Fast Scramblers (0808.2096v1)

Abstract: We consider the problem of how fast a quantum system can scramble (thermalize) information, given that the interactions are between bounded clusters of degrees of freedom; pairwise interactions would be an example. Based on previous work, we conjecture: 1) The most rapid scramblers take a time logarithmic in the number of degrees of freedom. 2) Matrix quantum mechanics (systems whose degrees of freedom are n by n matrices) saturate the bound. 3) Black holes are the fastest scramblers in nature. The conjectures are based on two sources, one from quantum information theory, and the other from the study of black holes in String Theory.

• The paper’s main contribution is the conjecture that matrix-based quantum systems scramble information in logarithmic time.
• It employs quantum circuit models and ADS/CFT correspondence to demonstrate rapid thermalization mechanisms.
• The work implies that black holes, as described in string theory, are nature’s fastest scramblers, deepening insights into quantum gravity.

Fast Scramblers: Quantum Information, Matrix Models, and Black Holes

The paper "Fast Scramblers" by Yasuhiro Sekino and Leonard Susskind explores the rapid thermalization of information in quantum systems, specifically focusing on matrix quantum mechanics and its potential to achieve the limits of scrambling time. The work presents a set of conjectures, supported by insights from both quantum information theory and string theory, which postulate the fast scrambling capabilities of certain quantum systems, including black holes.

Key Conjectures and Results

1. Logarithmic Scrambling Time: The authors conjecture that the quickest scramblers possess a scrambling time that scales logarithmically with the number of degrees of freedom, $N$. This is contrasted with classical diffusion processes where thermalization typically scales as a polynomial function of the system size.
2. Matrix Quantum Mechanics: It is posited that systems represented by $n \times n$ matrix degrees of freedom (matrix quantum mechanics) effectively saturate the logarithmic bound in scrambling time.
3. Black Holes as Fast Scramblers: The authors assert that black holes, within the framework of string theory, are the fastest scramblers known in nature, supporting this with evidence from the dynamics of D0-brane black holes.

These conjectures are informed by classical notions of thermalization in spin systems and quantum circuits, in addition to the semiclassical physics of black holes. The notion of scrambling is vital as it pertains to the process through which quantum information becomes distributed over a system such that it can no longer be easily accessed or localized.

Mechanisms and Models

To support these assertions, the paper explores several models and mechanisms:

• Quantum Circuits: The authors reference the work of Hayden and Preskill, which uses quantum circuits to simulate scrambling via parallel random unitary operations on q-bits, achieving logarithmic time scaling. This model helps establish a conceptual framework for scrambling without directly relying on Hamiltonian systems.
• Matrix Models: The discussion transitions to matrix models like M(atrix) theory, which is tied to M-theory and string theory. The paper posits that sets of hermitian matrices describing these systems can scramble added information efficiently, achieved through the intrinsic chaotic dynamics represented by these matrix structures.
• ADS/CFT Correspondence: Further exploration into the ADS/CFT context demonstrates how conformal field theory on a unit sphere aligns with swift scrambling predictions, reflecting on the interplay between large N Yang-Mills theory and its gravitational duals.

Black Hole Complementarity and Quantum Cloning

The authors review thought experiments that examine the implications of fast scrambling in black holes for black hole complementarity, addressing the potential for quantum cloning. The fast scrambling nature of black holes, when paired with the Hayden-Preskill framework, aligns with the principles of complementarity, reminding that an observer cannot witness the simultaneous cloning of quantum states.

Implications and Future Directions

The implications of this work for quantum information theory and black hole physics are profound. Understanding scrambling at these theoretical limits can elucidate aspects of quantum gravity and the holographic principle. Moreover, these ideas potentially impact the field of quantum computing, where controlling and exploiting such rapid scrambling might bear applications in error correction and quantum algorithm optimization.

The paper suggests that future investigations could refine the interplay between discrete quantum systems studied in information theory and continuous Hamiltonian systems within theoretical physics, seeking deeper insights into the mechanics of fast scrambling across diverse physical regimes.

Overall, "Fast Scramblers" provides a compelling argument backed by theoretical reasoning and models, proposing substantial implications for our understanding of information dynamics in both isolated quantum systems and astrophysical objects like black holes. Through this multi-faceted approach, the authors offer a fertile ground for further research and discovery in the nexus of quantum information theory, high-energy physics, and cosmology.