Crucial role of quantum entanglement in bulk properties of solids

We demonstrate that two well-established experimental techniques of condensed-matter physics, neutron-diffraction scattering and measurement of magnetic susceptibility, can be used to detect and quantify macroscopic entanglement in solids. Specifically, magnetic susceptibility of copper nitrate (CN) measured in 1963 cannot be described without presence of entanglement. A detailed analysis of the spin correlations in CN as obtained from neutron-scattering experiment from 2000 provides microscopic support for this interpretation and gives the value for the amount of entanglement. We present a quantitative analysis resulting in the critical temperature of 5 K in both, completely independent, experiments below which entanglement exists.

Figure 1

The temperature dependence of spin correlation function $⟨{\mathbf{S}}_0\bullet {\mathbf{S}}_{{\mathbf{d}}_1}⟩$ for neighboring sites in CN. The figure is taken from Ref. 17. The open circles correspond to the temperatures at which measurements were performed, the solid line was obtained from fits in Ref. 17. The dashed line is from this work. It distinguishes temperature ranges with and without entanglement in CN. The critical temperature is around $T_c^{exp}\approx 5\mathrm{K}$ and 0.25 on the $y$ axis is the maximal value for $⟨{\mathbf{S}}_0\bullet {\mathbf{S}}_{{\mathbf{d}}_1}⟩$ achievable with separable states. For $T<T_c^{exp}$ concurrence is given by $C=-2⟨{\mathbf{S}}_0\bullet {\mathbf{S}}_{{\mathbf{d}}_1}⟩-(1∕2)$ and has the same functional dependence on temperature as given in the figure. For $T⩾T_c^{exp}$, $C$ vanishes.

Figure 2

The temperature dependence of magnetic susceptibility of powder CN (triangles) and a single-crystal CN measured at low-field parallel (open squares) and perpendicular (open circles, crosses, filled circles) to the monoclinic $b$ axis. The data and the figure are from Ref. 19. The solid curve is the theoretical curve for a dimer rescaled for the amount of noise estimated from the experiment. This noise is computed as the ratio of the maximal experimental value (averaged over crystal data) and the maximal theoretical value. The dashed curve represents the macroscopic entanglement witness (6) and is rescaled in exactly the same way. The intersection point of this curve and the experimental one defines the temperature range (left from the intersection point) with entanglement in CN. The critical temperature is around $T_c^{exp}\approx 5\mathrm{K}$. Note that the entanglement witness will cut the experimental curve (and hence yield a critical temperature) independently of a particular rescaling procedure.