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arXiv:0705.3534 · Physical Review A

Circulant states with positive partial transpose

Dariusz Chruściński, Andrzej Kossakowski

Published Sep 10, 2007• 48 citations
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Abstract

We construct a large class of quantum $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure---that is why we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well-known examples of PPT states from the literature and gives rise to a huge family of completely new states.

DOI
10.1103/physreva.76.032308

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We construct a large class of quantum d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure---that is why we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well-known examples of PPT states from the literature and gives rise to a huge family of completely new states.
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We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of U(d)\mathrm{U}(d)U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.