Quantum, classical, and total amount of correlations in a quantum state

We give an operational definition of the quantum, classical, and total amounts of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work (noise) that is required to erase (destroy) the correlations: for the total correlation, we have to erase completely, for the quantum correlation we have to erase until a separable state is obtained, and the classical correlation is the maximal correlation left after erasing the quantum correlations. In particular, we show that the total amount of correlations is equal to the quantum mutual information, thus providing it with a direct operational interpretation. As a by-product, we obtain a direct, operational, and elementary proof of strong subadditivity of quantum entropy.

Figure 1

One can locally implement the CPTP maps $T_A$ and $T_B$ using ancillas and unitaries. These unitaries rotate the initial pure state $\mid \psi ⟩$ to a pure state $\mid \zeta ⟩=(U_A\otimes U_B)\left({\mid 0⟩}_a\mid \psi ⟩{\mid 0⟩}_b\right)$, which hence has the same entanglement as $\mid \psi ⟩$. The conjecture is thus a statement about the pure state $\zeta$: relative to $\zeta$, it states that $I(A_1:B_1)⩽S\left(A_1{A}_2\right)$.

Figure 2

Starting from $\rho$, this figure illustrates the different objectives one has when considering (i) the total correlations, (ii) the quantum correlations, and (iii) the classical correlations. For this purpose we have ignored the subtleties of the asymptotics, and symbolize the noise required to go from one point in state space to another by their distance. Then for (i) we seek the shortest way (minimal noise) from $\rho$ to the manifold of product states (and we expect the target $\pi$ to be $\approx {\rho }_A\otimes {\rho }_B$); for (ii) we seek instead the shortest way from $\rho$ to the convex set of separable states, and going to the optimal point ${\sigma }_1$ and from there on to a product state ${\pi }_1$ may in total yield a suboptimal erasure procedure. Finally, for (iii), we want to go from $\rho$ to a separable state ${\sigma }_2$ of maximal correlation $(=\text{distance from product states})$. Even if the transition from $\rho$ to ${\sigma }_2$ is done by a local randomizing map, it could be that the noise cost is significantly larger than that of going from $\rho$ to ${\sigma }_1$. For the pure state $\rho =\psi$ we have argued in Sec. 3C that all three optimal paths coincide, and that in fact $E_{\mathrm{er}}\left(\psi \right)=C{\ell }_{\mathrm{er}}^{*}\left(\psi \right)=(1∕2)C_{\mathrm{er}}\left(\psi \right)=E\left(\psi \right)$.