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The unique and often-weird properties of quantum mechanics allow aninformation carrier to propagate through multiple trajectories of quantumchannels simultaneously. This ultimately leads us to quantum trajectories withan indefinite causal order of quantum channels. It has been shown thatindefinite causal order enables the violation of bottleneck capacity, whichbounds the amount of the transferable classical and quantum information througha classical trajectory with a well-defined causal order of quantum channels. Inthis treatise, we investigate this beneficial property in the realm of bothentanglement-assisted classical and quantum communications. To this aim, wederive closed-form capacity expressions of entanglement-assisted classical andquantum communication for arbitrary quantum Pauli channels over classical andquantum trajectories. We show that by exploiting the indefinite causal order ofquantum channels, we obtain capacity gains over classical trajectory as well asthe violation of bottleneck capacity for various practical scenarios.Furthermore, we determine the operating region where entanglement-assistedcommunication over quantum trajectory obtains capacity gain against classicaltrajectory and where the entanglement-assisted communication over quantumtrajectory violates the bottleneck capacity.

The Entanglement-Assisted Communication Capacity over Quantum Trajectories