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The Quantum Eigenvalue Problem and Z-Eigenvalues of Tensors

The quantum eigenvalue problem arises in the study of the geometric measure of the quantum entanglement. In this paper, we convert the quantum eigenvalue problem to the Z-eigenvalue problem of a real symmetric tensor. In this way, the theory and algorithms for Z-eigenvalues can be applied to the quantum eigenvalue problem. In particular, this gives an upper bound for the number of quantum eigenvalues. We show that the quantum eigenvalues appear in pairs, i.e., if a real number $λ$ is a quantum eigenvalue of a square symmetric tensor $Ψ$, then $-λ$ is also a quantum eigenvalue of $Ψ$. When $Ψ$ is real, we show that the entanglement eigenvalue of $Ψ$ is always greater than or equal to the Z-spectral radius of $Ψ$, and that in several cases the equality holds. We also show that the ratio between the entanglement eigenvalue and the Z-spectral radius of a real symmetric tensor is bounded above in a real symmetric tensor space of fixed order and dimension.