Quantum Discord Cannot Be Shared

Bohr proposed that the outcome of a measurement becomes objective and real, and, hence, classical, when its results can be communicated by classical means. In this work we revisit Bohr’s postulate using modern tools from quantum information theory. We find a full confirmation of Bohr’s idea: if a measurement device is in a nonclassical state, the measurement results cannot be communicated perfectly by classical means. In this case some part of the information in the measurement apparatus is lost in the process of communication: the amount of this lost information turns out to be the quantum discord. The information loss occurs even when the apparatus is not entangled with the system of interest. The tools presented in this work allow us to generalize Bohr’s postulate: we show that for pure system-apparatus states quantum communication does not provide any advantage when measurement results are communicated to more than one recipient. We further demonstrate the superiority of quantum communication to two recipients on a mixed system-apparatus state and show that this effect is fundamentally different from quantum state cloning.

Figure 1

Ideal classical communication to a distant recipient. Left part of the figure shows the initial situation: the system $\mathcal{S}$ is correlated with the apparatus $\mathcal{A}$. The recipient $\mathcal{R}$ is initially not correlated with $\mathcal{S}\mathcal{A}$. Right part of the figure shows the ideal final setup after the application of LOCC protocol between $\mathcal{A}$ and $\mathcal{R}$. In the ideal case the recipient $\mathcal{R}$ obtains the full information which was initially present in the apparatus $\mathcal{A}$.

Figure 2

Distributing information to two recipients. Left part of the figure shows the initial situation: The apparatus $\mathcal{A}$ is initially correlated with the system $\mathcal{S}$, while two recipients ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ are not correlated with $\mathcal{S}\mathcal{A}$. The final situation after the application of a general quantum channel $\Lambda$ is shown in the right part of the figure. In the desired final case both recipients have the same information about the system, $I(\mathcal{S}\mathrm{\text{:}}{\mathcal{R}}_1)=I(\mathcal{S}\mathrm{\text{:}}{\mathcal{R}}_2)$.

Figure 3

Main figure shows the maximal amount of information attainable by classical means $I^c$ (solid line) and the mutual information $I(\mathcal{S}\mathrm{\text{:}}{\mathcal{R}}_1)=I(\mathcal{S}\mathrm{\text{:}}{\mathcal{R}}_2)$ for the “optimal state-dependent cloning” (dashed line) for the state given in Eq. (3) as a function of $\theta$. The inset shows the difference $I^c-I(\mathcal{S}\mathrm{\text{:}}{\mathcal{R}}_1)$. For ${\theta }^{\prime }<\theta <\pi /4$ with ${\theta }^{\prime }\approx 0.093\pi$ quantum communication outperforms any LOCC protocol, see main text for details.