Title | Inequalities for the ranks of multipartite quantum states |

Publication Type | Journal Article |

Year of Publication | 2014 |

Authors | Cadney, J, Huber, M, Linden, N, Winter, A |

Journal | Linear Algebra and its Applications |

Volume | 452 |

Pagination | 153 - 171 |

Date Published | 07/2014 |

ISSN | 00243795 |

Abstract | We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other \alpha-R\'enyi entropies for α∈(0,1)∪(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α∈(0,1) is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of \alpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., \alpha=1) and 0-R\'enyi entropy are exceptionally interesting measures of entanglement in the multipartite setting. |

DOI | 10.1016/j.laa.2014.03.035 |

Short Title | Linear Algebra and its Applications |