Quantum entanglement capacity with classical feedback

For any quantum discrete memoryless channel, we define a quantity called quantum entanglement capacity with classical feedback (E_B), and we show that this quantity lies between two other well-studied quantities. These two quantities—namely the quantum capacity assisted by two-way classical communication (Q₂) and the quantum capacity with classical feedback (Q_B)—are widely conjectured to be different: There exists a quantum discrete memoryless channel for which Q₂>Q_B. We then present a general scheme to convert any quantum error-correcting codes into adaptive protocols for this newly defined quantity of the quantum depolarizing channel, and illustrate with the repetition code and Shor code. We contrast the present notion with entanglement purification protocols by showing that, whilst the Leung-Shor protocol can be applied directly, recurrence methods need to be supplemented with other techniques but at the same time offer a way to improve the aforementioned repetition code. For the quantum depolarizing channel, we prove a formula that gives lower bounds on the quantum capacity with classical feedback from any E_B protocols. We then apply this formula to the E_B protocols that we discuss to obtain lower bounds on the quantum capacity with classical feedback of the quantum depolarizing channel.