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arXiv:quant-ph/0212114 · Physical Review A

Entanglement in SU(2)-invariant quantum spin systems

John Schliemann

Published Jul 11, 2003• 66 citations
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Abstract

We analyze the entanglement of SU(2)-invariant density matrices of two spins ${S}_{1},$ ${S}_{2}$ 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 ${S}_{1}=S,$ ${S}_{2}=1/2$ 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.

DOI
10.1103/physreva.68.012309

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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 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 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 discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices P+\mathfrak{P}_ +P+​ . Since elements of P+\mathfrak{P}_ +P+​ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, P+∈R15\mathfrak{P}_ + \in \mathbb{R}^{15}P+​∈R15 . We define P+\mathfrak{P}_ +P+​ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.
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.
We establish a novel procedure to analyze the entanglement properties of extremal density matrices depending on the parameters of a finite dimensional Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could exhibit Kramers degeneracy. This is done through the extremal density matrix formalism, which allows to extend the conventional variational principle to mixed states. By applying the positive partial transpose criterion in terms of the Correlation and Schlienz-Mahler matrices on the extremal density matrices, we demonstrate that it is possible to reach both pure and mixed entangled states, changing properly the parameters of the Hamiltonian. For time-reversal invariant Hamiltonians, the extremal pure states can be entangled or not and we prove that they are not time-reversal invariants. For extremal mixed states we have in general 5 possible cases: three of them are entangled and the other two separable.
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.