Multipartite Entanglement Measure and Complete Monogamy Relation

Abstract

Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure, a genuine MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM and a tightly complete monogamy relation for the genuine MEM. Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis $q$-entropy of entanglement, Rényi $\alpha$-entropy of entanglement, the convex-roof extension of negativity and negativity, respectively. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis $q$-entropy of entanglement are genuine MEMs, (ii) multipartite extensions of Rényi $\alpha$-entropy of entanglement, negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs, and (iii) all these multipartite extensions are completely monogamous and the ones which are defined by the convex-roof structure (except for the Rényi $\alpha$-entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, we find a class of tripartite states that one part can maximally entangled with other two parts simultaneously according to the definition of maximally entangled mixed state (MEMS) in [Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that there is no MEMS and that the only MES is the pure MES.

arXiv comment18 pages, 2 figures

References in corpus (91)

• Quantum entanglement
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Mixed State Entanglement and Quantum Error Correction
• A computable measure of entanglement
• Three qubits can be entangled in two inequivalent ways
• Concentrating Partial Entanglement by Local Operations
• Entanglement of a Pair of Quantum Bits
• Distributed Entanglement
• Quantifying Entanglement
• The logarithmic negativity: A full entanglement monotone that is not convex
• Entanglement Measures and Purification Procedures
• Beating the channel capacity limit for linear photonic superdense coding
• Universal state inversion and concurrence in arbitrary dimensions
• General Monogamy Inequality for Bipartite Qubit Entanglement
• Geometric measure of entanglement and applications to bipartite and multipartite quantum states
• Entanglement monotones
• Monogamy of entanglement and other correlations
• Superadditivity of communication capacity using entangled inputs
• Entanglement Measures under Symmetry
• "Squashed Entanglement" - An Additive Entanglement Measure
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Measure of genuine multipartite entanglement with computable lower bounds
• Monogamy Inequality in terms of Negativity for Three-Qubit States
• Normal forms and entanglement measures for multipartite quantum states
• Maximally entangled mixed states of two qubits
• Monotones and invariants for multi-particle quantum states
• Is Entanglement Monogamous?
• Experimental Quantum Teleportation of a Two-Qubit Composite System
• Maximally entangled mixed states in two qubits
• Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity
• Monogamy of nonlocal quantum correlations
• I-concurrence and tangle for isotropic states
• Constructing N-qubit entanglement monotones from anti-linear operators
• The polynomial invariants of four qubits
• Convex-roof extended negativity as an entanglement measure for bipartite quantum systems
• Maximally multipartite entangled states
• Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems
• Are general quantum correlations monogamous?
• General monogamy relation for the entanglement of formation in multiqubit systems
• The maximally entangled set of multipartite quantum states
• Security proof for cryptographic protocols based only on the monogamy of Bell's inequality violations
• Entanglement monogamy relations of qubit systems
• Monogamy of quantum correlations - a review
• Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states
• Maximally entangled mixed states: Creation and concentration
• The maximally entangled symmetric state in terms of the geometric measure
• Multipartite entanglement measures
• All Maximally Entangled Four Qubits States
• Loss-Induced Limits to Phase Measurement Precision with Maximally Entangled States
• On the permutationally invariant part of a density matrix and nonseparability of N-qubit states
• Unitarity of black hole evaporation in final-state projection models
• Strong monogamy conjecture for multiqubit entanglement: The four-qubit case
• Monogamy of entanglement of formation
• Measure of multipartite entanglement with computable lower bounds
• Bell inequalities tailored to maximally entangled states
• Should Entanglement Measures be Monogamous or Faithful?
• Monogamy of Correlations vs. Monogamy of Entanglement
• Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof
• General Monogamy of Tsallis $q$-Entropy Entanglement in Multiqubit Systems
• On polynomial invariants of several qubits
• Disentangling Theorem and Monogamy for Entanglement Negativity
• An additive and operational entanglement measure: conditional entanglement of mutual information
• Subadditivity of q-entropies for q>1
• Polynomial invariants for discrimination and classification of four-qubit entanglement
• The structure of Renyi entropic inequalities
• Monogamy of entanglement without inequalities
• Limitations to sharing entanglement
• Monogamy equalities for qubit entanglement from Lorentz invariance
• Detection of genuinely entangled and non-separable $n$-partite quantum states
• Renormalization group contraction of tensor networks in three dimensions
• Monogamy of the Entanglement of Formation
• Multipartite entanglement measure for all discrete systems
• Evolution and Symmetry of Multipartite Entanglement
• Anisotropic invariance and the distribution of quantum correlations
• Experimental Observation of a Topological Phase in the Maximally Entangled State of a Pair of Qubits
• Generation and purification of maximally-entangled atomic states in optical cavities
• Are all maximally entangled states pure?
• Elemental and tight monogamy relations in nonsignalling theories
• Strong monogamy inequalities for four qubits
• Monogamy inequality for any local quantum resource and entanglement
• Possibility of generalized monogamy relations for multipartite entanglement beyond three qubits
• Distribution of entanglement and correlations in all finite dimensions
• Maximally Entangled States via Mutual Unbiased collective Bases
• Exponentially many entanglement and correlation constraints for multipartite quantum states
• Polynomial Monogamy Relations for Entanglement Negativity
• Constructing UMEB from maximally entangled basis
• Conditions for monogamy of quantum correlations in multipartite systems
• Monogamy of particle statistics in tripartite systems simulating bosons and fermions
• Quantum Teleportation of Light
• Strict entanglement monotonicity under local operations and classical communication
• Additivity of entanglement of formation via entanglement-breaking space

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• Genuine multipartite entanglement measure
• Thermodynamic Signatures of Genuinely Multipartite Entanglement
• Multipartite entanglement measures: a review
• Multipartite entanglement in the 1-D spin-$\frac{1}{2}$ Heisenberg Antiferromagnet
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• When is a Genuine Multipartite Entanglement Measure Monogamous?
• Multipartite entanglement detection based on generalized state-dependent entropic uncertainty relation for multiple measurements
• Quantum Fisher information-based detection of genuine tripartite entanglement
• Complete monogamy of the multipartite quantum mutual information
• Charged particles interaction in both a finite volume and a uniform magnetic field II: topological and analytic properties of a magnetic system
• Complete Genuine Multipartite Entanglement Monotone
• Monogamy and trade-off relations for correlated quantum coherence
• Squashed Entanglement from Generalized Rindler Wedge
• Linear semi-infinite programming approach for entanglement quantification
• Probing multipartite entanglement, coherence and quantum information preservation under classical Ornstein-Uhlenbeck noise
• Upper bound of holographic entanglement entropy combinations
• Entanglement of Assistance as a measure of multiparty entanglement
• Multipartite Monogamy of Entanglement for Three Qubit States
• Measure of tripartite quantum correlation in multiqubit symmetric pure state
• Complete $k$-partite entanglement measure
• Measure of set imaginarity
• Partitewise Entanglement