Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0608151 · Journal of Physics A Mathematical and General

Separability and entanglement of identical bosonic systems

Xiaohong Wang, Shao-Ming Fei, Ke Wu

Published Aug 18, 2006• 2 citations
arXivHTMLPDF

Abstract

We investigate the separability of arbitrary n -dimensional multipartite identical bosonic systems. An explicit relation between the dimension and the separability is presented. In particular, for n = 3, it is shown that the property of PPT (positive partial transpose) and the separability are equivalent for tripartite systems.
DOI
10.1088/0305-4470/39/36/l01

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.

arXiv:quant-ph/0010096•Physical Review A•Mar 21, 2001•130 citations

Separability properties of tripartite states with UxUxU-symmetry

Tilo Eggeling, Reinhard F. Werner

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.

arXiv:quant-ph/0003008•Mar 2, 2000•2 citations

Entanglement in the Dicke subspace

Aabhas Gulati, Ion Nechita, Clément Pellegrini

We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.

quant-pharXiv:2602.15800•Quantum•Feb 18, 2026•1 citation

Separability problem for multipartite states of rank at most 4

Lin Chen, Dragomir Ž. Djoković

One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most 4 which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank 2 or 3 is separable and has length at most 4. For separable states of rank 4, we show that they have length at most 6. It is six only for some qubit–qutrit or multiqubit states. It turns out that any PPT entangled state of rank 4 is necessarily supported on a 3⊗3 or a 2⊗2⊗2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most 4: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3⊗3 or 2⊗2⊗2 system contains a product vector if and only if its Plücker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such an expression was already known in the latter case.

arXiv:1301.2372•Journal of Physics A Mathematical and Theoretical•Jun 13, 2013•30 citations

Separability of a mixture of Dicke states

Nengkun Yu

The structural relation between multipartite entanglement and symmetry is one of the central mysteries of quantum mechanics. In this paper, we study the separability of quantum states in the bosonic system. We show that a mixture of multiqubit Dicke states is separable if and only if its partial transpose is positive semidefinite, which confirms the hypothesis of Wolfe and Yelin [E. Wolfe and S. F. Yelin, Phys. Rev. Lett. 112, 140402 (2014)]. We generalize this result to a class of bosonic states in the $d\ensuremath{\bigotimes}d$ system; and for general $d$, we determine its separability is NP-hard although verifiable conditions for separability are easily derived when $d=3,4$.

arXiv:1501.02957•Physical Review A•Dec 16, 2016•43 citations

The distillability of entanglement of bipartite reduced density matrices of a tripartite state

Yize Sun, Lin Chen

Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.

Journal of Physics A Mathematical and Theoretical•Jun 1, 2020•4 citations
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 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.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.
One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most 4 which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank 2 or 3 is separable and has length at most 4. For separable states of rank 4, we show that they have length at most 6. It is six only for some qubit–qutrit or multiqubit states. It turns out that any PPT entangled state of rank 4 is necessarily supported on a 3⊗3 or a 2⊗2⊗2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most 4: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3⊗3 or 2⊗2⊗2 system contains a product vector if and only if its Plücker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such an expression was already known in the latter case.
The structural relation between multipartite entanglement and symmetry is one of the central mysteries of quantum mechanics. In this paper, we study the separability of quantum states in the bosonic system. We show that a mixture of multiqubit Dicke states is separable if and only if its partial transpose is positive semidefinite, which confirms the hypothesis of Wolfe and Yelin [E. Wolfe and S. F. Yelin, Phys. Rev. Lett. 112, 140402 (2014)]. We generalize this result to a class of bosonic states in the $d\ensuremath{\bigotimes}d$ system; and for general ddd, we determine its separability is NP-hard although verifiable conditions for separability are easily derived when d=3,4d=3,4d=3,4.
Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.