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Swapping and singlet projection as generic entanglement witnesses for quantum Heisenberg spin-$s$ systems

Guang-Ming Zhang, Xiaoguang Wang

Published Jan 19, 2006• 0 citations
OpenAlex

Abstract

Based on the SU($N$) representation of group theory, we derive the generalized SU($N$) invariant swapping operator for the quantum Heisenberg spin-$s$ systems ($N=2s+1$). 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($N$) representation and its conjugate one. For general SU(2) invariant bipartite spin-$s$ 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.

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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.
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.
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.
In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-1/21/21/2 particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form ⟨ψ∣Su^(1)Sv^(2)∣ψ⟩\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle⟨ψ∣Su^(1)​Sv^(2)​∣ψ⟩, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of 2×22\times22×2 complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.
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 study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the SSS-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of R ⁣⊗ ⁣R′R\!\otimes\!R'R⊗R′ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, EndSU(N)(N ⁣⊗ ⁣N)=Span{I,S}\mathrm{End}_{\mathrm{SU}(N)}(N\!\otimes\!N)=\mathrm{Span}\{\mathbb{I},\mathbb{S}\}EndSU(N)​(N⊗N)=Span{I,S}, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving ddd-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, E⋆(2)=34E_\star^{(2)}=\tfrac{3}{4}E⋆(2)​=43​ for SU(2)SU(2)SU(2) and E⋆(3)≃0.91E_\star^{(3)}\simeq0.91E⋆(3)​≃0.91, independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift E⋆(N)E_\star^{(N)}E⋆(N)​, suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.