Controlled Hawking Process by Quantum Information Masahiro Hotta Tohoku University arXiv: ブラックホールと量子エネルギーテレポーテーショ ン 改め

Black Hole Entropy Problem However, in 1974, S. Hawking argued that black holes emit thermal radiation by taking account of quantum effect of fields. This essentially establishes the belief that black holes are thermo-dynamical existence, which have entropy proportional to event horizon area. In classical theory, black holes are considered to absorb matter, but emit nothing. By virtue of the no-hair theorem, a black holes was regarded as a candidate of the final state of matter. Usually thermo-dynamical systems have statistical dynamical description using entropy concept. However, the statistical dynamical picture of black holes has not been established yet. This is called the black hole entropy problem. thermal radiation gravitational constant

In this presentation, we shed light on the problem by using a gedanken experiment of quantum energy teleportation (QET). The argument does not require un-established physics of quantum gravity. By using only semi-classical analysis, we can make significant statements about memory of black holes. In classical description of black holes, after the outside spacetimes are stabilized, they forget any memories except mass, charge and angular momentum (no-hair theorem). However, we can show that even after the outside spacetime become static, black holes still remember in a quantum mechanical way what they swallowed. Thermal relaxation process of black holes is also suggested, which time scale is of order of the inverse of black hole temperature.

The QET protocols attain energy transfer by Local Operations and Classical Communication (LOCC) without breaking causality and local energy conservation.

Nothing! Let us consider two empty boxes placed separately.

energy Let us infuse energy to one of the boxes.

Nothing! Something happens inside the box and outputs a numerical number 0 or 1 to be announced to the other box. energy Abracadabra 0 ! (or 1! )

In the other box, some operation is performed dependent on the announced abracadabra (0 or 1). After that, energy can be extracted from the box which was empty at the beginning. energy Abracadabra 0 ! (or 1! ) teleported energy! Scientific magic tricks can achieve this amazing magic !!

“Nothing” POINT: WHAT is “NOTHING” in Quantum Physics ? 無

NOTHING in Quantum Physics =GROUND STATE with Quantum Fluctuation (Quantum Fluctuation) GROUND STATE entanglement

○ Spin Chains ○ Trapped Ions ○ Massless Quantum Fields ○ Quantum Hall Edge Currents ○ 1 Dim. BEC (Trapped Atoms) ○ Carbon Nano Tubes ○ SQUID Chains ： QET protocols can be considered in a lot of quantum systems including ←Today’s talk Today, let us focus on the protocol for quantum fields in near-horizon regions of black holes.

Let us grasp a general idea of this QET mechanism ! M.Hotta, J. Phys. Soc. Jp. 78, (2009) M.Hotta, Phys.Lett.A372(2008)5671 M.Hotta, Phys.Rev.D78(2008)045006

Amplitude of fluctuation Quantum Fluctuation in the Ground State of Many-Body System AliceBob The ground state has many components of quantum fluctuation as superposition of states. In the above figure, red and blue lines simply describe those different components. spin chain

Amplitude of fluctuation Amount of energy increases on average ! If an unitary operation independent of Alice’s measurement result acts on Bob’s qubit, the blue-lined component may become suppressed, but the red-lined component becomes large. Thus, on average, positive amount of energy must be infused in the fluctuation.

Amplitude of fluctuation AliceBob Quantum Fluctuation in the Ground State entanglement In QET case, we use the ground-state entanglement between fluctuation around Alice and fluctuation around Bob.

Amplitude of fluctuation Measurement by AInformation around B via entanglement Alice’s measurement specifies the value of α and its corresponding fluctuation component. In the figure, the blue-lined component is selected and the red-lined component vanishes. Because of the entanglement, Alice’s measurement result α includes information about fluctuation around Bob.

Amplitude of fluctuation Local Unitary Operation by B By getting information about α from Alice, Bob knows how the fluctuation in front of Bob behaves. Bob can choose an appropriate unitary operation in order to suppress the fluctuation.

Amplitude of fluctuation A B Extraction of Energy from Spin Chain Local Operation of Bob dependent of measurement results By squeezing the fluctuation locally, Bob can obtain energy from the spin chain. The extracted energy was hidden in Bob’s region from the start ! Thus no energy carrier is hired in the QET protocol !!

Measurement information Impact to Black Hole Entropy Problem ? 1bit A B QET in BH spacetime = Controlled Hawking Process by Measurement Information

b=0 or 1 Horizon t=0 t=T unitary operation and energy extraction measurement and energy injection positive energy flux negative energy flux Controlled Hawking Process by Measurement Information

positive energy flux If B does not perform the operation, ….

positive energy flux negative energy flux If B performs the operation, the horizon recedes ! The black-hole entropy decreases !

Horizon （１） For N fields, A performs the measurement. measurement results The result-dependent wave packets are absorbed by the black hole. The amount of energy of the wave packets does not depends on the measurement result.

Shifted Horizon extracted energy （２） B performs result- dependent operations for N fields. absorbed information about the measurement result The wave packets with negative energy are absorbed by the black hole and recede the horizon.

Horizon If B uses incorrect information for the operations, the amount of energy extracted by QET decreases.

The amount of energy infused by the first measurement is independent of the measurement result. Thus, the outside spacetimes of black holes are also independent of the result from the classical theoretical point of view. This implies that the classical black holes do no remember the detailed information of falling matters. However, the black holes still remember what they swallowed in a quantum mechanical way. Via the QET protocol, we can extract energy from the black holes by using correct measurement results. If the operations is based on incorrect information of the measurement result, the amount of extracted energy decreases. Note that the amount of energy decreases in time. This describes thermal relaxation of the black holes. The time scale of the relaxation is of order of inverse of the black hole temperature, as naturally expected. Brief Summary

In the later, let me explain details by use of equations !

In order to discuss classical behaviors of horizon shift of black holes explicitly, let us adopt a two- dimensional solvable model, CGHS model, for simplicity. The obtained results are essentially the same as those of spherically symmetric black holes of Einstein gravity in higher dimensions.

Classical CGHS MODEL Black Hole solution Event Horizon

Using the light-cone coordinates, Because the horizons stay at, the black hole geometries can be described by the flat metric in near- horizon regions.

Exact Solution with Falling Matter Quantum fields in the classical black-hole spacetime can be analytically solved. Falling-Matter ModeOut-Going Mode

large-black-hole and near-horizon limit Even if the incoming energy flux exists, the geometry is described by the flat metric near the horizon in these coordinates.

It is possible to observe horizon-shift explicitly using rescaled coordinates such that Solution in the New Coordinates:

The outside spacetime becomes static soon after the matter is absorbed ! Horizon-Shift

Hamiltonian energy density vacuum state Horizon

Quantum Energy Teleportation in Black-Hole Spacetime The thermal equilibrium states of quantum fields in the black-hole spacetime are described by the entangled Hartle-Hawking state. Near the horizon, this state is reduced into the Minkowski vacuum state in the large-mass limit. Let us consider a QET protocol in the near-horizon region in order to investigate quantum properties of black holes.

In the near-horizon region, we get 1-bit information about quantum fluctuation of a field by interacting with a two- level spin and measuring the spin. This indirect measument is described by the following measurement operators: NOTE:

Post-Measurement States of Quantum Fields: The amount of energy for the state is independent of the measurement result and given by This energy is infused by the measurement device. The wave packets with this energy fall into the black hole and increase the mass and entropy of the black hole. Because the amount of energy is independent of the measurement result, the outside spacetime is the same, independent of the result.

Horizon

Classical treatment of spacetime is valid for the large N limit.

After time T passes, perform an operation dependent on the measurement result for the quantum fluctuation. real parameter fixed later Post-Measurement State

Average Energy after the Operation

By setting This implies that positive energy is transferred from the quantum fluctuation to the outside. Simultaneously, wave packets with negative energy are generated during the operation and fall into the black hole. This decreases the mass and entropy of the black hole.

Horizon

extracted energy

If wrong information about the measurement result is adopted, the operation does not extract but gives energy to the flctuation. When the errors happen for ¼ of N fields, the amount of energy extracted from the black hole becomes zero on average.

Horizon

The amount of energy infused by the first measurement is independent of the measurement result. Thus, the outside spacetimes of black holes are also independent of the result from the classical theoretical point of view. This implies that the classical black holes do no remember the detailed information of falling matters. However, the black holes remember what they swallow in a quantum mechanical way. Via the QET protocol, we can extract energy from the black holes by using correct measurement results. If the operations is based on incorrect information of the measurement result, the amount of extracted energy decreases. Note that the amount of energy decreases in time. This describes thermal relaxation of the black holes. The time scale of the relaxation is of order of inverse of the black hole temperature, as naturally expected. Come back to Summary