Heat capacity as an indicator of entanglement

We demonstrate that the presence of entanglement in macroscopic bodies (e.g., solids) in thermodynamical equilibrium could be revealed by measuring heat capacity. The idea is that if the system was in a separable state, then for certain Hamiltonians heat capacity would not tend asymptotically to zero as the temperature approaches absolute zero. Since this would contradict the third law of thermodynamics, one concludes that the system must contain entanglement. The separable bounds are obtained by minimalization of the heat capacity over separable states and using its universal low-temperature behavior. Our results open up a possibility to use standard experimental techniques of solid-state physics—namely, heat-capacity measurements—to detect entanglement in macroscopic samples.

Figure 1

The minimal variance per site ${\Delta }^2\left(H_{\text{Ising}}\right)/N$ of the Hamiltonian of a transverse Ising ring versus the magnetic field $B$ $\left(J=k=1\right)$. The minimization of the variance is performed over two-translation invariant product states ${\left(|{\varphi }_1⟩|{\varphi }_2⟩\right)}^{\otimes N/2}$.

Figure 2

(Color online) The specific heat per spin (in units $\left|J\right|=k=1$) of a transverse antiferromagnetic Ising chain versus the temperature $T$ and the magnetic field $B$.[17] The values below the red line indicate the region of the $\left(T,B\right)$ space, where entanglement exists. There the variance of the Hamiltonian is lower than for any separable state.

Figure 3

The specific heat per spin (in units $\left|J\right|=k=1$) of a transverse antiferromagnetic Ising chain versus the temperature $T$ for $B=2$. The gray line, $0.4197/T^2$, is our entanglement witness.