Separability of mixed Dicke states: an NP-hard optimization problem

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in

C^d\otimes C^d

(symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for

d \leq 4

. Furthermore, for

d \geq 5

, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on

N

-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

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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.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:2602.15800QuantumFeb 18, 20261 citation

Separability of diagonal symmetric states: a quadratic conic optimization problem

Jordi Tura, Albert Aloy, Ruben Quesada +2 more

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in C d ⊗ C d (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for d ≤ 4 . Furthermore, for d ≥ 5 , we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on N -partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in C d ⊗ C d (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for d ≤ 4 . Furthermore, for d ≥ 5 , we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on N -partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

arXiv:1706.09423QuantumJan 12, 201846 citations

Necessary and sufficient condition of separability for D -symmetric diagonal states

A. Rutkowski, Michał Banacki, Marcin Marciniak

For multipartite states, we consider a notion of $D$ symmetry. For a system of $N$ qubits, it coincides with the usual permutational symmetry. In the case of $N$ qudits ($d\ensuremath{\ge}3$), the $D$ symmetry is stronger than the permutational one. For the space of all $\mathrm{D}$ -symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of $D$ -symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if $N$ is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for $D$ -invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.For multipartite states, we consider a notion of $D$ symmetry. For a system of $N$ qubits, it coincides with the usual permutational symmetry. In the case of $N$ qudits ($d\ensuremath{\ge}3$), the $D$ symmetry is stronger than the permutational one. For the space of all $\mathrm{D}$-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of $D$-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if $N$ is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for $D$-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.

arXiv:1810.05522Physical Review AFeb 8, 20197 citations

Entanglement and nonlocality in diagonal symmetric states of N qubits

Ruben Quesada, Swapan Rana, Anna Sanpera

We analyze entanglement and nonlocal properties of the convex set of symmetric $N$ -qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.We analyze entanglement and nonlocal properties of the convex set of symmetric $N$-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.

arXiv:1701.03399Physical Review AApr 24, 201718 citations

Four-qubit entangled symmetric states with positive partial transpositions

Jordi Tura, Remigiusz Augusiak, P. Hyllus +3 more

We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical--half-numerical method that allows us to construct multipartite PPT entangled symmetric states (PPTESSs) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing us to search for extremal elements in the convex set of bipartite PPT states [Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESSs and show that generically they have ranks $(5,7,8)$ . Finally, we provide an exhaustive characterization of these states with respect to their separability properties.We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical--half-numerical method that allows us to construct multipartite PPT entangled symmetric states (PPTESSs) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing us to search for extremal elements in the convex set of bipartite PPT states [Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESSs and show that generically they have ranks $(5,7,8)$. Finally, we provide an exhaustive characterization of these states with respect to their separability properties.

arXiv:1203.3711Physical Review AJun 22, 201252 citations

Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits

Christopher Popp, Beatrix C. Hiesmayr

With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states [Formula: see text] are determined to be separable while [Formula: see text] are bound entangled and only [Formula: see text] remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states [Formula: see text] are determined to be separable while [Formula: see text] are bound entangled and only [Formula: see text] remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.

arXiv:2205.11405Scientific ReportsJul 21, 202214 citations

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in

C^d\otimes C^d

d \leq 4

. Furthermore, for

d \geq 5

, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on

N

For multipartite states, we consider a notion of

D

symmetry. For a system of

N

qubits, it coincides with the usual permutational symmetry. In the case of

N

qudits ($d\ensuremath{\ge}3$), the

D

symmetry is stronger than the permutational one. For the space of all

\mathrm{D}

-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of

D

-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if

N

is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for

D

-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.

We analyze entanglement and nonlocal properties of the convex set of symmetric

N

-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.

(5,7,8)

. Finally, we provide an exhaustive characterization of these states with respect to their separability properties.