Controlled Hawking process by quantum energy teleportation

In this paper, a new quantum mechanical method to extract energy from black holes with contracting horizons is proposed. The method is based on a gedanken experiment on quantum energy teleportation, which has been recently proposed in quantum information theory. We consider this quantum energy teleportation protocol for $N$ massless fields in near-horizon regions of large-mass black holes with near-horizon geometry described by the Minkowski metric. For each field, a two-level spin is strongly coupled with the local quantum fluctuation outside the horizon during a short time period. After the measurement of $N$ fields, $N$-bit information is obtained. During the measurement, positive-energy wave packets of the fields form and then fall into the black hole. The amount of excitation energy is independent of the measurement result. After absorption of the wave packets and increase of the black-hole mass, a measurement-result-dependent local operation of the $N$ fields is performed outside the horizon. Then, accompanying the extraction of positive energy from the quantum fluctuation by the operation, negative-energy wave packets of the fields form and then fall into the black hole, decreasing the black-hole mass. This implies that a part of the absorbed positive energy emitted from the measurement devices is effectively retrieved from the black hole via the measurement results.

Figure 1

Spacetime diagram of the QET process in the near-horizon region. The future horizon stays at $x^{-}=0$. At $t=0$, quantum fluctuation of $\widehat{f}$ in the spatial region $[x_1,x_2]$ is indirectly measured by a spin probe. A wave packet with positive energy $E_A$ is generated by the measurement device; thereafter, it falls into the future horizon. At $t=T$, the unitary operation $U_b$, which is dependent on the measurement result $b$, is performed; this operation generates a wave packet with negative energy $-|E_B|$, which falls into the horizon by extracting positive energy from the field.

Figure 2

Spacetime diagram of the horizon shift in the $X$ coordinates. The initial horizon $H_I$ stays at $X^{-}=0$. By absorbing the positive-energy flux generated by the measurement, the horizon moves outwards. $H_F$ denotes the shifted horizon. After the absorption of negative-energy flux generated by $U_b$, the horizon recedes. The shifted horizon is denoted by $H_{\mathrm{QET}}$. The positions of the horizons are fixed soon after the wave packets are swallowed. The argument does not change at all if the falling energy flux interacts with matter within the black hole. In the diagram, right-moving matter, which is represented by dashed lines, collides with the falling positive-energy flux inside the horizon.

Figure 3

Schematic diagram of the measurement process of QET. The $N$ cylinders in the figure represent the measurement devices used to detect the quantum fluctuation of the $N$ fields. In this figure, the $N$-bit output of the devices is $D_k=(101\cdots 0)$. Each detector creates a wave packet with positive energy $E_A$ that is independent of $b$. The wave packets are absorbed by the horizon, which is represented by the shaded region to the left.

Figure 4

Schematic diagram of a unitary operation process that is dependent on the measurement result. The $N$ cubes in the figure represent the operation devices of the unitary operation for the $N$ fields. After the operation, a wave packet with negative energy $-|E_B|$ is emitted from each device and is absorbed by the horizon. Before this, the horizon has already absorbed the wave packets corresponding to $D_k=(101\cdots 0)$ that are generated by the measurement. Local energy conservation assures extraction of positive energy $+|E_B|$ from each field by the device.

Figure 5

Schematic diagram of a wrong operation on the quantum fields. In the figure, a unitary operation corresponding to an $N$-bit data $(011\cdots 1)$ that is different from the correct data $(101\cdots 0)$ is performed. Devices with wrong bit numbers do not generate negative-energy flux, but generate positive-energy flux with $+|E_B^{\prime }|=3{\eta }^2/4\xi$. When the wrong bit number per $N$ exceeds $1/4$, there is no energy gain from the black holes on average by the QET process.