Relations among $k$-ME concurrence, negativity, polynomial invariants, and tangle

Abstract

The $k$-ME concurrence as a measure of multipartite entanglement (ME) unambiguously detects all $k$-nonseparable states in arbitrary dimensions, and satisfies many important properties of an entanglement measure. Negativity is a simple computable bipartite entanglement measure. Invariant and tangle are useful tools to study the properties of the quantum states. In this paper we mainly investigate the internal relations among the $k$-ME concurrence, negativity, polynomial invariants, and tangle. Strong links between $k$-ME concurrence and negativity as well as between $k$-ME concurrence and polynomial invariants are derived. We obtain the quantitative relation between $k$-ME ($k$=$n$) concurrence and negativity for all $n$-qubit states, give a exact value of the $n$-ME concurrence for the mixture of $n$-qubit GHZ states and white noise, and derive an connection between $k$-ME concurrence and tangle for $n$-qubit W state. Moreover, we find that for any $3$-qubit pure state the $k$-ME concurrence ($k$=2, 3) is related to negativity, tangle and polynomial invariants, while for $4$-qubit states the relations between $k$-ME concurrence (for $k$=2, 4) and negativity, and between $k$-ME concurrence and polynomial invariants also exist. Our work provides clear quantitative connections between $k$-ME concurrence and negativity, and between $k$-ME concurrence and polynomial invariants.

References in corpus (38)

• 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
• Entanglement of a Pair of Quantum Bits
• Distributed Entanglement
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Nonlinear atom interferometer surpasses classical precision limit
• Universal state inversion and concurrence in arbitrary dimensions
• Entanglement monotones
• Four qubits can be entangled in nine different ways
• Global entanglement in multiparticle systems
• Decoherence and multipartite entanglement
• Classification of multi-qubit mixed states: separability and distillability properties
• Quantum teleportation via a W state
• Monotones and invariants for multi-particle quantum states
• Distillability and partial transposition in bipartite systems
• Bell's theorem with and without inequalities for the three-qubit Greenberger-Horne-Zeilinger and W states
• Quantifying entanglement resources
• Convex-roof extended negativity as an entanglement measure for bipartite quantum systems
• Computing Local Invariants of Qubit Systems
• On multi-particle entanglement
• On local invariants of pure three-qubit states
• Can the states of the W-class be suitable for teleportation
• Optimal Controlled Teleportation
• On the permutationally invariant part of a density matrix and nonseparability of N-qubit states
• Measure of multipartite entanglement with computable lower bounds
• Local unitary equivalence and entanglement of multipartite pure states
• Negativity and strong monogamy of multi-party quantum entanglement beyond qubits
• Local Unitary Classification of Arbitrary Dimensional Multipartite Pure States
• Detection of genuinely entangled and non-separable $n$-partite quantum states
• Efficient $k$-separability criteria for mixed multipartite quantum states
• Separability criteria for several classes of $n$-partite quantum states
• Entanglement classification of 2 X 2 X 2 X d quantum systems via the ranks of the multiple coefficient matrices
• Quantum coherences, K-way negativities and multipartite entanglement
• Entanglement Classification with Algebraic Geometry
• Polynomial invariants of degree 4 for even-$n$ qubits and their applications in entanglement classification

Cited by in corpus (4)

• Parameterized multipartite entanglement measures
• Quantifying multipartite quantum states by ($k+1$)-partite entanglement measures
• Entanglement hierarchies in multipartite scenarios
• Optimal convex approximations of quantum states based on fidelity