arXiv:quant-ph/0003008

Separability properties of tripartite states with UxUxU-symmetry

Tilo Eggeling, Reinhard F. Werner

Published Mar 2, 2000• 2 citations

Abstract

We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition.

DOI: 10.48550/arxiv.quant-ph/0003008

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arXiv:quant-ph/0003008

Separability properties of tripartite states with UxUxU-symmetry

Tilo Eggeling, Reinhard F. Werner

Published Mar 2, 2000• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0003008

Related papers

Separability properties of tripartite states with U ⊗ U ⊗ U symmetry

Tilo Eggeling, Reinhard F. Werner

We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.

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We extend the classification of mixed states of quantum systems composed of arbitrary numbers of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives $1+18+1$ partial separability classes in the tripartite case contrary to a former $1+8+1$ . Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.We extend the classification of mixed states of quantum systems composed of arbitrary numbers of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives $1+18+1$ partial separability classes in the tripartite case contrary to a former $1+8+1$. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.

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State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say,

{\mathcal{B}}^{int}(ABC)

] is a strict subset of the set of states having positive partial transposition across the three bipartite cuts [say,

{\mathcal{P}}^{int}(ABC)

] for all the tripartite Hilbert spaces ${\mathbb{C}}_{A}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}_{B}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}_{C}^{{d}_{3}}$ with $min{{d}_{1},{d}_{2},{d}_{3}}\ensuremath{\ge}2$. The claim is proved by constructing a state belonging to the set

{\mathcal{P}}^{int}(ABC)

but not to

{\mathcal{B}}^{int}(ABC)

. For ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}3}$ with $d\ensuremath{\ge}3$, the construction follows from a specific type of multipartite unextendible product bases. However, such a construction is not possible for ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}3}$ since bipartite systems ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ do not allow any unextendible product bases for arbitrary

n

[Phys. Rev. Lett. 82, 5385 (1999)]. For the three-qubit system, we therefore come up with a different construction.

1+18+1

partial separability classes in the tripartite case contrary to a former

1+8+1

. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.