Accelerating Observers Reshape Quantum Complexity, New Study of Krylov Growth Reveals

How does quantum information spread through the fabric of spacetime when the observer doing the measuring is not sitting still? That deceptively simple question sits at the heart of a new theoretical study published in The European Physical Journal C, which for the first time builds a complete framework for Krylov complexity in non-inertial quantum systems. Krylov complexity has become one of the most influential diagnostic tools in modern theoretical physics, offering a way to quantify how a simple operator evolves, spreads and scrambles information across a system’s degrees of freedom. Until now, however, nearly all applications of the formalism assumed a static observer, with the vacuum state, the operator algebra and the generator of time evolution all anchored to a single inertial frame. The new work by Ming-Qi Ma, Shi-Cheng Liu, Lei-Hua Liu and Hai-Qing Zhang dismantles that assumption and asks what happens when acceleration itself becomes part of the story.

The physical motivation comes from one of the strangest results in relativistic quantum field theory: the Unruh effect. A uniformly accelerated observer does not perceive the vacuum of flat spacetime as empty. Instead, when the field is decomposed into Rindler modes, the natural modes of an accelerated frame, the Minkowski vacuum appears as a thermal bath of particles. This is not a mathematical trick; it has real consequences for entanglement, quantum communication and the behavior of quantum fields near horizons. The authors of the new study recognized that if the very concept of a particle depends on the observer’s motion, then the Krylov description of operator growth, which is built on top of a chosen basis of states, should inherit that observer dependence. Their central question was whether Krylov spreading is an observer-invariant quantity or something that changes when the frame changes.

To answer it, the team started from a two-mode massless scalar field that looks maximally entangled to inertial observers, with one observer, Alice, detecting one mode while her accelerated partner, Bob, detects the other. When Bob’s mode is mapped onto Rindler modes, the Minkowski vacuum expands into a two-mode squeezed state spanning the left and right Rindler wedges, a textbook consequence of the Fulling–Davies–Unruh effect. The squeezing is characterized by Bogoliubov coefficients, and the resulting state can be identified as a generalized Perelomov coherent state. The crucial conceptual move was to expand this state in the basis of correlated Rindler pair-number states, states containing n pairs of particles in the two Rindler wedges, and then to prove that this pair-number basis is exactly the Lanczos basis, the special one-dimensional chain on which Krylov complexity is defined.

This identification is the paper’s central structural result. In the closed single-pair sector, governed by an SU(1,1) algebra, the Hamiltonian acting on the pair-number basis produces a semi-infinite tridiagonal matrix, precisely the form required by the Lanczos algorithm. The hopping amplitude along the Krylov chain is set by the pair-production strength g, while a detuning parameter u2 creates a level-dependent onsite energy. Because each Lanczos level coincides with a definite number of correlated Rindler pairs, the Krylov spread complexity reduces exactly to the mean number of Rindler pairs: C_K equals the squared magnitude of the time-dependent Bogoliubov coefficient beta. In other words, within this exactly solvable benchmark, operator growth in Krylov space is literally the production and accumulation of correlated particle pairs as seen by the accelerated observer. The abstract measure of complexity acquires a tangible relativistic meaning.

The dynamics that follow from this identity divide cleanly into three regimes, controlled by the dimensionless ratio of detuning to coupling. When the detuning is weaker than the pair-production amplitude, the Krylov wave packet spreads hyperbolically along the pair-number chain, growing without bound as the observer’s proper acceleration increases. Detuning slows this growth relative to the standard Unruh case but never halts it. At the critical threshold, where detuning exactly matches the coupling, the complexity grows quadratically with the acceleration-controlled squeezing parameter, marking the boundary between unbounded and bounded behavior. The authors emphasize that unlike many chaotic systems, where Krylov complexity grows exponentially and then saturates, this Rindler pair chain in the weak-detuning regime grows indefinitely without ever saturating, a distinctive signature of the non-inertial setting.

The most striking phenomenon emerges in the strong-detuning regime. There, the effective frequency governing the Bogoliubov evolution becomes imaginary, converting hyperbolic growth into bounded trigonometric oscillation. The Krylov complexity acquires a rigorous upper bound inversely proportional to the excess detuning, and the probability distribution over the Krylov chain remains exponentially concentrated near low levels. The authors call this detuning-induced Krylov localization: the operator wave packet is dynamically confined, unable to penetrate deeper into the chain no matter how large the acceleration becomes. The Krylov entropy, which in this closed sector is completely determined by the complexity through a geometric probability distribution, inherits the same three-regime structure, remaining strictly bounded in the protected regime. Acceleration alone, the study concludes, does not dictate the fate of information scrambling; acceleration and detuning jointly govern it.

Perhaps the most consequential finding is that this elegant correspondence between complexity and particle production is not a universal law of non-inertial quantum dynamics. When the authors extended the framework to a strictly quadratic, multimode Bogoliubov Hamiltonian with two inequivalent Rindler wave-packet pairs, the exact identity broke down. The reason is structural: in the multimode case, each fixed total-pair-number sector contains several orthogonal mode-distribution states, so the Lanczos vectors generically cease to be eigenstates of the total pair-number operator. Using short-time expansions, the team derived explicit analytic expressions for the discrepancy. With unequal detunings in the two channels, the difference between Krylov complexity and the mean pair number appears at fourth order in time, proportional to the product of the couplings and the squared detuning difference. With unequal pair-production amplitudes at resonance, the deviation first appears at sixth order, because the first two nontrivial Lanczos vectors still carry definite pair numbers while the third mixes sectors.

Numerical simulations of the two-channel Hamiltonian confirmed the analytic predictions and revealed the physical content of the mismatch. The Krylov level counts the number of orthogonal dynamical directions generated by repeated action of the full multimode Hamiltonian, whereas the mean pair number counts only the average yield of produced pairs. The Krylov construction is therefore sensitive not just to how many particles are created but to how the dynamics are distributed among distinct Gaussian pair-production channels, information that particle counting alone cannot access. The correspondence is restored in two fine-tuned limits: the single-channel case and the fully symmetric case where both channels share identical couplings and detunings, in which the collective generators close into a single SU(1,1) algebra and the Lanczos coefficients revert to the universal form of the symmetric benchmark. This identifies the single-pair SU(1,1) model as an exactly solvable, observer-adapted reference point whose deviations signal genuinely new Krylov information.

The implications reach well beyond accelerated observers. The authors point to interacting quantum fields in curved spacetime, where particle production and Bogoliubov mixing in dynamical gravitational backgrounds have been studied extensively but a Krylov description remains largely unexplored. In such settings, Krylov spreading may encode information about gravitational back-reaction, nonlinear mode coupling and operator growth beyond Gaussian particle production. Potential applications include quantum information channels in curved spacetime, moving-cavity systems and relativistic quantum metrology, where acceleration and gravity directly affect measurement precision. The framework also invites integration with open-system Krylov complexity, multi-seed formalisms and entanglement-based diagnostics of operator growth. What the study establishes, above all, is a conceptual shift: complexity in quantum field theory is not a frame-independent absolute but a quantity woven together with the observer’s motion, the choice of vacuum and the mode decomposition, and its richest behavior appears precisely when those choices are allowed to differ.

Subject of Research: Krylov complexity and operator growth in non-inertial (accelerated) quantum field systems

Article Title: Krylov complexity in non-inertial quantum systems

Article References: Krylov complexity in non-inertial quantum systems. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16271-1

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16271-1

Keywords: Krylov complexity, Unruh effect, Rindler spacetime, Bogoliubov transformation, operator growth, quantum information, non-inertial frames, Lanczos algorithm, SU(1,1) symmetry, quantum field theory, particle production, relativistic quantum information