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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 $ρ$ of a four qubit pure state $\left\vert Ψ_{4}\right\rangle $, the non-Hermitian matrix $ρ\widetildeρ$ where $\widetildeρ$ $=\left( σ_{y} \otimesσ_{y}\right) ρ^{\ast}\left( σ_{y}\otimesσ_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in state $ρ$ but also about three tangles involving the selected pair as well as four-way correlations of the pair of qubits in $\left\vert Ψ_{4}\right\rangle $. To extract information about tangles of a four-qubit state $\left\vert Ψ_{4}\right\rangle $, the coefficients in the characteristic polynomial of matrix $ρ\widetildeρ$ are analytically expressed in terms of $2\times2$ matrices of state coefficients. Four-tangles distinguish between different types of entangled four-qubit pure states.