Krylov complexity of thermal state in early universe

Imagine the early universe not just as a place of exploding stars and expanding space, but as a giant, chaotic kitchen where the recipe for everything we see today was being cooked up.

This paper by Tao Li and Lei-Hua Liu is like a detailed report from a "Quantum Chef" trying to measure how complicated the cooking process gets over time. They use a new tool called Krylov Complexity to track this.

Here is the story of their findings, broken down into simple concepts:

1. The Tool: Measuring "Messiness"

In quantum physics, things can get incredibly messy and scrambled. Think of a drop of ink falling into a glass of water. At first, it's a distinct drop; soon, it spreads out until the whole glass is a uniform color. You can't get the drop back. This spreading out is called chaos or scrambling.

The authors use Krylov Complexity to measure exactly how "spread out" or "scrambled" the universe's quantum state has become.

Low Complexity: The universe is neat, organized, and predictable (like a drop of ink).
High Complexity: The universe is a tangled, chaotic mess (like the ink fully mixed in).

2. The Three Stages of the Cosmic Kitchen

The paper looks at three specific eras of the early universe, treating the universe as a system that interacts with its environment (an "open system"), rather than a closed box.

Stage A: Inflation (The Explosive Mixer)

What happened: The universe expanded faster than the speed of light. It was a period of intense, rapid growth.
The Complexity: The authors found that during this time, the "messiness" (complexity) grew exponentially.
The Analogy: Imagine a high-speed blender turned on full blast. The ingredients are being thrown around so violently that the complexity of the mixture skyrockets instantly. The universe was acting like a strongly dissipative system—it was losing energy and scrambling information at a furious rate.

Stage B: Radiation Dominated (The Cooling Pot)

What happened: The inflation stopped, and the universe was filled with a hot soup of particles (radiation).
The Complexity: The frantic growth stopped. The complexity saturated (leveled off) and started to oscillate around a steady value.
The Analogy: The blender is turned off, but the pot is still hot. The ingredients are still moving, but they aren't being violently thrown around anymore. They are just swirling in a stable pattern. This happened because of preheating—a process where the energy from the inflation field was transferred to create new particles, stabilizing the chaos.

Stage C: Matter Dominated (The Settling Soup)

What happened: The universe cooled down enough for matter (like atoms) to form and dominate.
The Complexity: The complexity stayed flat and stable, similar to the Radiation era.
The Analogy: The soup has cooled down. The particles are settling into a calm, predictable arrangement. The universe has transitioned from a "strongly chaotic" state to a "weakly dissipative" state. It's still a system, but it's much more orderly now.

3. The Big Discovery: A Change in Personality

The most exciting finding of this paper is that the universe changed its "personality" over time.

Early Universe (Inflation): It was a wild, chaotic, high-energy system that scrambled information rapidly.
Later Universe (Radiation & Matter): It became a calmer, more stable system.

The authors discovered that the "engine" driving this change was the inflationary potential (the energy field that drove the expansion). When they included this field in their math, they saw that the universe naturally transitioned from a state of extreme chaos to a state of stability.

4. Why Does This Matter?

Think of the universe as a library.

During Inflation, someone was frantically shuffling every book in the library, mixing them up so fast that the order was lost (High Complexity).
During Radiation and Matter eras, the shuffling stopped. The books settled into a new, stable arrangement. The library is still complex, but it's no longer getting more chaotic every second.

This paper gives us a new way to look at the history of the cosmos using the language of information theory. It tells us that the universe wasn't just expanding; it was evolving from a state of maximum chaos into a state of manageable order, driven by the creation of particles and the cooling of space.

In a nutshell: The early universe was a chaotic blender that eventually settled into a calm pot, and this paper proves that the "messiness" of the universe followed a specific, predictable path from wild chaos to stable order.

1. Problem Statement

The paper addresses the evolution of quantum complexity in the early universe, specifically focusing on the Krylov complexity (K-complexity) and Krylov entropy of thermal states. While previous studies have examined circuit complexity in cosmology, research on Krylov complexity has been limited, often restricted to the inflationary epoch or utilizing closed-system methodologies.

Key challenges identified by the authors include:

Thermal Nature: The early universe involves high temperatures and particle production, making the state a mixed thermal state rather than a pure state.
System Boundaries: Treating the universe as a strictly closed system is theoretically problematic in cosmology due to the lack of asymptotic flat boundaries (making ADM energy ill-defined) and the explicit time-dependence of the metric, which leads to non-conservation of global energy.
Dynamical History: Existing analyses often fail to capture the full dynamical transition from the inflationary era through the Radiation-Dominated (RD) and Matter-Dominated (MD) eras, particularly the effects of preheating and the breakdown of the slow-roll approximation.

2. Methodology

The authors employ a dual-methodology approach, comparing closed-system and open-system frameworks to analyze the evolution of thermal states across three cosmological epochs: Inflation, RD, and MD.

A. Theoretical Framework

Thermal State Purification: Since Krylov complexity is typically defined for pure states, the authors use the Thermofield Double (TFD) formalism. They purify the thermal density matrix $\hat{\rho}$ $\overset{ρ}{^}$ into a two-mode pure state $|\Psi\rangle$ $∣Ψ ⟩$ by doubling the Hilbert space ( $H \otimes H_{anc}$ $H \otimes H_{an c}$ ).
- The state is identified as a two-mode squeezed vacuum, linking the squeezing parameter $r_k$ to the effective temperature $T$ via $\beta\omega = -\ln(\tanh^2 r_k)$ .
Cosmological Model: They utilize the FLRW metric and model the scale factor evolution $a(\eta)$ $a (η)$ across the three eras.
- Inflation: Modeled with a slow-roll approximation where the potential term is negligible compared to geometric terms.
- RD/MD Eras: The authors explicitly include the inflaton potential $V(\phi)$ (approximated as quadratic $V \approx \frac{1}{2}m^2\phi^2$ during preheating) which becomes significant after inflation.
Hamiltonian Construction: Unlike standard approaches using Mukhanov-Sasaki variables (which mix metric and scalar perturbations), the authors use a spatially flat gauge and perturb the inflaton field directly. This allows for the explicit retention of potential-dependent mass terms in the Hamiltonian (Eq. 35), crucial for analyzing post-inflationary dynamics.

B. Computational Approaches

Closed-System Methodology:
- Uses the standard Lanczos algorithm to generate an orthonormal Krylov basis.
- Calculates Krylov complexity ( $K$ ) and K-entropy ( $S_K$ ) based on the probability distribution of the wave function amplitudes in the Krylov basis.
- Assumes unitary evolution.
Open-System Methodology:
- Treats the universe as an open system due to particle production and energy non-conservation.
- Introduces a Lindbladian superoperator $\mathcal{L}_0$ acting on the Krylov basis, which includes a diagonal term $c_n$ representing dissipation.
- Derives generalized wave function amplitudes $\phi_n$ dependent on dissipation coefficients $\mu_1$ and $\mu_2$ .
- Analyzes the dissipative strength ( $\mu_2$ ) to characterize the system's chaotic nature.

3. Key Contributions

Unified Evolutionary Analysis: The paper provides the first comprehensive analysis of Krylov complexity spanning the inflationary, RD, and MD eras, bridging the gap between early inflationary dynamics and late-time thermalization.
Open-System Formalism for Cosmology: It rigorously applies open quantum system theory to the early universe, demonstrating that the universe transitions from a strongly dissipative system (inflation) to a weakly dissipative one (RD/MD).
Role of the Inflaton Potential: By avoiding the Mukhanov-Sasaki variable, the authors isolate the specific contribution of the inflationary potential, showing it is the primary driver for the saturation of complexity in post-inflationary eras.
Thermal State Treatment: Successfully extends Krylov complexity analysis to thermal states using the TFD purification scheme within an open-system context.

4. Key Results

A. Evolution of Effective Temperature ( $T_k$ )

Inflation: The effective temperature increases monotonically, indicating rapid energy injection.
RD/MD Eras: The temperature oscillates around a constant baseline. This saturation is driven by the coherent oscillation of the inflaton field during preheating.

B. Krylov Complexity ( $K$ ) and Entropy ( $S_K$ )

Inflationary Era: Both $K$ and $S_K$ exhibit exponential growth, signaling chaotic behavior and information scrambling. The universe acts as a strongly dissipative system ( $\mu_2 \gg 1$ ).
RD and MD Eras: Both quantities saturate to nearly constant values.
- This saturation is attributed to particle production via preheating, which drives the system toward a stable asymptotic state.
- The system transitions to a weakly dissipative regime ( $\mu_2 \ll 1$ ).
Consistency: The open-system results converge with the closed-system results in the weak dissipation limit, validating the methodology.

C. Lanczos Coefficients and Chaos

The Lanczos coefficients $b_n$ grow linearly with $n$ ( $b_n \approx \alpha n$ ) during inflation, satisfying the bound for maximal chaos ( $\lambda_L = 2\alpha$ ).
In the RD and MD eras, the growth rate $\alpha$ diminishes, and the chaotic intensity is suppressed due to the breakdown of slow-roll conditions and the dominance of the potential term.

5. Significance

Quantum Information Perspective on Cosmology: The study offers a novel framework to understand early universe dynamics through the lens of quantum information theory, specifically linking complexity growth to the dissipative nature of cosmic expansion.
Dissipative Transition: It establishes a clear dynamical transition in the universe's nature: from a strongly dissipative, chaotic inflationary phase to a weakly dissipative, stabilized post-inflationary phase.
Resolution of Theoretical Issues: By treating the universe as an open system, the authors circumvent the theoretical difficulties associated with defining global energy conservation in non-asymptotically flat spacetimes.
Implications for Chaos: The results suggest that while the early universe was a maximally chaotic system during inflation, the subsequent particle production acts as a mechanism to "tame" this chaos, leading to the stable thermal states observed in the later universe.

In conclusion, the paper demonstrates that Krylov complexity serves as a robust probe for the thermodynamic and chaotic evolution of the early universe, revealing a distinct phase transition driven by the interplay between the inflationary potential and particle production.