More on complexity of operators in quantum field theory

Abstract

Recently it has been shown that the complexity of SU($n$) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten $p$-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as $k$-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU($n$) groups.

arXiv commentModified the Sec. 4.1, where we offered a powerful proof: if (1) the ket vector and bra vector in quantum mechanics contain same physics, or (2) adding divergent terms to a Lagrangian will not change underlying physics, then complexity in quantum mechanics must be bi-invariant!

References in corpus (32)

• Complexity Equals Action
• Localized shocks
• Quantum Computation as Geometry
• Circuit complexity in quantum field theory
• Towards Complexity for Quantum Field Theory States
• Comments on Holographic Complexity
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• On the Time Dependence of Holographic Complexity
• Noether charge, black hole volume, and complexity
• Complexity of Formation in Holography
• Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
• Quantum Computation vs. Firewalls
• Complexity Growth for AdS Black Holes
• Circuit complexity for free fermions
• Quantum Mechanics in Phase Space
• Circuit complexity in interacting QFTs and RG flows
• Holographic Complexity in Vaidya Spacetimes II
• Circuit Complexity for Coherent States
• Complexity as a novel probe of quantum quenches: universal scalings and purifications
• On Volumes of Subregions in Holography and Complexity
• Subsystem Complexity and Holography
• Black holes, complexity and quantum chaos
• Divergences in Holographic Complexity
• Holographic Complexity of Einstein-Maxwell-Dilaton Gravity
• Holographic Spacetimes as Quantum Circuits of Path-Integrations
• Holographic subregion complexity under a thermal quench
• Principles and symmetries of complexity in quantum field theory
• Complexity of the AdS Soliton
• Surface Counterterms and Regularized Holographic Complexity
• On the time dependence of holographic complexity in a dynamical Einstein-dilaton model
• Holographic Subregion Complexity in Einstein-Born-Infeld theory
• Complexity of operators generated by quantum mechanical Hamiltonians

Cited by in corpus (31)

• Time Evolution of Complexity: A Critique of Three Methods
• Chaos and Complexity in Quantum Mechanics
• Subregion Action and Complexity
• Complexity change under conformal transformations in AdS${}_3$/CFT${}_2$
• Circuit Complexity across a Topological Phase Transition
• Complexity of Holographic Superconductors
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• Circuit Complexity and 2D Bosonisation
• Time evolution of the complexity in chaotic systems: concrete examples
• Surface/State correspondence and $T\overline{T}$ deformation
• Geometry and Complexity of Path Integrals in Inhomogeneous CFTs
• Modified "complexity equals action" conjecture
• Holographic complexity growth in a FLRW universe
• Magnetotransport and Complexity of Holographic Metal-Insulator Transitions
• Time dependence of complexity for Lovelock black holes
• Revisit on holographic complexity in two-dimensional gravity
• Holographic complexity of charged Taub-NUT-AdS black holes
• Generalized Volume-Complexity For Two-Sided Hyperscaling Violating Black Branes
• Circuit Complexity of Knot States in Chern-Simons theory
• Holographic complexity of the electromagnetic black hole
• Holographic Complexity of LST and Single Trace $T\bar{T}$
• Stringy Effects and the Role of the Singularity in Holographic Complexity
• Nielsen complexity of coherent spin state operators
• Holographic Complexity in a Charged Supersymmetric Black Holes
• Complexity and scaling in quantum quench in $1+1$ dimensional fermionic field theories
• Complexity growth for topological black holes by holographic method
• Holographic Complexity of LST and Single Trace $T\bar{T}$, $J\bar{T}$ and $T\bar{J}$ deformations
• New Topological Gauss-Bonnet Black Holes in Five Dimensions
• Real space circuit complexity as a probe of phase diagrams
• Time-dependent Hamiltonians and Geometry of Operators Generated by Them
• Switchback effect of holographic complexity in multiple-horizon black holes