The structure of the state spaces of bipartite $N\ensuremath{\bigotimes}N$ quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin $j$ which transform according to an irreducible representation of the rotation group. A positive map $\ensuremath{\vartheta}$ is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation ${\ensuremath{\vartheta}}_{2}=I\ensuremath{\bigotimes}\ensuremath{\vartheta}$ acting on the composite system can be expressed in terms of the invariant $6\text{\ensuremath{-}}j$ symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of $4\ensuremath{\bigotimes}4$ systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive map is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of $4\ensuremath{\bigotimes}4$ systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.The structure of the state spaces of bipartite $N\ensuremath{\bigotimes}N$ quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin $j$ which transform according to an irreducible representation of the rotation group. A positive map $\ensuremath{\vartheta}$ is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation ${\ensuremath{\vartheta}}_{2}=I\ensuremath{\bigotimes}\ensuremath{\vartheta}$ acting on the composite system can be expressed in terms of the invariant $6\text{\ensuremath{-}}j$ symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of $4\ensuremath{\bigotimes}4$ systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive map is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of $4\ensuremath{\bigotimes}4$ systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.

arXiv:quant-ph/0503079Physical Review AJun 23, 200538 citations

Entanglement in Bipartite Quantum States

Ved Prakash Gupta, Prabha Mandayam, Viakalathur Shankar Sunder

Lecture notes in physicsJan 1, 20150 citations

Bound Entanglement for Bipartite and Tripartite Quantum Systems

Hui Zhao, Sha Guo

International Journal of Theoretical PhysicsMar 17, 20151 citation

Asymptotic relative entropy of entanglement for orthogonally invariant states

K. Audenaert, Bart De Moor, K. G. H. Vollbrecht +1 more

For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ${E}_{R}$ is additive, the Rains bound is equal to ${E}_{R}.$ For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ${E}_{R}$ is additive, the Rains bound is equal to ${E}_{R}.$

arXiv:quant-ph/0204143Physical Review ASep 18, 200283 citations

A complete basis of generalized Bell states

Сыч Денис Васильевич, Gerd Leuchs

A generalization of the Bell states and Pauli matrices to dimensions which are powers of 2 is considered. A basis of maximally entangled multidimensional bipartite states (MEMBS) is chosen very similar to the standard Bell states and constructed of only symmetric and antisymmetric states. This special basis of MEMBS preserves all basic properties of the standard Bell states. We present a recursive and non-recursive method for the construction of MEMBS and discuss their properties. The antisymmetric MEMBS possess the property of rotationally invariant exclusive correlations which is a generalization of the rotational invariance of the antisymmetric singlet Bell state.A generalization of the Bell states and Pauli matrices to dimensions which are powers of 2 is considered. A basis of maximally entangled multidimensional bipartite states (MEMBS) is chosen very similar to the standard Bell states and constructed of only symmetric and antisymmetric states. This special basis of MEMBS preserves all basic properties of the standard Bell states. We present a recursive and non-recursive method for the construction of MEMBS and discuss their properties. The antisymmetric MEMBS possess the property of rotationally invariant exclusive correlations which is a generalization of the rotational invariance of the antisymmetric singlet Bell state.

New Journal of PhysicsJan 7, 200990 citations

j

which transform according to an irreducible representation of the rotation group. A positive map $\ensuremath{\vartheta}$ is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation ${\ensuremath{\vartheta}}_{2}=I\ensuremath{\bigotimes}\ensuremath{\vartheta}$ acting on the composite system can be expressed in terms of the invariant $6\text{\ensuremath{-}}j$ symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of $4\ensuremath{\bigotimes}4$ systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive map is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of $4\ensuremath{\bigotimes}4$ systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.

{E}_{R}

is additive, the Rains bound is equal to

{E}_{R}.