Strong Monogamy Conjecture for Multiqubit Entanglement: The Four-Qubit Case

We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular, we define a set of monogamy inequalities sharpening the conventional Coffman-Kundu-Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.

Figure 1

Strong monogamy of four-qubit entanglement.

Figure 2

(a) Lower bound to the residual four-tangle ${\tau }_{q_1|q_2|q_3|q_4}^{(4)\text{low}}$ versus the parameter $a$ (here assumed real) for the normal-form states: $|G_{abc}^2⟩$ with $c=b+1=a$ (red solid line), $|G_{ab}^3⟩$ with $b=a/4$ (green dashed line), $|G_{ab}^4⟩$ with $b=a/2$ (blue dotted line), $|G_a^5⟩$ (magenta dot-dashed line), $|G_a^6⟩$ (black dot-dot-dashed line). The residuals stay non-negative for general choices of the parameters $a$, $b$, $c$. (b) Lower bound to the residual four-tangle ${\tau }_{q_i|q_j|q_k|q_l}^{(4)\text{low}}$ versus the one-tangle ${\tau }_{q_i|(q_j{q}_k{q}_l)}^{(1)}$ for $8\times 10^6$ random four-qubit pure states, with four partitions tested per state. Each point is gray scaled according to the SLOCC class of the state, from 90% gray (darkest, class 1) to 20% gray (lightest, class 8). The solid line is saturated by GHZ states. All the data points are above the horizontal axis, verifying the SM inequality (9).