How to Build the Thermofield Double State

Abstract

Given two copies of any quantum mechanical system, one may want to prepare them in the thermofield double state for the purpose of studying thermal physics or black holes. However, the thermofield double is a unique entangled pure state and may be difficult to prepare. We propose a local interacting Hamiltonian for the combined system whose ground state is approximately the thermofield double. The energy gap for this Hamiltonian is of order the temperature. Our construction works for any quantum system satisfying the Eigenvalue Thermalization Hypothesis.

Figures (8)

• Figure 1: Spectrum of $H_{\text{TFD}}$ in energy window. K=N=50.
• Figure 2: Gap as a function of N for $H_{\text{TFD}}$ of dimension $\text{N}^2 \times \text{N}^2$ and K=50.
• Figure 3: Components of the ground state (in red) and the first excited state (in blue) of the full Hamiltonian in (a) symmetric subspace and (b) the complement of the symmetric subspace. We have set K=N=50.
• Figure 4: Low-lying eigenvalue spectrum for $H_{\text{TFD}}$ of dimension $N^2 \times N^2$ with $N=100$, $K=2$. The gap is finite as seen in Figure \ref{['fig:GapvsNK2']}.
• Figure 5: Components of the ground state (in red) and the first excited state (in blue) of the full Hamiltonian in (a) symmetric subspace and (b) the complement of the symmetric subspace. We have set N=100, K=2.