Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the quantumness) of the state. This extends previous results connecting entanglement to spectral properties related to the state. We show that if the smallest tensor eigenvalue is negative, the state is detected as entangled. While for spin-1 states the positivity of the smallest tensor eigenvalue is equivalent to separability, we show that for higher values of the angular momentum there is a correlation between entanglement and the value of the smallest tensor eigenvalue.
We give a compact, frame-independent characterization of the set of classical second-order moments for a single spin-1 particle. Defining the moment matrix M = 2Q + (1/3) I, we show that a moment pair (s, Q) arises from a positive mixture of spin-coherent states if and only if M is positive semidefinite, M minus ss^T is positive semidefinite, and the trace of M equals one. These necessary and sufficient matrix conditions delimit the classical moment region and yield simple, basis-free witnesses of higher-order tensor nonclassicality, such as bounds on Tr(Q^2). A constructive proof of sufficiency is given in the appendix.
Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.
The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
Multipartite quantum scenarios are a significant and challenging resource in quantum information science. Tensors provide a powerful framework for representing multipartite quantum systems. In this work, we introduce the role of tensor-generated matrices that can broadly be defined as the relationships between an
m-th order
n-dimensional tensor and an
n-dimensional square matrix. Through these established connections, we demonstrate that the classification of the tensor-generated matrix as an
H-matrix implies the original tensor is also an
H-tensor. We also explore various similar properties exhibited by both the original tensor and the tensor-generated matrix, including weak irreducibility, weakly chained diagonal dominance, and (strong) symmetry. These findings provide a method to transform intricate tensor problems into matrices in specific contexts, which is especially pertinent due to the NP-hard complexity of the majority of tensor problems. Subsequently, we explore the application of tensor-generated matrices in analyzing the classicality of spin states. Leveraging the tensor representation, we introduce classicality criteria for (strongly) symmetric spin-
j states, which potentially provide fresh perspectives on the study of multipartite quantum resources. Finally, we extend classical matrix eigenvalue inclusion sets to higher-order tensor
H-eigenvalues, a task that is typically challenging for higher-order tensors. Consequently, we propose representative tensor
H-eigenvalue inclusion sets, such as modified Brauer's Ovals of Cassini sets, Ostrowski sets, and
S-type inclusion sets.
A sufficient condition for a quantum state of a system of spin-1/2 particles to admit a local hidden variable description, i.e., to be classical, is the separability of the density matrix characterizing its state, but not all classical states are separable. This leads one to infer that separability and classicality are two different concepts. These concepts are examined here in the framework of a criterion for identifying the classicality of a system of spin-1/2 particles based on the concept of joint quasiprobability (JQP) for the eigenvalues of spin components [Puri, J. Phys. A 29, 5719 (1996)]. The said criterion identifies a state as classical if a suitably defined JQP of the eigenvalues of spin components in three suitably chosen orthogonal directions is non-negative. In agreement with other approaches, the JQP-based criterion also leads to the result that all nonfactorizable pure states of two spin-1/2 particles are nonclassical. In this paper it is shown that the application of the said criterion to mixed states suggests that the states it identifies as classical are also separable and that there exist states which, identified as classical by other methods, may not be identified as classical by the criterion as it stands. However, results in agreement with the known ones are obtained if the criterion is modified to identify as classical also those states for which the JQP of the eigenvalues of the spin components in two of the three prescribed orthogonal directions is non-negative. The validity of the modified criterion is confirmed by comparing its predictions with those arrived at by other methods when applied to several mixed states of two spin-1/2 particles and the Werner-like state of three spin-1/2 particles [T\`oth and Ac\`{\i}n, Phys. Rev. A 74, 030306(R) (2006)]. The JQP-based approach, formulated as it is along the lines of the
P-function approach for identifying classical states of the electromagnetic field, offers a unified approach for systems of an arbitrary number of spin-1/2 particles and the possibility of linking classicality with the nature of the measurement process.