The Holography of Spread Complexity: A Story of Observers

Abstract: We propose a holographic description of spread complexity and its rate in 2D CFTs, building on the pioneering work \cite{Caputa:2024sux}. By exploiting the $SL(2,\mathbb{R})$ symmetry, we construct the Krylov basis and demonstrate that its non-uniqueness gives rise to ambiguities analogous to those in quantum complexity. Within the AdS/CFT correspondence, we identify the spread complexity as the energy measured by a bulk observer, with its rate corresponding to the radial momentum. These results suggest a novel holographic interpretation of complexity ambiguity: rather than arising from infinitely many gravitational observables, it may reflect the existence of infinitely many observers, each measuring complexity from their own perspective.

• The paper demonstrates that spread complexity in 2D CFTs corresponds to the energy measured by a bulk observer using holographic principles.
• It employs SL(2,R) symmetry to construct a Krylov basis, revealing that distance complexity remains invariant under these transformations.
• The study links conformal symmetry generators with conserved AdS quantities, offering a novel geometric interpretation of quantum complexity.

The Holography of Spread Complexity: A Story of Observers

This paper proposes an intriguing holographic description of spread complexity and its rate within two-dimensional conformal field theories (2D CFTs). Drawing from established theoretical frameworks, this research introduces creative ways of understanding how quantum complexity can be mapped onto holographic principles via the AdS/CFT correspondence.

Overview and Key Findings

The authors begin by acknowledging the foundational complexities inherent in AdS/CFT correspondence, notably the challenge of equating quantum complexity with holographic duals due to the intrinsic ambiguities rooted in observer-dependent variables. Central to their approach is the exploitation of the $SL(2, \mathbb{R})$ symmetry to construct a Krylov basis that encapsulates these ambiguities.

The paper asserts that spread complexity in the holographic context is essentially the energy as gauged by a bulk observer, and its rate is analogous to the measured radial momentum. Notably, this interpretation suggests that the ambiguities in complexity arise not from countless gravitational observables but potentially from the varying perspectives of infinitely many observers measuring complexity in distinct manners.

A notable contribution of this research is the definition of distance complexity as an invariant quantity under $SL(2, \mathbb{R})$ transformations, providing a novel geometric understanding of the spread complexity. The notion of state distance elaborated in this work casts a much-needed geometric perspective on the measurement of complexity within this holographic framework.

Practical and Theoretical Implications

This study delivers several practical implications for how quantum states and their complexities could be visualized and computed within holographic models. By forging a connection between the expectation values of conformal symmetry generators and conserved quantities tied to corresponding Killing vectors in AdS spacetime, the work offers new insights that may facilitate accurate geometrical interpretations of quantum complexity through holography.

This paper also presents polyvalent theoretical implications and applications, notably in exploring and potentially resolving complexities associated with the holographic model. The articulation of observer-dependence in measuring complexity provides a conceptual shift in understanding holographic ambiguity that could propel further research into gravitational observables and their duality with quantum complexities.

Future Directions

Speculating on future developments in AI and holography, this paper posits several research avenues. This encompasses the potential generalization of its proposed models to higher-dimensional design and even application to the complexities within black hole physics and cosmos modeling. Furthermore, the extension of Krylov complexity studies to accommodate deformation due to both chaotic and accelerated motion in diverse holographic models presents exciting possibilities worth exploring.

The holographic paradigm shifts introduced here beckon more comprehensive analyses and modeling to ascertain the roles these newly defined complexities play within broader quantum mechanics and relativity frameworks. The insights gained could pave the way toward breakthroughs in understanding the very fabric of spacetime and its myriad entwinements with quantum information theory.

In summary, this paper opens up new vistas for both classical and quantum complexity studies, offering rich potential for future explorations in holography and relativity.