Tensor eigenvalues and entanglement of symmetric states

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.

Figure 1

Probability distribution of the smallest tensor eigenvalue ${\lambda }_{min}$ for random states on the border of the classical domain, with $j=2$ (black dots), $j=3$ (red crosses), and $j=4$ (blue solid line). These states are the closest classical states to random mixed states and were determined with the quadratic and linear algorithms described in Sec. 4.

Figure 2

Probability distribution of the quantumness $Q\left(\rho \right)$ (3) for states having a positive smallest tensor eigenvalue smaller than ${10}^{-5}$. The states are created by mixing a random mixed state with the maximally mixed state according to (35) and decreasing $a$ until the smallest tensor eigenvalue is close to zero. The numerical uncertainty of the quantumness is of the order of ${10}^{-4}$. The black line with circles, red line with crosses, and blue solid line correspond to spin sizes $j=2,3,4$. These states are all entangled but nevertheless have a positive definite tensor representation.

Figure 3

The quantumness (3) as a function of the smallest tensor eigenvalue (12) for $\sim 60$ 000 randomly generated mixed spin-$j$ states. The top panel corresponds to spin size $j=2$, the middle panel corresponds to $j=3$, and the bottom panel corresponds to spin size $j=4$. There is a clear correlation between the amount of quantumness and the magnitude of the negative smallest tensor eigenvalue; however, this correlation gets weaker for $j=3$ and even weaker for $j=4$.

Figure 4

The quantumness (3) of $\sim 60$ 000 randomly generated spin-6 mixed states as a function of their smallest tensor eigenvalue (12). For this system size there is almost no correlation between the magnitude of the smallest tensor eigenvalue and the quantumness.