Entanglement between collective operators in a linear harmonic chain

We investigate entanglement between collective operators of two blocks of oscillators in an infinite linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. On the one hand, this approach of “physical blocks” meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them, i.e., where in contrast to usually studied “mathematical blocks” not every possible measurement is allowed. On the other, this formalism naturally allows the generalization to blocks which may consist of several noncontiguous regions. We quantify entanglement between the collective operators by a measure based on the Peres-Horodecki criterion and show how it can be extracted and transferred to two qubits. 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 bipartite entanglement between collective operators. Allowing the blocks to consist of a periodic sequence of subblocks, we verify that entanglement scales at most with the total boundary region. We also apply the approach of collective operators to scalar quantum field theory.

Figure 1

Two blocks of a harmonic chain $A$ and $B$. Each block consists of $n$ oscillators and the blocks are separated by $d$ oscillators.

Figure 2

Degree of collective entanglement $\epsilon$ for two blocks of oscillators as a function of their size $n$. (a) The blocks are neighboring $(d=0)$ and entanglement exists for all $n$ and coupling strengths $\alpha$. Plotted are $\alpha =0.99$ (diamonds), $\alpha =0.9$ (squares), and $\alpha =0.5$ (triangles). (b) The same for two blocks which are separated by one oscillator $(d=1)$. The two blocks are unentangled for $n=1$ but can be entangled, if one increases the block size $(n>1)$, although none of the individual pairs between the blocks is entangled.

Figure 3

Two periodic blocks of a harmonic chain $A$ and $B$. Each block can consist of $m$ subblocks with $s$ oscillators each, separated by $d$ oscillators. In the picture $d=1$, $m=2$, $s=3$, and the number of oscillators per block is $n=ms=6$.

Figure 4

Degree of collective entanglement $\epsilon$ for two periodic blocks of oscillators as a function of their total size $n$. (a) Neighboring one-particle subblocks ($d=0$, $s=1$). Entanglement monotonically increases with $n$ and becomes larger as the coupling strength $\alpha$ increases. Plotted are $\alpha =0.99$ (diamonds), $\alpha =0.9$ (squares), and $\alpha =0.5$ (triangles). (b) The coupling is fixed to $\alpha =0.99$ for this and the subsequent figures. There is no separation, $d=0$. Plotted are the cases $s=1,2,5$. For fixed $n$ the entanglement is more or less proportional to the number of boundaries, i.e., inversely proportional to the subblock size $s$. (c) and (d) correspond to the cases $d=1$ and $d=2$, respectively. The dependence on the size of the subblocks is more complicated as there is a tradeoff between having a large number of boundaries (i.e., small $s$) and the fact that one should have large subblocks as individual separated oscillators are not entangled.