Stabilizer Complexity of Hawking Radiation Demonstrates Exponential Growth Past the Page Transition

The enduring puzzle of how information escapes black holes receives fresh scrutiny in new research exploring the complexity of Hawking radiation, the thermal radiation predicted to emanate from these cosmic objects. Ritam Basu, Onkar Parrikar, and Suprakash Paul, all from the Tata Institute of Fundamental Research, alongside Harshit Rajgadia, investigate this phenomenon using the concept of ‘stabilizer complexity’, a measure of how difficult it is to simulate the radiation classically. Their calculations reveal a crucial shift in complexity around the ‘Page transition’, the point at which the black hole is predicted to begin releasing information, demonstrating a dramatic increase in the resources needed for simulation. By developing both specific models and a universal formula for calculating this complexity, the team offers new insights into the fundamental mechanisms governing information retrieval from black holes and advances our understanding of quantum gravity.

Researchers interpret Wigner negativity as a measure of computational complexity, effectively quantifying the difficulty of simulating systems in various simplified models of evaporating black holes. They calculated the negativity of Hawking radiation and demonstrated that it remains constant before the Page transition, but increases exponentially afterwards. The team also derived a universal formula for Wigner negativity, independent of the specific computational basis used, by directly calculating it using established gravitational models.

Holographic Complexity and Quantum Gravity Links

A substantial body of work explores the intersection of quantum gravity, black holes, quantum information theory, and computational complexity. Early investigations laid the groundwork for understanding how to map information between gravity and quantum systems, connecting entanglement structure to the geometry of spacetime. Key developments addressed the black hole information paradox, proposing mechanisms for information preservation through concepts like random tensor networks and coarse-grained black hole interiors. A major focus has emerged on quantum complexity, with researchers investigating how the complexity of quantum states relates to black hole horizons and the second law of thermodynamics.

Recent research explores the use of tensor networks to construct holographic states and reconstruct bulk geometry, offering new insights into the emergence of spacetime. Investigations into quantum chaos and scrambling seek to understand the relationship between these phenomena and black holes. Emerging areas of study include the concept of “magic” in quantum states, its connection to computational complexity, and the use of ensemble methods to study black holes. Krylov subspaces are also being utilized as a tool for analyzing quantum dynamics and complexity. This rapidly evolving field aims to understand the fundamental relationship between quantum gravity, black holes, quantum information, and computational complexity.

Wigner Negativity Tracks Black Hole Evaporation Complexity

Scientists have made a breakthrough in understanding the complexity of Hawking radiation emitted by evaporating black holes, utilizing Wigner negativity as a measure of computational complexity. Their work demonstrates that the negativity of Hawking radiation exhibits distinct behaviour before and after the Page transition. Before the transition, the negativity remains at a constant level, but it grows exponentially once the Page transition is surpassed. The team derived a universal formula for Wigner negativity, independent of the computational basis used, by directly calculating it using established gravitational models.

Further investigation using a dynamical model of black hole evaporation confirmed this universal behaviour, revealing an initial spike in negativity followed by a settling down to the same predicted values. Expanding beyond specific models, researchers proposed a geometric formula for Wigner negativity applicable to more general holographic states, linking complexity to the size of the entanglement wedge. This formula links negativity to the difference between the areas of extremal surfaces, with a larger area difference corresponding to exponentially higher negativity. The presence of a specific region in the entanglement wedge directly correlates with exponentially high complexity, offering a new perspective on the relationship between geometry and computational complexity in the context of black holes. These findings provide a powerful new tool for characterizing the complexity of quantum systems and exploring the fundamental nature of black hole evaporation.

Wigner Negativity Tracks Black Hole Complexity

This research investigates the complexity of Hawking radiation emitted from an evaporating black hole, employing a measure known as Wigner negativity. Wigner negativity serves as a proxy for the computational difficulty of simulating the radiation, effectively quantifying its complexity. The team calculated this negativity in simplified models of black hole evaporation and demonstrated that it remains low before the Page time, but increases dramatically afterwards. Importantly, they derived a general formula that accurately predicts the Wigner negativity across different stages of evaporation. Further analysis using a dynamical model of black hole evaporation confirmed this universal behaviour, revealing an initial spike in negativity followed by a settling down to the same predicted values.

The researchers also proposed a geometric interpretation of Wigner negativity applicable to more general holographic states, linking complexity to the size of the entanglement wedge. Their findings suggest that the complexity of Hawking radiation grows exponentially with time after the Page transition, indicating a fundamental shift in the information content of the emitted particles. While the calculations rely on approximations within the models used, this work provides valuable insights into the information paradox and the nature of quantum gravity.