Teleportation through the wormhole

$\mathrm{ER}=\mathrm{EPR}$ allows us to think of quantum teleportation as communication of quantum information through spacetime wormholes connecting entangled systems. The conditions for teleportation render the wormhole traversable so that a quantum system entering one end of the ERB will, after a suitable time, appear at the other end. Teleportation requires the transfer of classical information outside the horizon, but the classical bit-string carries no information about the teleported system; the teleported system passes through the ERB leaving no trace outside the horizon. In general the teleported system will retain a memory of what it encountered in the wormhole. This phenomenon could be observable in a laboratory equipped with quantum computers.

Figure 1

Schematic illustration of a quantum circuit for the time-evolution operator $U(t)$.

Figure 2

Time evolution operator for a precursor $U^{†}(t)AU(t)$ Note that the operator $U(t)$ acts on $N$ qubits and $U^{†}(t)$ acts on $N+1$ qubits. The arrows on the right indicate the direction of time-flow [9].

Figure 3

Six and eight qubit scrambling operators.

Figure 4

A trivial way to teleport through the TFD state: The teleportee qubit is combined with one of the entangled qubits and a measurement is carried out in the Bell basis $|\theta ⟩$. The result is sent to Bob by classical communication of two bits. Bob applies the single qubit operator $S^{\theta }$ and recovers the teleportee.

Figure 5

A circuit for teleporting a single qubit through an entangled state of $N+N$ qubits. At this stage the unitary operator $Z$ on Bob’s side is unspecified.

Figure 6

Definition of the symbol $V_{I,j}^{\theta ,\beta }$. It may be thought of as a matrix connecting the $N$ qubit state $I$ to the $N$ qubit state $\alpha$, $j$.

Figure 7

Our choice of $Z^{\theta }$. The complexity of $Z$ is only of order $N$ due to the switchback effect.

Figure 8

Teleportation circuit for acting on Bob’s side at $t_R=0$.

Figure 9

Alice measures two qubits and sends the output $\theta$ to Bob in the form of a classical 2-bit message. Bob applies the unitary $Z^{\theta }$ at $t=0$. to recover the message qubit on his side.

Figure 10

Definition of the $N$ qubit unitary operator $S^{\theta }(t)$. One of the qubits on the right side loops from the bottom of $V^{†}$ to the bottom of $U^{†}(t)$. We’ll call it the “looping qubit”.

Figure 11

Schematic circuit diagram for $S^{\theta }(0)$.

Figure 12

Schematic circuit diagram for $S^{\theta }(t_{*}/2)$.

Figure 13

Schematic circuit diagram for $S^{\theta }(t_{*})$.

Figure 14

The complexity of $S^{\theta }(t)$ reaches a minimum of $N^{1/2}$ at $t=-t_{*}/2$. The light blue line shows the fictitious extrapolation of the complexity to $\mathcal{C}\sim 1$ at $t=t_{*}$.

Figure 15

Bob can accomplish the same thing by applying $S^{\theta }$ at an earlier time $-t_{*}$.

Figure 16

Assuming $S$ is a simple operator of relatively low complexity, it sends out a shock wave that interrupts the signal and shifts it back in time so that it comes out at $t=0$.

Figure 17

Boosting the previous figure allows the coupling to act at $t=0$. The picture now closely resembles the geometry described in [5, 6].

Figure 18

This figure is the same as Fig. 17 except that the shockwave turns back toward the horizon at time $t=-t_{*}/2$.

Figure 19

The initial TFD state is modified by the insertion of a photon.

Figure 20

This figure is the same as Fig. 16 with the addition of an extra photon added to the TFD state. The extra photon is shown as a broken green line.

Figure 21

The figures are adapted from the second reference in [4]. They illustrate the five steps described below: (1) Alice combines her share of the mediator, $A$, with the teleportee $T$, (2) The combined system ($\mathbf{A}\mathbf{T}$) evolves for a scrambling time, (3) Alice performs a complete set of measurements on ($\mathbf{A}\mathbf{T}$) in the computational basis, (4) Alice sends the classical outcome to Bob, (5) Bob applies a unitary rotation, thereby transforming the state of his black hole to $|\mathrm{\Phi }⟩$.

Figure 22

Circuit diagram for large-system teleportation.

Figure 23

Circuit diagram for the teleportation of a black hole.

Figure 24

Message in nontraversable wormhole.

Figure 25

Encoding and minimal complexity decoding.

Figure 26

Decoding by Bob.

Figure 27

(a) $t>\frac{1}{2}{t}_{*}$ (b) $t<\frac{1}{2}{t}_{*}$.

Figure 28

(a) $t<\frac{3}{4}{t}_{*}$ (b) $t>\frac{3}{4}{t}_{*}$.