Measuring Entanglement in Condensed Matter Systems

We show how entanglement may be quantified in spin and cold atom many-body systems using standard experimental techniques only. The scheme requires no assumptions on the state in the laboratory, and a lower bound to the entanglement can be read off directly from the scattering cross section of neutrons deflected from solid state samples or the time-of-flight distribution of cold atoms in optical lattices, respectively. This removes a major obstacle which so far has prevented the direct and quantitative experimental study of genuine quantum correlations in many-body systems: The need for a full characterization of the state to quantify the entanglement contained in it. Instead, the scheme presented here relies solely on global measurements that are routinely performed and is versatile enough to accommodate systems and measurements different from the ones we exemplify in this work.

Figure 1

Lower bound $E$ on the entanglement—as measured in terms of the best separable approximation (BSA)—for a thermal state $\widehat{\varrho }=\exp (-\beta \widehat{H})/Z$, $\beta J=J/k_{B}T=1$, of the Heisenberg model in Eq. (4). For every $\mathit{q}$, $E(\mathit{q})$ provides a lower bound to the BSA. The square lattice with open boundary conditions and lattice constant $a=1$ has $30\times 30$ lattice sites, and $⟨\widehat{S}(\mathit{q})⟩$ was obtained using the generalized directed loop QMC algorithm [29] of the ALPS package [21]. Inset shows $E(\frac{n\pi }{16},\frac{n\pi }{16})$, $n=0,4,6,7$ (top to bottom), as a function of the temperature. Lines are a guide to the eye. Note that the BSA is upper bounded by unity, a bound that $E(\mathit{q})$ saturates at low temperatures and $\mathit{q}=\mathbf{0}$.

Figure 2

Lower bound $E(x,y)$ on the entanglement for a thermal state $\widehat{\varrho }=\exp (-\widehat{H}/k_{B}T)/Z$ with constant filling factor $⟨{\widehat{n}}_{\mathit{i}}⟩=1$ of the Bose-Hubbard model in Eq. (13). The three-dimensional cubic lattice with periodic boundary conditions and lattice constant $a=1$ has $10\times 10\times 10$ lattice sites and $⟨\widehat{n}(x,y)⟩$ was obtained using the same numerical code as for Fig. 1. Left-hand plot shows $E(x,y)$ as in Eq. (8) for $\beta U=1/5$, $J/U=0.01$. Right-hand plot shows $E(\frac{n\pi }{64},\frac{n\pi }{64})$, $n=64,48,44,36,34,33$ (top to bottom) as a function of the temperature (black lines) and of the tunneling amplitude $J/U$ (gray lines). Lines are guides to the eye.