Entanglement between smeared field operators in the Klein-Gordon vacuum

Quantum field theory is the application of quantum physics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. One of the most distinct features of quantum physics with respect to classical physics is entanglement or the existence of strong correlations between subsystems that can even be spacelike separated. In quantum fields, observables restricted to a region of space define a subsystem. While there are proofs on the existence of local observables that would allow a violation of Bell’s inequalities in the vacuum states of quantum fields as well as some explicit but technically demanding schemes requiring an extreme fine-tuning of the interaction between the fields and detectors, an experimentally accessible entanglement witness for quantum fields is still missing. Here we introduce smeared field operators which allow reducing the vacuum to a system of two effective bosonic modes. The introduction of such collective observables is motivated by the fact that no physical probe has access to fields in single spatial (mathematical) points but rather smeared over finite volumes. We first give explicit collective observables whose correlations reveal vacuum entanglement in the Klein-Gordon field. We then show that the critical distance between the two regions of space above which two effective bosonic modes become separable is of the order of the Compton wavelength of the particle corresponding to the massive Klein-Gordon field.

Figure 1

Two blocks of a harmonic chain A and B. In this example, each block consists of $n=5$ oscillators and the blocks are separated by $d=3$ oscillators. The physical length $L$ of the blocks and their separation $D$ are related by $D=\frac{d}{n}L$.

Figure 2

Three optimal (maximizing the entanglement measure $\epsilon$) profiles for fixed physical size of the regions $L=\sqrt{2}$. The number $n_{\mathrm{crit}}$ of oscillators in one block is the smallest that allows entanglement for the given number $d$ of oscillators between the two blocks. For plot (a) $d=2$, $n_{\mathrm{crit}}=8$, for (b) $d=8$, $n_{\mathrm{crit}}=43$ and for (c) $d=16$, $n_{\mathrm{crit}}=88$. The plotted profiles are for the left block; the profiles for the block on the right have the same shape, mirrored.

Figure 3

Dependence of the critical number $n_{\mathrm{crit}}(d)$ of sites within the blocks on the number $d$ of sites between the blocks for different region sizes $L$. The critical value $n_{\mathrm{crit}}(d)$ is defined as the minimal number of sites that gives entanglement of the corresponding collective operators. For each $L$ a linear law is evident, the linear coefficient is defined as $1/C(L)$.

Figure 4

Dependence of the coefficient $C(L)$ on the region size $L$. In the graph $\frac{1}{C(L)}$ is compared with a linear law.

Figure 5

Two mirroring triangular profiles for the support’s size $L=4$, tip position $s=0.75$ and distance between the supports $D=2$.

Figure 6

Minimal length of the profiles’ support $L_{\min }$, for which entanglement appears, plotted as a function of their separation $D$. Inset: logarithmic plot of the same dependence.

Figure 7

(a) Degree of entanglement $\epsilon$ maximized over the size of the profiles’ support $L$ and the tip position $s$ as a function of the separation $D$, i.e. maximal available entanglement max[$\epsilon$] as a function of the separation of the profiles $D$. (b): Logarithmic plot of the same dependence.

Figure 8

Optimal and minimal length of the profiles’ supports, plot (a), and optimal position of the triangles’ tip, plot (b), as functions of the separation $D$ of the profiles’ support. By optimal values of parameters we understand such that maximize our entanglement measure $\epsilon$.

Figure 9

Entanglement as a function of the size of the profiles’ support $L$ for given separation of the profiles, $D=0.2$ for plot (a) and $D=0.1$ for plot (b). To obtain each point on the plots, the degree of entanglement $\epsilon$ was maximized over the tip position $s$.

Figure 10

Entanglement as a function of the tip position $s$ for a few chosen sizes of profiles’ supports, $L$, and the separation of the profiles: $D=0$ for plot (a) and $D=0.15$ for plot (b). $L_{\mathrm{opt}}$ indicates the value of $L$ parameter giving maximal entanglement for the relevant value of the separation parameter $D$, (compare Fig. 8).

Figure 11

Results for the symmetrical triangles, i.e. profiles (29) with $s=\frac{1}{2}$. Plot (a): for given separation $D$, we maximize $\epsilon$ over the size of the supports $L$ keeping $s=\frac{1}{2}$. In this way, we obtain the maximal available entanglement for symmetrical profiles as a function of the separation. Plot (b): optimal lengths of the symmetric and general profiles. By optimal we understand those that maximize the degree of entanglement $\epsilon$.