Concentrating Tripartite Quantum Information

We introduce the concentrated information of tripartite quantum states. For three parties Alice, Bob, and Charlie, it is defined as the maximal mutual information achievable between Alice and Charlie via local operations and classical communication performed by Charlie and Bob. We derive upper and lower bounds to the concentrated information, and obtain a closed expression for it on several classes of states including arbitrary pure tripartite states in the asymptotic setting. We show that distillable entanglement, entanglement of assistance, and quantum discord can all be expressed in terms of the concentrated information, thus revealing its role as a unifying informational primitive. We finally investigate quantum state merging of mixed states with and without additional entanglement. The gap between classical and quantum concentrated information is proven to be an operational figure of merit for mixed state merging in the absence of additional entanglement. Contrary to the pure state merging, our analysis shows that classical communication in both directions can provide an advantage for merging of mixed states.

Figure 1

Concentrating information in tripartite quantum states. (a) The initial situation: Alice, Bob, and Charlie share a quantum state ${\rho }^{ABC}$; additionally, Charlie has access to a quantum register $R$. Bob and Charlie perform LOCC, aiming to maximize the mutual information between Alice and Charlie. The final state shared by Alice and Charlie is illustrated in (b).