State Exchange with Quantum Side Information

We consider a quantum communication task between two users Alice and Bob, in which Alice and Bob exchange their respective quantum information by means of local operations and classical communication assisted by shared entanglement. Here, we assume that Alice and Bob may have quantum side information, not transferred, and classical communication is free. In this work, we derive general upper and lower bounds for the least amount of entanglement which is necessary to perfectly perform this task, called the state exchange with quantum side information. Moreover, we show that the optimal entanglement cost can be negative when Alice and Bob make use of their quantum side information. We finally provide conditions on the initial state for the state exchange with quantum side information which give the exact optimal entanglement cost.

Figure 1

Illustration of state exchange protocol $\mathcal{E}$ with QSI. Starting from an initial state $|\psi {⟩}_{A{C}_{\mathrm{A}}B{C}_{\mathrm{B}}R}$ of Alice, Bob, and a referee ($R$), Alice and Bob exchange their parts $C_{\mathrm{A}}$ and $C_{\mathrm{B}}$, exploiting their respective QSI $A$ and $B$. The ancillary systems $E_{\mathrm{A}}^{\mathrm{in}}$ and $E_{\mathrm{B}}^{\mathrm{in}}$ represent an initial entanglement consumed for the exchanging task, while $E_{\mathrm{A}}^{\mathrm{out}}$ and $E_{\mathrm{B}}^{\mathrm{out}}$ indicate entanglement generated from the task.

Figure 2

Upper bounds $u_1({\psi }_{\lambda })$, $u_2({\psi }_{\lambda })$ and lower bounds $l_1({\psi }_{\lambda })$, $l_2({\psi }_{\lambda })$ to the optimal entanglement cost $e_{\mathrm{opt}}$ for the specific initial state $|{\psi }_{\lambda }⟩$ of Eq. (4) with $0\le \lambda \le 1$.