He–Vidal monogamy conjecture for the negativity of entanglement

Let $\lvert\psi\rangle$ be a normalised complex vector in the tensor product Hilbert space

$$\mathcal H=\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C.$$

For a subsystem, $\operatorname{Tr}_B$ and $\operatorname{Tr}_C$ denote the corresponding partial traces, and $\Gamma$ denotes partial transpose. He–Vidal conjecture. The inequality

$$N^2(\operatorname{Tr}_B\lvert\psi\rangle\langle\psi\rvert^\Gamma)+N^2(\operatorname{Tr}_C\lvert\psi\rangle\langle\psi\rvert^\Gamma)\leq N^2(\lvert\psi\rangle\langle\psi\rvert^\Gamma)$$

holds for every such $\lvert\psi\rangle$. This asserts monogamy of entanglement for the negativity in arbitrary three-partite systems; the source notes that the property had been proved in specific cases, including three qubits for concurrence, but gives no resolution of the general conjecture.