Faithful geometric measures for genuine tripartite entanglement

Abstract

We present a faithful geometric picture for genuine tripartite entanglement of discrete, continuous, and hybrid quantum systems. We first find that the triangle relation ${\mathcal{E}}_{i|jk}^{\alpha }\le {\mathcal{E}}_{j|ik}^{\alpha }+{\mathcal{E}}_{k|ij}^{\alpha }$ holds for all subadditive bipartite entanglement measure $\mathcal{E}$, all permutations under parties $i,j,k$, all $\alpha \in [0,1]$, and all pure tripartite states. It provides a geometric interpretation that bipartition entanglement, measured by ${\mathcal{E}}^{\alpha }$, corresponds to the side of a triangle, of which the area with $\alpha \in (0,1)$ is nonzero if and only if the underlying state is genuinely entangled. Then, we rigorously prove the non-obtuse triangle area with $0<\alpha \le 1/2$ is a measure for genuine tripartite entanglement. Useful lower and upper bounds for these measures are obtained, and generalizations of our results are also presented. Finally, it is significantly strengthened for qubits that, given a set of subadditive and non-additive measures, some state is always found to violate the triangle relation for any $\alpha >1$, and the triangle area is not a measure for any $\alpha >1/2$. Hence, our results are expected to aid significant progress in studying both discrete and continuous multipartite entanglement.

arXiv commentAccepted as a Letter at Phys. Rev. A. Comments are welcome

References in corpus (43)

• Quantum entanglement
• Entanglement of Formation of an Arbitrary State of Two Qubits
• A computable measure of entanglement
• Gaussian Quantum Information
• Three qubits can be entangled in two inequivalent ways
• Distributed Entanglement
• Entanglement detection
• Quantifying Entanglement
• Quantum Resource Theories
• Universal state inversion and concurrence in arbitrary dimensions
• Continuous variable quantum information: Gaussian states and beyond
• Entanglement monotones
• Extremal entanglement and mixedness in continuous variable systems
• Advances in High Dimensional Quantum Entanglement
• Generalized Schmidt decomposition and classification of three-quantum-bit states
• Global entanglement in multiparticle systems
• Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
• Measure of genuine multipartite entanglement with computable lower bounds
• Multipartite entanglement in three-mode Gaussian states of continuous variable systems: Quantification, sharing structure and decoherence
• Measuring Gaussian quantum information and correlations using the Renyi entropy of order 2
• Quantum marginal problem and N-representability
• A Note on Invariants and Entanglements
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• Monogamy of quantum correlations - a review
• Multipartite entanglement measures
• A Triangle Governs Genuine Tripartite Entanglement
• One-qubit reduced states of a pure many-qubit state: polygon inequalities
• Schur complement inequalities for covariance matrices and monogamy of quantum correlations
• Disentangling Theorem and Monogamy for Entanglement Negativity
• Efficient entanglement criteria for discrete, continuous, and hybrid variables
• Subadditivity of q-entropies for q>1
• The structure of Renyi entropic inequalities
• Generalized Monogamy Relations of Concurrence for N-qubit Systems
• Relative entropy and squashed entanglement
• Gaussian quantum marginal problem
• Genuine multipartite entanglement measure
• Entanglement Polygon Inequality in Qubit Systems
• Thermodynamic Signatures of Genuinely Multipartite Entanglement
• Concurrence triangle induced genuine multipartite entanglement measure
• How entangled can a multi-party system possibly be?
• Characterizing entanglement with global and marginal entropic measures
• Tripartite entanglement measure under local operations and classical communication
• Multipartite Entanglement: A Journey Through Geometry

Cited by in corpus (4)

• Monogamy-of-entanglement-inspired protocol to quantify bipartite entanglement using spin squeezing
• The polygon relation and subadditivity of entropic measures for discrete and continuous multipartite entanglement
• Geometry of two-body correlations in three-qubit states
• Tripartite entanglement and tripartite steering in three-qubit pure states induced by vacuum--one-photon superpositions