Handy elementary algebraic properties of the geometry of entanglement

Howard A. Blair, Paul M. Alsing

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure states are characterized in terms of the separation hypersurface.

Original languageEnglish (US)
Title of host publicationQuantum Information and Computation XI
DOIs
StatePublished - Aug 12 2013
EventQuantum Information and Computation XI - Baltimore, MD, United States
Duration: May 2 2013May 3 2013

Publication series

NameProceedings of SPIE - The International Society for Optical Engineering
Volume8749
ISSN (Print)0277-786X
ISSN (Electronic)1996-756X

Other

OtherQuantum Information and Computation XI
CountryUnited States
CityBaltimore, MD
Period5/2/135/3/13

Keywords

  • Computational complexity
  • Entanglement measure
  • Hypermatrix
  • Separable state

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics
  • Computer Science Applications
  • Applied Mathematics
  • Electrical and Electronic Engineering

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  • Cite this

    Blair, H. A., & Alsing, P. M. (2013). Handy elementary algebraic properties of the geometry of entanglement. In Quantum Information and Computation XI [874905] (Proceedings of SPIE - The International Society for Optical Engineering; Vol. 8749). https://doi.org/10.1117/12.2015994