Activation of nonmonogamous multipartite quantum states

The monogamy relation for quantum correlations is not satisfied by all measures for all multiparty quantum states. We prove that an arbitrary quantum state which is nonmonogamous for negativity will become monogamous if a finite number of copies of the same state is provided. We refer to this as activation of nonmonogamous states. We also show that multiple copies of a state satisfy monogamy for negativity if it does so for a single copy. The results are true for all quantum states of an arbitrary number of parties. Moreover, we find that for a three-qubit pure state, we can find another state such that they individually violate the monogamy relation for negativity and when taken together they can satisfy the three-party monogamy relation. This holds for all three-qubit pure states except $1\%$ states from the W class, while it is true for all three-qubit pure states when a four-party monogamy relation is used to check for their activation. We connect monogamy of negativity with genuine multipartite entanglement.

Figure 1

The two scenarios of activation. (a) The multicopy case. Schematic diagram of ${\delta }_{\mathcal{Q}}({\rho }_{1_1{2}_1{3}_1}^1\otimes {\rho }_{1_2{2}_2{3}_2}^2\otimes \cdots )$, where the three-party monogamy score is considered. For activation, we consider two scenarios: (1) when all ${\rho }^i$ are the same and (2) when they are different. In both the cases, activation occurs when ${\delta }_{\mathcal{Q}}({\rho }_{1_1{2}_1{3}_1}^1\otimes {\rho }_{1_2{2}_2{3}_2}^2\otimes \cdots )\ge 0$ although ${\delta }_{\mathcal{Q}}\left({\rho }^i\right)<0$. (b) A schematic representation of the four-party monogamy score of ${\delta }_{\mathcal{Q}}({\rho }_{123}\otimes {\sigma }_{1^{\prime }{2}^{\prime }{3}^{\prime }})$, with 3 and $3^{\prime }$ located on separate locations. We look for positivity of ${\delta }_{\mathcal{Q}}({\rho }_{123}\otimes {\sigma }_{1^{\prime }{2}^{\prime }{3}^{\prime }}$) with ${\delta }_{\mathcal{Q}}\left({\rho }_{123}\right)<0$ and ${\delta }_{\mathcal{Q}}\left({\sigma }_{1^{\prime }{2}^{\prime }{3}^{\prime }}\right)<0$.

Figure 2

Histogram of probability for obtaining ${\delta }_{\mathcal{N}}\left({\rho }_{123}^{\otimes m}\right)\ge 0$ against the minimum number of copies $m$. We Haar uniformly generate ${10}^7$ states from both the GHZ and the W classes [26]. We observe that approximately $88\%$ of states from the GHZ class, which do not satisfy monogamy when a single copy is provided, become monogamous when two copies are considered. For the W class, the same is $47\%$. (Inset) The same plot for only W-class states but for higher $m$. All the axes are dimensionless.

Figure 3

(a) Distribution of negativity monogamy score of three-qubit pure states, $|\psi \rangle$. On the horizontal axis, the value of the tick ($x_i$) indicates that the corresponding probability columns are for $x_i\le {\delta }_{\mathcal{N}}\left(\psi \right)<x_{i+1}$, where $x_{i+1}$ denotes the value of the next tick. A higher number of states from the GHZ class satisfy monogamy compared to the W-class states, confirming the results in Fig. 2. (b) A similar distribution is depicted for ${|\psi \rangle }^{\otimes 2}$. Other notations are the same. We see that almost all states from the GHZ class become monogamous when two copies are considered. The abscissa is in ebits and the ordinate is dimensionless.

Figure 4

Negativity monogamy score vs GGM. We Haar uniformly generate ${10}^7$ arbitrary three-qubit pure states from the GHZ (hollow circles) and the W classes (solid triangles) [51]. ${\delta }_{\mathcal{N}}$ is plotted along the abscissa while the $\mathcal{E}$ is plotted along the ordinate. The blue line corresponds to the generalized GHZ state. The horizontal axis is in ebits, where the vertical axis is dimensionless.

Figure 5

Negativity monogamy score of two copies of randomly generated states against GGM. All notations are same as in Fig. 4.