Entanglement-invertible channels

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].