Hierarchies of Geometric Entanglement

Abstract

We introduce a class of generalized geometric measures of entanglement. For pure quantum states of $N$ elementary subsystems, they are defined as the distances from the sets of $K$-separable states ($K=2,...,N$). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows to discriminate among the different contributions. The extended measures are applied to the study of entanglement in different classes of $N$-qubit pure states. These classes include $W$ and $GHZ$ states, and their symmetric superpositions; symmetric multi-magnon states; cluster states; and, finally, asymmetric generalized $W$-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between $GHZ$ and $W$ states, clarifying the original result of Wei and Goldbart that $W$ states possess a larger global entanglement than $GHZ$ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.

16 pages, 7 figures. Final version, to appear in Phys. Rev. A

References in corpus (8)

• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Multiphoton Quantum Optics and Quantum State Engineering
• Experimental Analysis of a 4-Qubit Cluster State
• Estimating entanglement measures in experiments
• Universal geometric entanglement close to quantum phase transitions
• Probability density function characterization of multipartite entanglement
• Operational Classification and Quantification of Multipartite Entangled States
• Connecting the generalized robustness and the geometric measure of entanglement

Cited by in corpus (9)

• Entanglement in neutrino oscillations
• Relative entropy of entanglement for certain multipartite mixed states
• Genuine multipartite entanglement of symmmetric Gaussian states: Strong monogamy, unitary localization, scaling behavior, and molecular sharing structure
• Scale invariance of entanglement dynamics in Grover's quantum search algorithm
• Entanglement quantification by local unitaries
• Maximally dense coding capable quantum states
• On entanglement in neutrino mixing and oscillations
• Macroscopic Observables Detecting Genuine Multipartite Entanglement and Partial Inseparability in Many-Body Systems
• Entanglement Measures for Intermediate Separability of Quantum States