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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 U 14 U 23 describing two bilinear interactions: we show that entanglement between the noninteracting subsystems 12 and 34 may lead to invariance of ͉⌿͘ under the action of thedoi:10.1103/physreva.80.032329 fatcat:t6yjtzguabegxlvnf5rowdtqte