutrits in noisy channels
• Separability, entanglement and full families of commuting normal matrices

Cited by in corpus (19)

• Four-qubit entangled symmetric states with positive partial transpositions
• Entangled symmetric states of N qubits with all positive partial transpositions
• Separability of diagonal symmetric states: a quadratic conic optimization problem
• Three-by-three bound entanglement with general unextendible product bases
• Family of bound entangled states on the boundary of Peres set
• Facial structures for various notions of positivity and applications to the theory of entanglement
• Qubit-qudit states with positive partial transpose
• Description of rank four PPT entangled states of two qutrits
• Separability problem for multipartite states of rank at most four
• Bound entangled singlet-like states for quantum metrology
• Properties and construction of extreme bipartite states having positive partial transpose
• Geometry for separable states and construction of entangled states with positive partial transposes
• Existence of product vectors and their partial conjugates in a pair of spaces
• PPT-inducing, distillation-prohibiting, and entanglement-binding quantum channels
• On witnessing arbitrary bipartite entanglement in a measurement device independent way
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
• On the state space structure of tripartite quantum systems
• Global geometric difference between separable and Positive partial transpose states
• Bound entanglement in symmetric random induced states