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 $|{\Psi }_4⟩$, 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 $|{\Psi }_4⟩$. To extract information about tangles of a four-qubit state $|{\Psi }_4⟩$, 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.

20 pages, Three figures, An expanded new version with additional sections on classification of four-qubit states and entanglement transfer. This version is an application to four-qubits of the general formalism reported in a recently uploaded letter "Beyond the entanglement of qubit pair in a mixed state"

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