Entanglement quantification made easy: Polynomial measures invariant under convex decomposition

Abstract

Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are only available in few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-two states obeying such condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and show that several representative classes of four-qubit pure states have marginals that enjoy this property.

5 pages, 1 figure; to appear in Phys. Rev. Lett

References in corpus (14)

• Quantifying entanglement resources
• Constructing N-qubit entanglement monotones from anti-linear operators
• Entangled three-qubit states without concurrence and three-tangle
• All entangled states are useful for channel discrimination
• All Maximally Entangled Four Qubits States
• Evaluation of convex roof entanglement measures
• Strong monogamy conjecture for multiqubit entanglement: The four-qubit case
• Quantum steering ellipsoids, extremal physical states and monogamy
• Three-tangle for mixtures of generalized GHZ and generalized W states
• An operational interpretation for multipartite entanglement
• Tangles of superpositions and the convex-roof extension
• Maximally entangled three-qubit states via geometric measure of entanglement
• Three-Tangle for Rank-3 Mixed States: mixture of Greenberger-Horne-Zeilinger, W and flipped W states
• Classification of GHZ-type, W-type and GHZ-W-type multiqubit entanglements

Cited by in corpus (20)

• Convex geometry of quantum resource quantification
• $N$ identical particles and one particle to entangle them all
• Strong monogamy inequalities for four qubits
• Geometry of entanglement in the Bloch sphere
• Entangling three identical particles via spatial overlap
• Center-of-mass interpretation for bipartite purity analysis of $N$-party entanglement
• Geometric approach to entanglement quantification with polynomial measures
• Entangled states in the role of witnesses
• Generalized W-state of four qubits with exclusively threetangle
• The hierarchies of "witnesses" and the properties and characterization of entangled states as super entanglement witnesses
• The nine ways of four qubit entanglement and their threetangle
• Exact zeros of entanglement for arbitrary rank-two mixtures: how a geometric view of the zero-polytope makes life more easy
• Polynomial measure of coherence
• Rabi oscillations, entanglement and teleportation in the anti-Jaynes-Cummings model
• Quantum Information Geometry in the Space of Measurements
• Entanglement Classification via Single Entanglement Measure
• A Quantum Algorithm for Testing Junta Variables and Learning Boolean Functions via Entanglement Measure
• Convex-roof entanglement measures of density matrices block diagonal in disjoint subspaces for the study of thermal states
• Preservation of dynamics in coupled cavity system using second order nonlinearity
• Threetangle in the XY-model class with a non-integrable field background