Entanglement-saving channels (Q2798674)

The text investigates quantum channels (completely positive normalised maps) with respect to the amount of entanglement with a spectator system they preserve. Specifically, consider a maximally entangled state of a bipartite system and apply a quantum channel \(\varphi\) to one subsystem, i.e. subject the bipartite system to the channel \(\varphi\otimes 1\).NEWLINENEWLINEThe extreme cases are realised if the resulting state is separable (no matter what the initial maximally entangled state was) -- \(\varphi\) is called \textit{entanglement breaking} -- or if the resulting state is still maximally entangled in which case \(\varphi\) is called \textit{entanglement preserving}. Intermediate cases are in particular \textit{entanglement saving} maps, those for which \(\varphi^n\) is not entanglement breaking and \textit{asymptotic entanglement saving}, those for which this even holds in the limit \(n\to\infty\).NEWLINENEWLINEThe text characterises such maps: According to Horodecki etal., entanglement breaking channels are characterised as measuring the state in a POVM and the reconstructing the state accordingly, i.e. \(\phi = \sum_i \rho_i tr(E_i\cdot)\) while this text characterised entanglement preserving channels to be unitary (with proof which is claimed to be original).NEWLINENEWLINEFor entropy saving channels it is demonstrated that these are those which have a semipositive fixed point or which have more than one (counting multiplicity) eigenvalue on the unit circle. Asymptotic entropy saving channels are those that have at least two non-commuting eigenstates with eigenvalue on the unit circle.NEWLINENEWLINEThe paper is set in the finite dimensional situation and works out specifically the two-dimensional case of qubits.