We give explicit expressions for canonical states labeling the vast majority of entanglement equivalent classes of symmetric states of qubits and efficient algorithms for reducing a given state to the representative of the class it belongs. This way, we achieve an almost complete classification under local unitary and local invertible transformations for symmetric states. The main tool is a technique introduced in this work, enabling to decompose in a unique way, spin symmetric states into a superposition of spin- $1/2$ coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization.We give explicit expressions for canonical states labeling the vast majority of entanglement equivalent classes of symmetric states of qubits and efficient algorithms for reducing a given state to the representative of the class it belongs. This way, we achieve an almost complete classification under local unitary and local invertible transformations for symmetric states. The main tool is a technique introduced in this work, enabling to decompose in a unique way, spin symmetric states into a superposition of spin-$1/2$ coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization.

arXiv:1402.0987Physical Review ANov 7, 201415 citations

Representations of symmetric states

M. Fannes, D. Testard, André F. Verbeure

Jan 1, 19820 citations

Decomposition of completely symmetric states states

Lilong Qian, Delin Chu

In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state $|x,x\rangle\langle x,x|$ , which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or $N$ . Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to $4$ or $N+1$ . We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state $|x,x\rangle\langle x,x|$, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or $N$. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to $4$ or $N+1$. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.

arXiv:1810.03125Oct 7, 20181 citation

Separable States with Unique Decompositions

Kil-Chan Ha, Seung-Hyeok Kye

arXiv:1210.1088Communications in Mathematical PhysicsMar 10, 201427 citations

Mixed states in one spatial dimension: Decompositions and correspondence with nonnegative matrices

Gemma de las Cuevas, Tim Netzer

We study six natural decompositions of mixed states in one spatial dimension: the matrix product density operator form, the local purification form, the separable decomposition (for separable states), and their three translational invariant analogs. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorizations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorization, the nonnegative factorization, and the positive semidefinite factorization. We also show that a symmetric version of these decompositions corresponds to the symmetric factorization, the completely positive factorization, and the completely positive semidefinite transposed factorization. We leverage this correspondence to characterize the six decompositions of mixed states.We study six natural decompositions of mixed states in one spatial dimension: the matrix product density operator form, the local purification form, the separable decomposition (for separable states), and their three translational invariant analogs. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorizations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorization, the nonnegative factorization, and the positive semidefinite factorization. We also show that a symmetric version of these decompositions corresponds to the symmetric factorization, the completely positive factorization, and the completely positive semidefinite transposed factorization. We leverage this correspondence to characterize the six decompositions of mixed states.

arXiv:1907.03664Journal of Mathematical PhysicsApr 1, 202013 citations

Separable decompositions of bipartite mixed states

Jun‐Li Li, Cong‐Feng Qiao

arXiv:1708.05336Quantum Information ProcessingMar 1, 20186 citations

1/2

coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization.

|x,x\rangle\langle x,x|

, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or

N

. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to

4

N+1

. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.