Internal Entanglement and External Correlations of Any Form Limit Each Other

We show a relation between entanglement and correlations of any form. The internal entanglement of a bipartite system, and its correlations with another system, limit each other. A measure of correlations, of any nature, cannot increase under local operations. Examples are the entanglement monotones, the mutual information that quantifies total correlations, and the Henderson-Vedral measure of classical correlations. External correlations, evaluated by such a measure, set a tight upper bound on the internal entanglement that decreases as they increase, and so does quantum discord.

Figure 1

Schematic representation of the systems and correlations considered.

Figure 2

Maximum internal entanglement as a function of external correlations, for a system $A$ consisting of two two-level systems, the entanglement of formation $E_f$, and the measure of total correlations $C^{(D_H)}$ (solid line). The maximum entanglement $E_f({\rho }_A)$ for classical-classical states $\rho$ is given by the dashed line. This line is also the maximum value of $E_f({\rho }_A)$ as a function of $C^{(D_B)}(\rho )$ for all states $\rho$.