Effect of a large number of parties on the monogamy of quantum correlations

Monogamy is a nonclassical property that restricts the sharability of quantum correlation among the constituents of a multipartite quantum system. Quantum correlations may satisfy or violate monogamy for quantum states, which was tested mainly for three-qubit states. Here we establish a sufficient condition for monogamy of arbitrary quantum correlation measures of states of an arbitrary number of parties, using which and further numerical results, we obtain evidence for monogamy of measures such as distillable entanglement and relative entropy of entanglement, which are physically important but mathematically intractable, for almost all quantum states of a moderate number of parties. The result is generic and holds for a large class of quantum correlation measures. Nonetheless, we identify important zero Haar measure classes of pure states that remain nonmonogamous with respect to quantum discord and quantum work deficit, irrespective of the number of qubits.

Figure 1

Percentage bar diagram for monogamy scores of entanglement measures for pure $n$-party states. ${10}^5$ random pure quantum states are generated for each $n$. All notations are the same as for Table 1.

Figure 2

Percentage bar diagram for monogamy scores of information-theoretic quantum correlation measures for pure $n$-party states. ${10}^5$ random pure quantum states are generated for each $n$. All notations are the same as in Table 2.

Figure 3

Top panel: Nonmonogamy of Dicke states with respect to quantum discord. For fixed $n(>6)$, ${\delta }_{\mathcal{D}}^{\leftarrow }$ decreases with increasing $r$ and it decays exponentially for large $n$. Bottom panel: Nonmonogamy of Dicke states with respect to quantum work deficit. For fixed $n(>6)$, the trajectory of ${\delta }_{▵}^{\leftarrow }$ resembles an anharmonic potential well. It decays with increasing $r$ (up to $r\le \left[\frac{n}{4}\right]$) and rises in the interval $\left[\frac{n}{4}\right]<r\le \left[\frac{n}{2}\right]$. In both the panels, the horizontal axes are in terms of the number of excitations in the corresponding Dicke state, while the vertical axis in the top panel is in bits and that in the bottom panel is in qubits.

Figure 4

Tangle $\left(\tau \right)$ for Dicke states is plotted against the number of excitations, $r$, present in the $n$-party system. The tangle vanishes for all Dicke states with $r=1$ (i.e., for the $W$ states) and remains positive for Dicke states with $r>1$. Tangle is measured in ebits.