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arXiv:1907.13290 · Journal of the Korean Physical Society

Demonstration of the Hayden-Preskill Protocol via Mutual Information

Jeong-Myeong Bae, Subeom Kang, Dong-han Yeom, Heeseung Zoe

Published Dec 1, 2019• 2 citations

Abstract

We construct the Hayden-Preskill protocol by using a system of spin-1/2 particles and demonstrate information flows of this system which can mimic black holes. We first define an analogous black hole A as a collection of such particles. Second, we take the particles from inside to outside the black hole to define an analogous system of Hawking radiation B as outside particles. When the black hole and the radiation have the maximum entanglement at the Page time, we take an entangled pair system C and D. The particles of C fall into the black hole while their counterparts of D remain outside. If we assume rapid mixing of the particle states in the black hole A ⋃ C , can the information of C rapidly escape from the black hole like a mirror? We numerically show that if we turn on the rapid mixing in the black hole, the original information of C rapidly escapes from the black hole to outside in the form of the mutual information between B and D. On the other hand, if the mixing between A and C is not enough, the information escapes slowly. Hence, we explicitly demonstrate the original conjecture of Hayden and Preskill. We emphasize that enough mixing is an essential condition to make the Hayden-Preskill protocol functionally work.

DOI: 10.3938/jkps.75.941

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We construct the Hayden-Preskill protocol by using a system of spin-1/2 particles and demonstrate information flows of this system which can mimic black holes. We first define an analogous black hole A as a collection of such particles. Second, we take the particles from inside to outside the black hole to define an analogous system of Hawking radiation B as outside particles. When the black hole and the radiation have the maximum entanglement at the Page time, we take an entangled pair system C and D. The particles of C fall into the black hole while their counterparts of D remain outside. If we assume rapid mixing of the particle states in the black hole A ⋃ C , can the information of C rapidly escape from the black hole like a mirror? We numerically show that if we turn on the rapid mixing in the black hole, the original information of C rapidly escapes from the black hole to outside in the form of the mutual information between B and D. On the other hand, if the mixing between A and C is not enough, the information escapes slowly. Hence, we explicitly demonstrate the original conjecture of Hayden and Preskill. We emphasize that enough mixing is an essential condition to make the Hayden-Preskill protocol functionally work.

The core of the black hole information paradox is whether information falling into the horizon is completely lost during Hawking radiation. Based on connectivity engineering, this paper constructs a one-dimensional spin network with an effective horizon structure. We divide the system into a black hole interior (A) and exterior radiation (B), introducing a cross-horizon tunneling coupling ε. Numerical simulations show that the fidelity of the initial state in region A decays rapidly and plateaus, exhibiting a local information loss illusion; however, the quantum mutual information between regions A and B increases significantly. Furthermore, decoding operations based on the Hayden-Preskill protocol elevate the recovery fidelity to above 0.7 (far exceeding the random baseline), and the maximum entanglement capacity approaches the theoretical maximum 2 ln dim(A). This model requires no gravitational degrees of freedom and relies purely on many-body unitary evolution, providing a minimal deterministic dynamical example for the microscopic implementation of information conservation in Hawking radiation.

We present two particular decoding procedures for reconstructing a quantum state from the Hawking radiation in the Hayden-Preskill thought experiment. We work in an idealized setting and represent the black hole and its entangled partner by

n

EPR pairs. The first procedure teleports the state thrown into the black hole to an outside observer by post-selecting on the condition that a sufficient number of EPR pairs remain undisturbed. The probability of this favorable event scales as

1/d_{A}^2

, where

d_A

is the Hilbert space dimension for the input state. The second procedure is deterministic and combines the previous idea with Grover's search. The decoding complexity is

\mathcal{O}(d_{A}\mathcal{C})

where

\mathcal{C}

is the size of the quantum circuit implementing the unitary evolution operator

U

of the black hole. As with the original (non-constructive) decoding scheme, our algorithms utilize scrambling, where the decay of out-of-time-order correlators (OTOCs) guarantees faithful state recovery.

We study the Hayden-Preskill thought experiment at finite temperature and obtain the decoupling condition that the information thrown into an old black hole can be extracted by decoding the Hawking radiation. We then consider the decoding Hayden-Preskill protocol at finite temperature assuming the observer outside the black hole who has the access to the full radiation and the unitary dynamics of the black hole. We also consider the cases when the Hawking radiation has noise and decoherence in the storage. The decoding probabilities and the corresponding fidelities are calculated. It is shown that for all the three cases we have considered, the decoding fidelities are less than unity in general. This result indicates that at finite temperature, the decoding strategy and the recovery algorithm are harder to realize than that at infinite temperature.

We revisit information retrieval from evaporating black holes in the Hayden-Preskill protocol, treating the black hole dynamics as Haar-random. We compute, down to the first exponentially suppressed terms, all integer-indexed Rényi mutual informations between a black hole, its radiation, and a reference that catalogues Alice's diaries. We find that dropping a diary into a young black hole effectively delays the Page time. We also compute the radiation : diary reflected Rényi entropies, and identify a technical reason why they cannot be continued to the reflected entropy by the replica trick.

A bstract We revisit information retrieval from evaporating black holes in the Hayden-Preskill protocol, treating the black hole dynamics as Haar-random. We compute, down to the first exponentially suppressed terms, all integer-indexed Rényi mutual informations between a black hole, its radiation, and a reference that catalogues Alice’s diaries. We find that dropping a diary into a young black hole effectively delays the Page time. We also compute the radiation: diary reflected Rényi entropies, and identify a technical reason why they cannot be continued to the reflected entropy by the replica trick.

In this paper, we consider the time evolution of entanglement asymmetry of the black hole radiation in the Hayden-Preskill thought experiment. We assume the black hole is initially in a mixed state since it is entangled with the early radiation. Alice throws a diary maximally entangled with a reference system into the black hole. After the black hole has absorbed the diary, Bob tries to recover the information that Alice thought should be destroyed by the black hole. In this protocol, we found that a U ( 1 ) symmetry of the radiation emerges before a certain transition time (the time when the vanishing entanglement asymmetry begins to grow). This emergent symmetry is exact in the thermodynamic limit and can be characterized by the vanishing entanglement asymmetry of the radiation. The transition time depends on the initial entropy and the size of the diary. What is more, when the initial state of the black hole is maximally mixed, this emergent symmetry survives during the whole procedure of the black hole radiation. We successfully explained this novel phenomenon using the decoupling inequality.