Entanglement quantification by local unitaries

Abstract

Invariance under local unitary operations is a fundamental property that must be obeyed by every proper measure of quantum entanglement. However, this is not the only aspect of entanglement theory where local unitaries play a relevant role. In the present work we show that the application of suitable local unitary operations defines a family of bipartite entanglement monotones, collectively referred to as "mirror entanglement". They are constructed by first considering the (squared) Hilbert-Schmidt distance of the state from the set of states obtained by applying to it a given local unitary. To the action of each different local unitary there corresponds a different distance. We then minimize these distances over the sets of local unitaries with different spectra, obtaining an entire family of different entanglement monotones. We show that these mirror entanglement monotones are organized in a hierarchical structure, and we establish the conditions that need to be imposed on the spectrum of a local unitary for the associated mirror entanglement to be faithful, i.e. to vanish on and only on separable pure states. We analyze in detail the properties of one particularly relevant member of the family, the "stellar mirror entanglement" associated to traceless local unitaries with nondegenerate spectrum and equispaced eigenvalues in the complex plane. This particular measure generalizes the original analysis of [Giampaolo and Illuminati, Phys. Rev. A 76, 042301 (2007)], valid for qubits and qutrits. We prove that the stellar entanglement is a faithful bipartite entanglement monotone in any dimension, and that it is bounded from below by a function proportional to the linear entropy and from above by the linear entropy itself, coinciding with it in two- and three-dimensional spaces.

13 pages, 3 figures. Improved and generalized proof of monotonicity of the mirror and stellar entanglement

References in corpus (13)

• General Monogamy Inequality for Bipartite Qubit Entanglement
• No-local-broadcasting theorem for quantum correlations
• Hierarchies of Geometric Entanglement
• Divergence of the entanglement range in low dimensional quantum systems
• Theory of ground state factorization in quantum cooperative systems
• Linking a distance measure of entanglement to its convex roof
• Separability and ground state factorization in quantum spin systems
• Entanglement of finite cyclic chains at factorizing fields
• Generic Entanglement and Standard Form for N-mode Pure Gaussian States
• Characterization of separability and entanglement in $(2\times D)$- and $(3\times D)$-dimensional systems by single-qubit and single-qutrit unitary transformations
• Determination of ground state properties in quantum spin systems by single qubit unitary operations and entanglement excitation energies
• Fidelity and Coherence Measures from Interference
• Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations