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Journal of Physics Conference Series

Permutations and quantum entanglement

Dariusz Chruściński, Andrzej Kossakowski

Published Mar 1, 2008• 1 citation
Publisher

Abstract

We construct a large class of quantum d⊗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 which is governed by by cyclic permutation from the symmetric group Sd-i. It turns out that partial transposition maps any such decomposition into another one corresponding to 'complementary' permutation. 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
10.1088/1742-6596/104/1/012002

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We construct a large class of quantum d ⊗ 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 way 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.
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.
We introduce a class of bipartite operators acting on H⊗H\mathcal{H} \otimes \mathcal{H}H⊗H (H\mathcal{H}H being an nnn-dimensional Hilbert space) defined by a set of nnn Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.
We introduce a class of bipartite operators acting on H⊗H (H being an n-dimensional Hilbert space) defined by a set of n Completely Different Permutations (CDPs). Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group, then the corresponding bipartite operator displays an additional property – the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of the so-called PPT (positive under partial transposition) states which are of great importance for quantum information. Using well-known relation between bipartite operators and linear maps, one can use also construct linear maps related to CDPs.