Information Geometry and Holographic Correlators (2110.15359v2)

Abstract: We explore perturbative corrections to quantum information geometry. In particular, we study a Bures information metric naturally associated with the correlation functions of a conformal field theory. We compute the metric of holographic four-point functions and include corrections generated by tree Witten diagrams in the bulk. In this setting, we translate properties of correlators into the language of information geometry. Cross terms in the information metric encode non-identity operators in the OPE. We find that the information metric is asymptotically AdS. Finally, we discuss an information metric for transition amplitudes.

• The paper demonstrates that perturbative Bures metric corrections encode non-identity operator contributions, clarifying the geometric structure of holographic correlators in CFTs.
• The authors apply tree-level Witten diagram computations to reveal 1/N² corrections, confirming the metric’s consistency with asymptotic AdS behavior.
• The study advances the use of information geometry to reinterpret CFT correlators and opens new avenues for exploring dualities in quantum gravity.

Information Geometry and Holographic Correlators: An Expert's Perspective

The paper "Information Geometry and Holographic Correlators" by Hardik Bohra, Ashish Kakkar, and Allic Sivaramakrishnan presents a technical exploration of perturbative corrections to quantum information geometry, focusing on the Bures information metric associated with correlators in conformal field theories (CFTs). The paper translates properties of correlators into the language of information geometry, providing new insights into how quantum information principles can be applied to effective field theory (EFT), particularly in the context of the AdS/CFT correspondence.

The authors compute the Bures information metric for holographic four-point functions, addressing corrections arising from tree-level Witten diagrams in the bulk. Key findings include the identification of cross terms within the information metric that encode the non-identity operators present in the operator product expansion (OPE). Notably, the paper demonstrates that the information metric reflects the asymptotic Anti-de Sitter (AdS) characteristic, a critical insight for understanding holographic CFTs and EFTs through an information-theoretical lens.

Key Contributions and Numerical Insights

• Bures Information Metric: The Bures information metric provides a measure of state distinguishability and is geometrically interpreted as the embedding of quantum correlators into an information space. The authors' computation shows that for four-point functions, the metric is governed by the ADE characteristics in the limit of small times, indicating deep connections between information geometry and the established structure of AdS/CFT correlators.
• Perturbative Corrections: The paper describes how perturbative expansions in the Bures metric terms manifest novel relationships between quantum information concepts and perturbatively corrected correlators. Particular attention is paid to the fact that the cross terms in the metric encode significant OPE information, leading to potential implications for the factorization properties of CFT correlators.
• Numerical Examples: Through illustrative computation, the work demonstrates perturbative corrections at order $1/N^2$ involving holographic CFTs, calculated via tree Witten diagrams. These numerical results confirm that these contributions preserve the essential properties of the Bures metric's AdS behavior as found in mean-field theories.

Implications and Future Directions

The research prominently suggests that information geometry can serve as a formal framework for understanding the behavior of CFT correlators with potential applications in quantum field theory and holography. The implications are multifaceted:

1. Conformal Structure: Information geometry provides an alternative representation that geometrizes conformal correlators, suggesting new organizational principles for CFT data that could potentially simplify or illuminate some of the classical hurdles in handling large sets of correlator data.
2. Quantum Information and Gravitation: The work adds to the growing body of evidence that quantum information theory concepts like the Bures distance can elucidate aspects of gravitational dynamics portrayed through holography.
3. Higher-Point Functions and Transition Amplitudes: The paper opens avenues for extending the analysis to more complex multi-operator states, higher-point functions, and transition amplitudes, including instigating a search for novel metric-influenced constraints or dualities.

In its entirety, the paper situates the Bures metric as a promising mechanism through which the rich structure of CFTs and their gravitational duals in AdS space can be probed. Continued exploration into metric formulations of CFT data and their computational counterparts may yield substantial advancements in our theoretical grasp of quantum field theories and beyond. The pursuit of these geometrical representations offers promise for deriving new bounds and establishing connections between quantum information measures and fundamental aspects of theoretical physics.