⚛️ general relativity

Krylov Complexity in Non-Inertial Quantum Systems

This paper establishes that in non-inertial quantum systems, the Krylov complexity corresponds exactly to the mean number of correlated Rindler pairs generated via Bogoliubov mixing, revealing three distinct dynamical regimes including a detuning-dominated localization where complexity remains confined to low Krylov levels.

Original authors: Ming-Qi Ma, Shi-Cheng Liu, Lei-Hua Liu, Hai-Qing Zhang

Published 2026-06-30

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CC BY 4.0

Original authors: Ming-Qi Ma, Shi-Cheng Liu, Lei-Hua Liu, Hai-Qing Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a movie, but instead of sitting in a comfortable chair in a quiet room, you are strapped into a rocket ship that is constantly speeding up. In the world of physics, this is called being a "non-inertial observer."

This paper explores what happens to the fundamental rules of quantum mechanics when you are in that speeding rocket. Specifically, the authors investigate a concept called Krylov complexity. To understand this, let's break it down with some everyday analogies.

The Big Picture: A New Way to Count "Chaos"

In standard physics (where you are sitting still), scientists measure how "complex" or "scrambled" a quantum system gets over time. They imagine a simple wave packet (a tiny bundle of energy) moving along a long, one-dimensional hallway. As time passes, this wave packet spreads out, visiting more and more rooms in the hallway. The further it gets, the more "complex" the system is considered to be. This hallway is called a Krylov chain.

Usually, this hallway is built based on a stationary observer's view of the universe. But the authors ask: What if the observer is accelerating? Does the hallway look different?

The Discovery: The "Rindler" Hallway

The authors found that for an accelerating observer, the universe looks different. Because of the acceleration, the vacuum of space (empty space) doesn't look empty anymore; it looks like it's filled with pairs of particles popping in and out of existence. This is known as the Unruh effect.

The paper claims that for this accelerating observer, the natural "hallway" (the Krylov chain) isn't built out of random rooms. Instead, it is built out of pairs of these particles.

The Analogy: Imagine a standard hallway has rooms numbered 1, 2, 3, 4...
The Accelerating Hallway: The rooms are now numbered by how many pairs of particles exist. Room 0 has zero pairs. Room 1 has one pair. Room 2 has two pairs, and so on.

The authors discovered a beautiful, simple rule: The "complexity" of the system is exactly equal to the average number of these particle pairs. If you have 5 pairs on average, your complexity is 5. It's a direct, one-to-one match.

The Three Regimes: How the Wave Packet Moves

The system is controlled by two competing forces:

The "Pair-Production" Force (g): This is the engine that creates new particle pairs, pushing the wave packet further down the hallway to higher numbers.
The "Detuning" Force (u²): This is like a brake or a friction that tries to keep the wave packet from moving too far.

Depending on which force wins, the system behaves in three distinct ways:

1. The "Hyperbolic" Regime (The Fast Runner)

When: The pair-production force is stronger than the brake.
What happens: The wave packet zooms down the hallway, visiting higher and higher numbers of particle pairs. The complexity grows rapidly and keeps growing forever. It's like a car with a gas pedal stuck to the floor; it never stops accelerating.

2. The "Critical" Regime (The Balanced Walker)

When: The pair-production force and the brake are perfectly balanced.
What happens: The wave packet still moves down the hallway, but it grows at a steady, moderate pace. It's the tipping point between running away and getting stuck.

3. The "Bounded" or "Protected" Regime (The Trapped Walker)

When: The brake (detuning) is stronger than the pair-production force.
What happens: This is the most surprising finding. The wave packet tries to move down the hallway, but the brake is so strong that it cannot go very far. It gets "trapped" in the low-numbered rooms (near zero pairs).
The Result: The complexity stops growing. It hits a ceiling and bounces back and forth. The authors call this Krylov localization. Even though the observer is accelerating, the system refuses to get "scrambled" or complex because the internal brake is too strong.

Why This Matters (According to the Paper)

The paper concludes that acceleration doesn't just change what particles you see; it changes the very structure of how complexity is measured.

If you are in a rocket, the "language" of complexity is the number of particle pairs.
By adjusting the internal "brakes" (detuning), you can control whether the system becomes infinitely complex or stays simple and trapped, regardless of how hard you accelerate.

In short, the authors have built a new map for how quantum information spreads in an accelerating universe, showing that sometimes, the system can be "locked" in a simple state, refusing to get complicated no matter how much you push it.

Technical Summary: Krylov Complexity in Non-Inertial Quantum Systems

Problem Statement
The paper addresses a fundamental gap in the application of Krylov complexity, a metric used to diagnose the spread of quantum information. While Krylov complexity has been extensively studied in static, inertial frameworks (e.g., conformal field theories, SYK models, and cosmological environments), its formulation in relativistic, non-inertial settings remains underexplored. Specifically, the authors question whether the Krylov description of operator growth, which relies on a specific operator basis and time-evolution generator, should inherently depend on the observer's frame. In relativistic quantum field theory, a uniformly accelerated observer perceives the Minkowski vacuum as a thermal state (Unruh effect) and decomposes fields into Rindler modes. The central problem is to formulate a Krylov framework that naturally accommodates this observer-dependent mode decomposition and to determine how acceleration influences the dynamics of operator spreading.

Methodology
The authors construct a Krylov framework for non-inertial systems by leveraging the algebraic structure of the Bogoliubov transformation between Minkowski and Rindler vacua.

Basis Construction: They identify the Rindler pair-number sectors ( $|n_k, n_{-k}\rangle$ ) as the natural Krylov basis for uniformly accelerating observers. This identification is motivated by the fact that the Klein-Gordon symplectic form, which defines the space of classical solutions, naturally yields an $SU(1, 1)$ sector when restricted to a single pair of modes.
Hamiltonian Formulation: The dynamics are governed by an effective Hamiltonian with an $SU(1, 1)$ group structure:
$H = 2u_2 K_0 + g(K_+ + K_-)$
Here, $g$ represents the pair-production amplitude (related to squeezing), and $u_2$ is a detuning parameter. The generators $K_0, K_+, K_-$ act on the Rindler pair-number basis $|n\rangle_K$ , mapping the Hamiltonian onto a semi-infinite tridiagonal Krylov chain.
Generalized Bogoliubov Transformation: Using the generalized Lanczos algorithm and the $SU(1, 1)$ algebra, the authors derive time-dependent Bogoliubov coefficients ( $\alpha_k(t), \beta_k(t)$ ) by solving the Heisenberg equations of motion for the operators. This allows them to express the wavefunction as a generalized Perelomov coherent state.
Complexity Calculation: The Krylov complexity ( $C_K$ ) is defined as the mean position of the wave packet along the Krylov chain: $C_K(t) = \sum n |\phi_n(t)|^2$ .

Key Contributions and Results
The paper establishes several exact analytical results connecting non-inertial effects to Krylov complexity:

Exact Equivalence: The primary result is the derivation that the Krylov complexity is exactly equal to the mean number of correlated Rindler pairs generated via Bogoliubov mixing:
$C_K(t) = |\beta_k(t)|^2$
This provides a direct physical interpretation: Krylov complexity in this context quantifies the dynamic production of Rindler pairs. Consequently, the complexity vanishes in the limit of zero acceleration ( $a \to 0$ ), highlighting its observer-dependent nature.
Three Dynamical Regimes: The competition between the pair-production parameter ( $g$ ) and the detuning parameter ( $u_2$ ), encapsulated in the discriminant $\Omega^2 = g^2 - u_2^2$ , separates the system into three distinct regimes based on the dimensionless parameter $\eta = u_2/g$ :
1. Hyperbolic Spreading ( $|\eta| < 1$ ): In the weak-detuning (supercritical) regime, the Krylov complexity grows indefinitely and hyperbolically. The operator wave packet penetrates deeper into the Krylov chain as acceleration increases, without saturation.
2. Critical Growth ( $|\eta| = 1$ ): At the critical threshold, the growth rate is suppressed compared to the zero-detuning case but remains unbounded, scaling as $C_K = r^2(a)$ .
3. Bounded Motion / Localization ( $|\eta| > 1$ ): In the strong-detuning (subcritical) regime, the dynamics transition from hyperbolic to trigonometric. The Krylov complexity becomes strictly bounded ( $C_K \leq \frac{1}{\eta^2 - 1}$ ). The probability distribution of the wave packet remains exponentially confined to low Krylov levels, manifesting as Krylov localization.
Krylov Entropy: The authors also derive the Krylov entropy ( $S_K$ ), showing it is a function of $C_K$ ( $S_K = (1+C_K)\ln(1+C_K) - C_K\ln C_K$ ). The entropy mirrors the three regimes, exhibiting bounded production in the localized regime.

Significance and Claims
The authors claim that this work provides a structural construction of Krylov complexity specifically for non-inertial quantum systems. The significance lies in:

Observer-Dependence: Demonstrating that the Krylov basis and the resulting complexity are not absolute but depend on the observer's acceleration. The Rindler pair-number sectors naturally emerge as the Krylov chain for accelerated observers.
Control Mechanism: Showing that non-inertial motion (via the interplay of acceleration and internal detuning) can be used to control the spreading of operators in Krylov space. Specifically, strong detuning can induce a form of "Krylov saturation" or localization, preventing the unbounded growth typically associated with chaotic or thermalizing systems.
Physical Interpretation: Providing an operator-level counterpart to relativistic quantum information effects, where the growth of complexity is directly linked to the production of entangled particle pairs.

The paper concludes that while acceleration reshapes the operator decomposition, the internal detuning parameter rigidly governs the resulting Krylov dynamics, offering a new mechanism for understanding information scrambling and localization in curved spacetime or accelerated frames.