One-shot quantum state exchange

Quantum state exchange is a quantum communication task in which two users exchange their respective quantum information in the asymptotic setting. In this paper, we consider a one-shot version of the quantum state exchange task, in which the users hold a single copy of the initial state, and they exchange their parts of the initial state by means of entanglement-assisted local operations and classical communication. We first derive lower bounds on the least amount of entanglement required for carrying out this task, and provide conditions on the initial state such that the protocol succeeds with zero entanglement cost. Based on these results, we study how the users deal with their symmetric information in order to reduce the entanglement cost. Moreover, we show that it is possible for the users to gain extra shared entanglement after this task.

Figure 1

Illustrations for three one-shot quantum state exchange protocols ${\mathcal{E}}_{\psi ,K,L}^1, {\mathcal{E}}_{\psi ,K,L}^{1|2}$, and ${\mathcal{E}}_{\psi ,K,L}^{12}$: In each illustration, Alice and Bob can apply local operations to their parts represented by circles, while they cannot apply any local operations to those depicted by squares. Shaded circles indicate the systems which are exchanged from the OSQSE protocols.

Figure 2

The graph of the function $max\{f_{{\psi }_1}^{\left(2\right)}\left(\alpha \right),f_{{\psi }_1}^{\left(3\right)}\left(\alpha \right)\}$ for a specific initial state $|{\psi }_1\rangle$ in Eq. (29). The maximum of the function is attained at the point ${\alpha }_0(\approx 3.362)$, leading to an improved converse bound compared to that in Ref. [2]. In the graph, ${\alpha }_0$ is represented as the yellow dashed line, and 1 is represented as the red dash-dotted line.

Figure 3

Illustration of the one-shot quantum state exchange protocol of $|{\varphi }_2\rangle$ in Eq. (41). (a) In order to exchange $A_1$ and $B_1$, Alice and Bob locally prepare an ebit each, and they apply Bell measurements to the shaded areas. (b) By performing local operations corresponding to the measurement outcomes, the parts $A_1$ and $B_1$ can be exchanged. At the same time, Alice and Bob can share two ebits.