Thermal state entanglement in harmonic lattices

We investigate the entanglement properties of thermal states of the harmonic lattice in one, two, and three dimensions. We establish the value of the critical temperature for entanglement between neighboring sites and give physical reasons. Further sites are shown to be entangled only due to boundary effects. Other forms of entanglement are addressed in the second part of the paper by using the energy as witness of entanglement. We close with a comprehensive diagram showing the different phases of entanglement versus complete separability and propose techniques to swap and tune entanglement experimentally.

Figure 1

Negativity $E_{\mathcal{N}}$ of a one-dimensional chain in thermal equilibrium over the temperature $T$ (expressed in multiples of the coupling $\omega$). The trapping potential is small compared to the coupling, $\delta ={10}^{-4}\omega$. The solid curves show the nearest neighbor entanglement for an increasing number of sites: $N=3$ (uppermost),$\dots ,51$ which converge to an asymptotic curve for $N\to \infty$ (lowest of the solid curves). The dashed curves show the next-nearest neighbor entanglement for an increasing number of sites, $N=3$ (uppermost),$\dots ,15$, which vanishes for further increasing $N$.

Figure 2

Entanglement $E_{\mathcal{N}}$ of nearest neighbors in a one-dimensional chain with $N=49$ sites, depicted vs temperature $T$. The different curves represent increasing ratios of the trapping potential versus the coupling strength, with the uppermost curve reaching $E_{\mathcal{N}}\approx 0.3$ at $T=0$ for $\delta /\omega ={10}^{-4}$, and lower curves for the parameters $\delta /\omega ={10}^{-7/2},\dots ,{10}^{-1/2},1$ (dashed),$\sqrt{10}$.

Figure 3

Entanglement of nearest neighbors in 1D (solid line, $N=31$), 2D (dashed, $N={31}^2$), and 3D (points, $N={31}^3$) is shown vs temperature. The trapping potential is $\delta /\omega ={10}^{-4}$ in all cases. For a discussion please refer to the main text. The dimension-independent temperature for any witnessed entanglement, $T_{\text{EW}}$, derived in Sec. 6 is also indicated and discussed in Sec. 7.

Figure 4

Exact scale relations of the different phases of entanglement and separability in the harmonic chain $\left(N=49\right)$, for varying trapping potential $\delta$ and vs the temperature $T$. Nearest neighbor entanglement exists in the dark shaded region up to the temperature threshold $T_{\text{NN}}$. Entanglement is witnessed by the energy in the hatched area, with threshold temperature $T_{\text{EW}}$. Additionally the temperature threshold $T_{\text{blocks}}$ (black point) for the existence of entanglement between two neighboring blocks of size $N/2$ from [1] (for a single value of $\delta$) is displayed. The chain passes into the light shaded region of complete separability at the temperature $T_{\text{crit}}$ from [3]. For $\delta =0$ the threshold values are displayed below the figure; for their discussion please refer to the text.