Entanglement-induced invariance in bilinear interactions

We point out a symmetry exhibited by pairs of entangled states and discuss its possible applications in quantum information. More specifically, we consider quadripartite systems prepared in bipartite product states of the form = 12 34 and let the uncorrelated subsystems 14 and 23 interact by a given unitary U14 U23 describing two bilinear interactions: we show that entanglement between the noninteracting subsystems 12 and 34 may lead to invariance of under the action of the unitary, i.e., make an eigenstate of U14 U23. We call this phenomenon entanglement induced transparency and investigate its occurrence both in continuous variable and qubit systems. We also discuss its possible applications to bath engineering, double swapping and remote inversion.