Quantum Channel Capacity of Traversable Wormhole

Abstract

We formulate the Gao-Jafferis-Wall traversable wormhole protocol as a quantum channel and compute its quantum channel capacity. We show that this capacity is governed by the time derivative of an out-of-time-ordered correlator, hence by operator size growth in the holographic dual, and that its growth is bounded above by the Einstein gravity limit. The channel capacity therefore provides a natural benchmark for quantum simulations of traversable wormholes.

Figures (8)

• Figure 1: The Penrose diagram of traversable wormhole. We use orange line to denote $a^{+}$, which is the null shift experienced by $\phi$ particles due to the shockwave created by $O$ particles along $X^{+}$ direction. In the diagram, we ignore the backreaction of $\phi$ particles on $O$ particles.
• Figure 2: Schematic process of a quantum channel. $M_r$, $A$, $B$, $E$ denotes the Hilbert space $\mathcal{H}_{M_r}, \ \mathcal{H}_A, \ \mathcal{H}_B,\ \mathcal{H}_E$ respectively. And $\mathcal{H}_{M_r}$ is the reference system to purify the state $\rho_A\in\mathcal{H}_A$. $U$ is the isometric extension related to the quantum channel. $\mathcal{H}_E$ is the environment.
• Figure 3: The OTOC configuration related to the coherent information, where $\epsilon$ is the Euclidean cutoff introduced for energy regularization.
• Figure 4: Left: Plot of the relation between the OTOC and $t$ with maximal chaos and stringy corrections, respectively, where $g=100$, $\epsilon=1$, $\Delta_\phi=\Delta_O=1$ and $4C=10^6$. Right: Plot of the relation between $C_Q(\mathcal{N})$ and $t$. The blue curves represent the OTOC and $C_Q(\mathcal{N})$ of JT gravity and the yellow curves represent the case with stringy correction ($a=1/2$).
• Figure 5: $n=2$ replicated geometry related to $\text{Tr}(\rho_L^2)$. The dashed purple lines represent the double trace operator, and the dashed blue lines represent the contraction between $\phi_R^a$s. The yellow lines represent the state $\rho_{\alpha}$. Here we have two different kinds of contractions of the $\phi$ particles. On the left, we have $(\text{Tr}\rho_{\alpha})^2$. In the right figure, the Wick contraction leads to $\text{Tr}\rho_{\alpha}^2$.