Generalised twirling of quantum channels
Imagine the situation in which Alice has to send a quantum state to Bob, but Bob is affected by a lack of knowledge about what transformation relates his reference frame to Alice's. As a consequence, the states that she sends automatically undergo an operation known as "twirling" and in general appear to Bob as different states. Twirling depends on Bob's knowledge, which is encoded in a probability distribution w(g) over a group of transformations G. Another consequence is that the channels that Alice applies to her states appear to Bob as different channels. Not all the channels that she can apply have a representation in Bob's reference frame. The main result of this paper is a theorem which determines the requirement for a channel in Alice's reference frame to make sense in Bob's, and it specifies how he perceives it. Additionally, we generalise the particular case of minimal knowledge (in which w(g)=1), to any partial knowledge. Due to the generality of the result, a possible application besides characterising the capabilities of the quantum communications between two parties, is to understand how any transformation that is analogous to some twirling (i.e. that is generated by unitary transformations) affects the operations that we may want to perform. With this work, we hope to contribute to a better understanding of quantum resource theory.