We investigate the detection of entanglement in $n$ -partite quantum states. We obtain practical separability criteria to identify genuinely entangled and nonseparable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine $n$ -partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiments.We investigate the detection of entanglement in $n$-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and nonseparable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine $n$-partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiments.

arXiv:1007.4493Physical Review ADec 14, 201049 citations

Symmetric states and dynamics of three quantum bits

Francesca Albertini, Domenico D’Alessandro

The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$ , of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$ , $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$ . The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$ . This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$ . As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

arXiv:2111.07208Quantum Information and ComputationMay 1, 20223 citations

Separability Criteria via Type‐II Discrete Cosine Transform

A. A. Xu, Baozhi Sun, Zhen Wang +1 more

ABSTRACT Quantum entanglement is a fundamental resource in quantum information science. We present three effective separability criteria for arbitrary bipartite quantum states based on the Type‐II discrete cosine transform. The proposed framework unifies and significantly generalizes the computable cross‐norm and realignment criterion (CNNR), the de Vicente criterion, as well as a wide range of existing separability criteria formulated within the Bloch representation framework. Moreover, our approach admits a natural extension to the multipartite setting via the multipartite correlation matrix. Detailed examples demonstrate that our criteria outperform the existing ones in entanglement detection.ABSTRACT Quantum entanglement is a fundamental resource in quantum information science. We present three effective separability criteria for arbitrary bipartite quantum states based on the Type‐II discrete cosine transform. The proposed framework unifies and significantly generalizes the computable cross‐norm and realignment criterion (CNNR), the de Vicente criterion, as well as a wide range of existing separability criteria formulated within the Bloch representation framework. Moreover, our approach admits a natural extension to the multipartite setting via the multipartite correlation matrix. Detailed examples demonstrate that our criteria outperform the existing ones in entanglement detection.

Advanced Quantum TechnologiesSep 1, 20260 citations

We investigate the detection of entanglement in

n

-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and nonseparable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine

n

-partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiments.

The unitary group acting on the Hilbert space

{\cal H}:=(C^2)^{\otimes 3}

of three quantum bits admits a Lie subgroup,

U^{S_3}(8)

, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space

{\cal H}

splits into three invariant subspaces of dimensions

4

2

and

2

respectively, each corresponding to an irreducible representation of

su(2)

. The subspace of dimension

4

is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under

U^{S_3}(8)

. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup

U^{S_3}(8)

. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin

\frac{1}{2}

particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.