Multipartite Generalization of Quantum Discord

A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.

Figure 1

The tripartite quantum discord and its decompositions. Definitions of quantities are given in Eqs. (8), (12), (13), and (15). The states are (a) Werner-GHZ states ${\rho }_{\mathrm{W}}=\mu |\psi ⟩⟨\psi |+(1-\mu )\mathbb{1}/8$, where $|\psi ⟩=(|000⟩+|111⟩)/\sqrt{2}$; (b) Werner-W states ${\rho }_{\mathrm{W}}$ defined the same as (a), but with $|\psi ⟩=(|001⟩+|010⟩+|001⟩)/\sqrt{3}$; (c) mixed Bell states $\rho =\mu |{\mathrm{\Phi }}_{AB}^{+}⟩⟨{\mathrm{\Phi }}_{AB}^{+}|+(1-\mu )|{\mathrm{\Phi }}_{AC}^{+}⟩⟨{\mathrm{\Phi }}_{AC}^{+}|$, where $|{\mathrm{\Phi }}_{AB}^{+}⟩=(|000⟩+|110⟩)/\sqrt{2}$, $|{\mathrm{\Phi }}_{AC}^{+}⟩=(|000⟩+|101⟩)/\sqrt{2}$; (d) tripartite quantum correlated states $\rho =\mu |000⟩⟨000|+(1-\mu )|+++⟩⟨+++|$. The optimization is performed by minimizing the expression Eq. (8) over all projection measurements ${\mathrm{\Pi }}_j\in \{\cos \theta |0⟩+e^{i\varphi }\sin \theta |1⟩,\sin \theta |0⟩-e^{i\varphi }\cos \theta |1⟩\}$ for the form Eq. (3), giving six parameters to optimize.

Figure 2

Distribution of various entropies during a measurement in bipartite and tripartite systems. (a),(d) The initial state before the measurement. (b) The final state after a measurement on subsystem $A$. The thick outline indicates the measured system. (c),(e) The change in entropy after a measurement on $A$. (f) The change in entropy after measuring both $A$ and $B$. Bipartite entropy contributions are defined according to $I_{A:B}(\rho )=S_A(\rho )+S_B(\rho )-S_{AB}(\rho )$, $J_{A:B}(\rho )=S_B(\rho )-S_{B|{\mathrm{\Pi }}^A}(\rho )=I_{A:B}({\rho }_{{\mathrm{\Pi }}^A})$, $S_{{\mathrm{\Pi }}^A|B}(\rho )\equiv S_{AB}({\rho }_{{\mathrm{\Pi }}^A})-S_B({\rho }_{{\mathrm{\Pi }}^A})$, $\delta S_{{\mathrm{\Pi }}^A}(\rho )=S_A({\rho }_{{\mathrm{\Pi }}^A})-S_A(\rho )$, $d_{A;B}(\rho )=S_{B|{\mathrm{\Pi }}^A}(\rho )-S_{B|A}(\rho )$. Tripartite entropy contributions are defined by $I_{A:B|C}(\rho )=S_{A|C}(\rho )-S_{A|BC}(\rho )$, $I_{A:B:C}(\rho )=I_{A:C}(\rho )-I_{A:C|B}(\rho )$, $\delta S_{B|{\mathrm{\Pi }}^A}(\rho )=S_{B|A}({\rho }_{{\mathrm{\Pi }}^{AB}})-S_{B|A}({\rho }_{{\mathrm{\Pi }}^A})$, ${\mathrm{\Delta }}_{B;{\mathrm{\Pi }}^A|C}(\rho )=J_{A:B|C}(\rho )-K_{A:B|C}(\rho )$, ${\mathrm{\Delta }}_{B;{\mathrm{\Pi }}^A;C}(\rho )=J_{A:B:C}(\rho )-K_{A:B:C}(\rho )$.