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Title: Entanglement quantification by local unitary operations

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 unitary operations 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 operator. To the action of each different local unitary operator there corresponds a different distance. We then minimize these distances over the sets of local unitary operations 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 operator for the associated mirror entanglement to be faithful, i.e., to vanish in and only in separable pure states. We analyze in detail the properties of one particularly relevant member of the family, the ''stellar mirror entanglement'' associated with the tracelessmore » local unitary operations with nondegenerate spectra and equispaced eigenvalues in the complex plane. This particular measure generalizes the original analysis of S. M. Giampaolo and F. 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.« less

Authors:
; ; ;  [1]; ;  [2]
  1. Dipartimento di Matematica e Informatica, Universita degli Studi di Salerno, CNISM, Unita di Salerno, and INFN, Sezione di Napoli-Gruppo Collegato di Salerno, Via Ponte don Melillo, I-84084 Fisciano (Italy)
  2. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD (United Kingdom)
Publication Date:
OSTI Identifier:
22038607
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; EIGENVALUES; ENTROPY; HILBERT SPACE; PURE STATES; QUANTUM ENTANGLEMENT; QUBITS; THREE-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL CALCULATIONS; UNITARITY

Citation Formats

Monras, A, Giampaolo, S M, Gualdi, G, Illuminati, F, Adesso, G, and Davies, G B. Entanglement quantification by local unitary operations. United States: N. p., 2011. Web. doi:10.1103/PHYSREVA.84.012301.
Monras, A, Giampaolo, S M, Gualdi, G, Illuminati, F, Adesso, G, & Davies, G B. Entanglement quantification by local unitary operations. United States. https://doi.org/10.1103/PHYSREVA.84.012301
Monras, A, Giampaolo, S M, Gualdi, G, Illuminati, F, Adesso, G, and Davies, G B. 2011. "Entanglement quantification by local unitary operations". United States. https://doi.org/10.1103/PHYSREVA.84.012301.
@article{osti_22038607,
title = {Entanglement quantification by local unitary operations},
author = {Monras, A and Giampaolo, S M and Gualdi, G and Illuminati, F and Adesso, G and Davies, G B},
abstractNote = {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 unitary operations 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 operator. To the action of each different local unitary operator there corresponds a different distance. We then minimize these distances over the sets of local unitary operations 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 operator for the associated mirror entanglement to be faithful, i.e., to vanish in and only in separable pure states. We analyze in detail the properties of one particularly relevant member of the family, the ''stellar mirror entanglement'' associated with the traceless local unitary operations with nondegenerate spectra and equispaced eigenvalues in the complex plane. This particular measure generalizes the original analysis of S. M. Giampaolo and F. 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.},
doi = {10.1103/PHYSREVA.84.012301},
url = {https://www.osti.gov/biblio/22038607}, journal = {Physical Review. A},
issn = {1050-2947},
number = 1,
volume = 84,
place = {United States},
year = {Fri Jul 15 00:00:00 EDT 2011},
month = {Fri Jul 15 00:00:00 EDT 2011}
}