2017/12/28 by Sebastian Gartzke, Andreas Osterloh
Computer Science · Mathematics · Physics and Astronomy · #Bipartite graph #Combinatorics #Discrete mathematics #Graph #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Polytope #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Qubit #Upper and lower bounds #quant-ph
paper · pdf · doi:10.1103/physreva.98.052307
published as Phys. Rev. A 98, 052307 (2018) · 8 pages, 3 figures, revtex4.1. Comments are welcome
arxiv created 2017/12/28 · openalex created_date 2018/01/05 · openalex publication_date 2018/11/06 · arxiv updated 2018/11/14 · openalex updated_date 2026/08/05
We single out a class of states possessing only the three-tangle but distributed all over four qubits. This is a three-site analog of states from the W class. The latter possess exclusively globally distributed pairwise entanglement as measured by the concurrence. We perform an analysis for four qubits, showing that such a state indeed exists. To this end we analyze specific states of four qubits for which all possible SL invariants vanish, and hence which are part of the SL null cone. Instead, they will possess a certain unitary invariant. In analyzing the three-tangle of rank-two reduced density matrices of these states, we manage to show that in this particular case we reach the convex roof exactly. As an interesting by-product this solution is extended in the rank-two case to a homogeneous polynomial SL-invariant measure of entanglement of degree 2m, if there are two states which correspond to an at most n-fold degenerate solution in the zero polytope for 0<n<m that can be combined with the convexified minimal characteristic curve at an (2m\ensuremath-n)-fold zero yielding a decomposition of \ensuremathρ. If more than one such state does exist in the zero polytope, a minimization must be performed. If no decomposition of \ensuremathρ is obtained in this way, it provides a better lower bound than the lowest convexified curve.