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arXiv:quant-ph/0506224 · Journal of Physics A Mathematical and General

State space structure and entanglement of rotationally invariant spin systems

Heinz‐Peter Breuer

Published Sep 28, 2005• 33 citations
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Abstract

We investigate the structure of SO (3)-invariant quantum systems which are composed of two particles with spins j 1 and j 2 . The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P J onto the eigenspaces of the total angular momentum J , and by certain invariant operators Q K which are built out of spherical tensor operators of rank K . It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6- j symbols. The operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the Q K -representation. These results are employed to obtain a complete characterization of spin systems with j 1 = 1 and arbitrary j 2 ⩾ 1. We prove that the Peres–Horodecki criterion of positive partial transposition (PPT) is necessary and sufficient for separability if j 2 is an integer, while for half-integer spins j 2 there always exist entangled PPT states (bound entanglement). We construct an optimal entanglement witness for the case of half-integer spins and design a protocol for the detection of entangled PPT states through measurements of the total angular momentum.

DOI
10.1088/0305-4470/38/41/013

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We investigate the structure of SO (3)-invariant quantum systems which are composed of two particles with spins j 1 and j 2 . The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P J onto the eigenspaces of the total angular momentum J , and by certain invariant operators Q K which are built out of spherical tensor operators of rank K . It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6- j symbols. The operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the Q K -representation. These results are employed to obtain a complete characterization of spin systems with j 1 = 1 and arbitrary j 2 ⩾ 1. We prove that the Peres–Horodecki criterion of positive partial transposition (PPT) is necessary and sufficient for separability if j 2 is an integer, while for half-integer spins j 2 there always exist entangled PPT states (bound entanglement). We construct an optimal entanglement witness for the case of half-integer spins and design a protocol for the detection of entangled PPT states through measurements of the total angular momentum.
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 jjj 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.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix rho\\rhorho is entirely determined by the diagonal elements of rho\\rhorho in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for rho\\rhorho having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
We analyze the entanglement of SU(2)-invariant density matrices of two spins S1,{S}_{1},S1​, S2{S}_{2}S2​ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic-spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with the eigenvalue of largest multiplicity being non-negative. The case S1=S,{S}_{1}=S,S1​=S, S2=1/2{S}_{2}=1/2S2​=1/2 can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki criterion turns out to be a sufficient condition for nonseparability. We also characterize SU(2)-invariant states of two spins of length 1.
We introduce entanglement witnesses for spin ensembles which detect genuine multipartite entanglement using only measurements of the total angular momentum. States that are missed by most other angular-momentum-based witnesses for spin ensembles, which include Greenberger-Horne-Zeilinger states and certain superpositions of Dicke states, can be effectively detected by our witness. The protocol involves estimating the probability that the total angular momentum is positive along equally spaced directions on a plane. Alternatively, one could measure along a single direction at different times, under the assumption that the total spins undergo a uniform precession. Genuine multipartite entanglement is detected when the observed score exceeds a separable bound. Exact analytical expressions for the separable bound are obtained for spin ensembles ${j}_{1}\ensuremath{\bigotimes}{j}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{j}_{N}$ such that the total spin is a half-integer and numerical results are reported for the other cases. Finally, we conjecture an expression for the separable bound when the total spin is not known, which is well supported by the numerical results.
Abstract A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.