Pseudo-Entropy in Free Quantum Field Theories

Pseudo-entropy is an interesting quantity with a simple gravity dual, which generalizes entanglement entropy such that it depends on both an initial and a final state. Here we reveal the basic properties of pseudo-entropy in quantum field theories by numerically calculating this quantity for a set of two-dimensional free-scalar field theories and the Ising spin chain. We extend the Gaussian method for pseudo-entropy in free-scalar theories with two parameters: mass $m$ and dynamical exponent $z$. This computation finds two novel properties of pseudo-entropy which we conjecture to be universal in field theories, in addition to an area law behavior. One is a saturation behavior and the other one is nonpositivity of the difference between pseudo-entropy and averaged entanglement entropy. Moreover, our numerical results for the Ising chain imply that pseudo-entropy can play a role as a new quantum order parameter which detects whether two states are in the same quantum phase or not.

Figure 1

$S({\tau }_A^{1|2})$ as a function of the size of the subsytem $N_A$. We set $N=200$ and $z_1=z_2=1$. The curves are $c_1\ln [(N/\pi )\sin (\pi N_A/N)]+c_0$, where $c_1\simeq 0.3333$ and $6.028<c_0<6.453$.

Figure 2

The upper plot shows the pseudo-entropy as a function of the subsystem size $N_A$ when we chose $m_1={10}^{-3}$ and $m_2={10}^{-5}$ for various values of $z_1=z_2$. The lower plot shows the pseudo-entropy when we set $z_1=3$ and $m_1=m_2={10}^{-5}$. We chose the total system $N=100$.

Figure 3

The upper graph shows the pseudo-entropy as a function of $m_2$ when we set $z_1=1$ and $m_1={10}^{-5}$. The lower graph depicts the pseudo-entropy as a function of $z_2$ when we set $m_1=m_2={10}^{-5}$.We chose $N_A=50$ and $N=\infty$.

Figure 4

The plots of the difference $\mathrm{\Delta }{S}_{12}$ as a function of $m_2-m_1$ (upper) and $z_2$ (lower). We set $m_1={10}^{-5}$ and $z_1=z_2$ in the upper graph. We chose $m_1=m_2={10}^{-5}$ in the lower graph.

Figure 5

Pseudo-entropy and average of entanglement for a single interval. Here, we choose $N=16$, $N_A=8$, $h_1=h_2=1$. We set $J_1=2$ (upper left), $J_1=1$ (upper right), and $J_1=1/2$ (lower). The horizontal axis is the value of $J_2$. The blue dots show the pseudo-entropy $S({\mathcal{T}}_A^{1|2})$ and the orange dots show the average of the entanglement entropy $[S({\rho }_A^1)+S({\rho }_A^2)]/2$.