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arXiv:quant-ph/0601129 · Journal of Physics A Mathematical and General

Spin swapping operator as an entanglement witness for quantum Heisenberg spin-ssystems

Guang-Ming Zhang, Xiaoguang Wang

Published Jun 14, 2006• 15 citations
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Abstract

Using the SU( N ) representation of the group theory, we derive the general form of the spin swapping operator for the quantum Heisenberg spin- s systems. We further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition. For SU(2) invariant states, it is shown that the expectation value of the spin swapping operator and its generalizations, the permutations, can be used as an entanglement witness, especially, for the formulation of observable conditions of entanglement.

DOI
10.1088/0305-4470/39/26/017

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Using the SU( N ) representation of the group theory, we derive the general form of the spin swapping operator for the quantum Heisenberg spin- s systems. We further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition. For SU(2) invariant states, it is shown that the expectation value of the spin swapping operator and its generalizations, the permutations, can be used as an entanglement witness, especially, for the formulation of observable conditions of entanglement.
Swapping and singlet projection as generic entanglement witnesses for quantum Heisenberg spin-sss systems
Based on the SU(NNN) representation of group theory, we derive the generalized SU(NNN) invariant swapping operator for the quantum Heisenberg spin-sss systems (N=2s+1N=2s+1N=2s+1). It has been demonstrated that the partial transposition of the swapping operator is equal to the singlet pairing projection in the tensor product space of the fundamental SU(NNN) representation and its conjugate one. For general SU(2) invariant bipartite spin-sss systems, the expectation value of the swapping operator is found to be the leading term in the negativity expression with respect to the corresponding density matrix. Generalized to the many-body SU(2) invariant states, we prove that the expectation values of the swapping operator and the singlet pairing projector can be regarded as two generic entanglement witnesses, which can be detectable in future experiments.
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 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.
Quantum entanglement, as an indispensable resource in quantum communication and quantum computation, is widely used in the field of quantum information. However, people's understanding on entanglement is quite limited both theoretically and experimentally. How to determine whether a given quantum state is entangled is still an important task. The entanglement witness is a kind of special self-adjoint operator, it can be used to determine whether a quantum state is an entangled state. This provides a new direction for the determination of entangled states. Entanglement witness has its own unique characteristics in various kinds of entanglement criterion. It is the most effective tool for detecting multipartite entanglement, and the most useful method to detect entanglement in experiments. In the background of quantum theory, we use theory of operators to make a thorough and systematic study of the construction of entanglement witness in this paper. First, from the definition of an entanglement witness, a general method is given to construct an entanglement witness. It is proved that when the maximal expectation CA of an observable A in the separable pure states is strictly less than its biggest eigenvalue max(A), the operator WC=CI-A is an entanglement witness provided that CA C max(A). Although the entanglement witness WCA can detect more entangled states than WC, but it is difficult to calculate the exact value of CA, and the estimate of the upper bound of CA is easier. Therefore, it is more convenient to construct entanglement witness WC than WCA. In quantum computation, a graph state is a special kind of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. Graph states play a crucial role in many applications of quantum information theory, such as quantum error correcting codes, measurement-based quantum computation, and quantum simulation. Consequently, a significant effort is devoted to the creation and investigation of graph states. In the last part of this paper, as applications of our method, a series of methods for constructing an entanglement witness is obtained in the stabilizer formalism. It is also proved that how entanglement witnesses can be derived for a given graph state, provided some stabilizing operators of the graph state are known. Especially, when A is made up of some stabilizing operators of a graph state, entanglement witness WCA becomes one in literature.
We discuss the detection of entanglement in interacting quantum spin systems. First, thermodynamic Hamiltonian-based witnesses are computed for a general class of one-dimensional spin-1∕21∕21∕2 models. Second, we introduce optimal bipartite entanglement observables. We show that a bipartite entanglement measure can generally be associated with a set of independent two-body spin observables whose expectation values can be used to witness entanglement. The number of necessary observables is ruled by the symmetries of the model. Illustrative examples are presented.