### 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 language | English (US) |
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Title of host publication | Quantum Information and Computation XI |

DOIs | |

State | Published - Aug 12 2013 |

Event | Quantum Information and Computation XI - Baltimore, MD, United States Duration: May 2 2013 → May 3 2013 |

### Publication series

Name | Proceedings of SPIE - The International Society for Optical Engineering |
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Volume | 8749 |

ISSN (Print) | 0277-786X |

ISSN (Electronic) | 1996-756X |

### Other

Other | Quantum Information and Computation XI |
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Country | United States |

City | Baltimore, MD |

Period | 5/2/13 → 5/3/13 |

### 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

*Quantum Information and Computation XI*[874905] (Proceedings of SPIE - The International Society for Optical Engineering; Vol. 8749). https://doi.org/10.1117/12.2015994