Monogamy constraints on entanglement of four-qubit pure states

Abstract

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 \(\left| \Psi _{4}\right\rangle \), the non-Hermitian matrix \(\rho {\widetilde{\rho }}\) where \({\widetilde{\rho }}\) \(=\left( \sigma _{y} \otimes \sigma _{y}\right) \rho ^{*}\left( \sigma _{y}\otimes \sigma _{y}\right) \), 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 \(\left| \Psi _{4}\right\rangle \). To extract information about tangles of a four-qubit state \(\left| \Psi _{4}\right\rangle \), the coefficients in the characteristic polynomial of matrix \(\rho {\widetilde{\rho }}\) 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.