Quantum Entanglement Capacity with Classical Feedback

TL;DR

This paper introduces the quantum entanglement capacity with classical feedback ($E_B$), explores its relation to other capacities, and provides methods to convert quantum error-correcting codes into adaptive protocols, offering new bounds for the depolarizing channel.

Contribution

It defines the $E_B$ capacity, relates it to existing capacities, and develops conversion schemes for quantum codes into adaptive protocols with new capacity bounds.

Findings

$E_B$ lies between $Q_2$ and $Q_B$ capacities.

Conversion schemes for quantum codes into adaptive protocols.

New lower bounds on quantum capacity with classical feedback for depolarizing channels.

Abstract

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 ($Q_2$) and the quantum capacity with classical feedback ($Q_B$) - are widely conjectured to be different: there exists quantum discrete memoryless channel for which $Q_2>Q_B$. 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 Cat (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 Cat 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 new lower bounds on the quantum capacity with classical feedback of the quantum depolarizing channel.