Irreversibility of Entanglement Loss

The action of a channel on a quantum system, when non trivial, always causes deterioration of initial quantum resources, understood as the entanglement initially shared by the input system with some reference purifying it. One effective way to measure such a deterioration is by measuring the loss of coherent information, namely the difference between the initial coherent information and the final one: such a difference is “small”, if and only if the action of the channel can be “almost perfectly” corrected with probability one. In this work, we generalise this result to different entanglement loss functions, notably including the entanglement of formation loss, and prove that many inequivalent entanglement measures lead to equivalent conditions for approximate quantum error correction. In doing this, we show how different measures of bipartite entanglement give rise to corresponding distance-like functions between quantum channels, and we investigate how these induced distances are related to the cb-norm.