Monogamy constraints on entanglement of four-qubit pure states

We report a set of monogamy constraints on one-tangle, two-tangles, three-tangles and four-way correlations of a general four-qubit pure state. It is found that given a two-qubit marginal state $\rho$ of a four qubit pure state $\mid {\mathrm{\Psi }}_4⟩$, the non-Hermitian matrix $\rho \tilde{\rho }$ where $\tilde{\rho }$ $=\left( \sigma_{y} \otimes\sigma_{y}\right) \rho^{\ast}\left( \sigma_{y}\otimes\sigma_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in state $\rho$ but also about three tangles involving the selected pair as well as four-way correlations of the pair of qubits in $\left\vert \Psi_{4}\right\rangle $. To extract information about tangles of a four-qubit state $\mid {\mathrm{\Psi }}_4⟩$, the coefficients in the characteristic polynomial of matrix $\rho \tilde{\rho }$ are analytically expressed in terms of $2\times 2$ matrices of state coefficients. Four-tangles distinguish between different types of entangled four-qubit pure states.