Entanglement between Collective Operators in a Linear Harmonic Chain

We investigate entanglement between collective operators of blocks of oscillators in a linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. This approach meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them. We quantify entanglement between the collective operators by a measure based on the Peres-Horodecki criterion and detect it by an uncertainty relation. Entanglement between two blocks is found even in the case where none of the oscillators from one block is entangled with an oscillator from the other, showing genuine multi-particle entanglement