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Circuit complexity of coupled four-mode squeezed states in two-field inflation

This paper investigates the circuit complexity of coupled curvature and entropy perturbations in two-field inflation, deriving a formula based on squeezing strengths and demonstrating that while interfield coupling alters the four-mode squeezed state, its impact on the resulting complexity is minimal.

Original authors: Tao Li, Hai-Bing Fu, Lei-Hua Liu

Published 2026-10-30

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CC BY 4.0

Original authors: Tao Li, Hai-Bing Fu, Lei-Hua Liu

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

In the earliest moments of our universe, a period of rapid expansion known as inflation stretched space itself, turning tiny quantum jitters into the seeds of galaxies. While the simplest models of this era imagine a single field driving the expansion, many theories suggest that multiple fields were active, interacting with one another as the cosmos grew. When these fields are present, they do not just evolve in isolation; they can exchange energy and information, creating a complex web of correlations between different types of fluctuations. One way physicists try to understand the structure of these quantum states is by measuring their "circuit complexity." This concept does not refer to the difficulty of a puzzle, but rather to the minimum amount of effort, or steps, required to build a specific quantum state starting from a simple, empty beginning. It is a measure of how much the state has changed and how intricate its internal connections have become.

A team of researchers has recently applied this idea to a scenario where two fields interact during inflation. They focused on a specific type of quantum state called a four-mode squeezed state. In the simplest case, a single field creates pairs of particles moving in opposite directions, a process that "squeezes" the uncertainty of their properties. When two fields are involved, the situation becomes richer: the fields can create pairs within themselves, but they can also create pairs that mix the two fields together. The researchers wanted to know if this mixing, or coupling, significantly changes the complexity of the resulting state. To find out, they started with the fundamental equations that describe how these fields move and interact. From these equations, they derived a complete description of the energy and motion of the system, accounting for every possible way the fields could influence each other.

Using this detailed description, the team calculated how the quantum state evolves over time. They tracked the growth of the "squeezing," which represents the amplification of these quantum fluctuations as they move beyond the scale of the observable universe. Their calculations showed that the complexity of the state increases significantly once the fluctuations cross a critical boundary known as the horizon. This growth happens in both scenarios: when the two fields are completely separate and when they are coupled together. The researchers found that while the coupling between the fields does alter the specific details of how the pairs are created, the overall measure of complexity remains surprisingly similar in both cases.

The study reveals a subtle but important point about how we measure quantum states. Even though the coupling changes the internal structure of the state—shifting how much of the "squeezing" comes from one field versus the other, and creating new mixed pairs—the total cost to build this state from scratch, according to the specific method used, barely changes. This suggests that different internal arrangements of the quantum state can result in nearly identical complexity values. The researchers confirmed this by running simulations with specific parameters, showing that the difference in complexity between the coupled and uncoupled models is very small, visible only on a very fine scale.

This finding does not mean that the interaction between the fields is unimportant or that the fields are not talking to each other. The coupling clearly changes the mathematical description of the state and the way the fields evolve. However, for the specific measure of complexity used in this study, which treats all possible ways of building the state as equally costly, the extra effort required to account for the mixing is minimal. The work provides a clear reference for future studies, showing that in certain models, the presence of multiple fields does not necessarily lead to a dramatic increase in the complexity of the quantum state, even though it fundamentally alters the nature of the correlations within it. The results offer a new perspective on how we quantify the information content of the early universe, highlighting that the way we choose to measure complexity can sometimes mask the intricate details of the underlying physics.

Technical Summary: Circuit Complexity of Coupled Four-Mode Squeezed States in Two-Field Inflation

Problem Statement
This study investigates the circuit complexity of primordial perturbations in two-field inflationary models. While inflation generates quantum fluctuations that evolve into squeezed Gaussian states, the specific complexity of preparing these states—particularly when curvature and entropy perturbations are coupled—remains to be fully characterized. Previous work has addressed single-field models or uncoupled perturbations, but the interplay between interfield coupling and the geometric distance required to prepare the resulting four-mode squeezed state requires analysis. The authors aim to determine how field-space coupling modifies the pair amplitudes of the state and whether these modifications significantly alter the circuit complexity compared to an uncoupled scenario.

Methodology
The authors adopt a geometric approach to quantum complexity, specifically utilizing the Nielsen circuit complexity framework for Gaussian states.

Hamiltonian Derivation: Starting from the quadratic action for coupled curvature ( $\xi$ ) and entropy ( $\psi$ ) perturbations, the authors derive the full Hamiltonian. This includes all diagonal and off-diagonal quadratic terms, capturing same-field and cross-field pair creation as well as interfield transfer terms. The system is treated in terms of canonically conjugate variables for each momentum pair $(k, -k)$ .
State Representation: The time-evolved state is represented as a four-mode squeezed state. The authors express this state using a symmetric pair-amplitude matrix $Z_k$ , which contains three complex entries: $Z_{11,k}$ and $Z_{22,k}$ (same-field pairs) and $Z_{12,k}$ (cross-field pairs).
Dynamics: By applying the Schrödinger equation to the Gaussian ansatz, the authors derive three coupled complex differential equations governing the evolution of the pair amplitudes $Z_{ij,k}$ . These equations depend on background parameters such as the Hubble rate, the turn rate of the trajectory ( $\lambda$ ), and the effective mass.
Complexity Calculation:
- The pair-amplitude matrix is decomposed via Takagi factorization to extract two principal squeezing strengths ( $r_{1k}, r_{2k}$ ) and associated angles. These strengths characterize the principal squeezing channels of the joint state, which do not necessarily align with the original field basis.
- The circuit complexity is calculated using the equal-weight Nielsen $F_2$ metric. For a vacuum reference state and unrestricted quadratic Gaussian gates, the complexity for a single momentum pair is derived as $C_{2,k} = \sqrt{2(r_{1k}^2 + r_{2k}^2)}$ .
- The authors note that under this equal-weight prescription, the complexity depends explicitly on the squeezing strengths but is independent of the squeezing angles when the strengths are fixed.
Numerical Simulation: The coupled equations are solved numerically for a quasi-de Sitter background. The results are compared against a matched uncoupled evolution (where the turn rate $\lambda$ is set to zero) to isolate the effects of interfield coupling.

Key Contributions

Full Hamiltonian Formulation: The paper provides the complete Hamiltonian for coupled two-field inflation, explicitly detailing all diagonal and off-diagonal terms required to describe the four-mode squeezed state.
Coupled Squeezing Dynamics: The authors derive the specific set of three coupled differential equations that govern the evolution of the pair amplitudes, demonstrating how a non-zero turn rate sources the initial growth of cross-field pair amplitudes.
Complexity Metric Application: The study applies the equal-weight Nielsen distance to a coupled four-mode system, establishing the formula $C_{2,k} = \sqrt{2(r_{1k}^2 + r_{2k}^2)}$ and clarifying its independence from squeezing angles under this specific cost function.
Decomposition of Principal Channels: The work distinguishes between the original field basis and the principal squeezing channels, showing that the latter are determined by the eigenvalues of the full pair-amplitude matrix and evolve dynamically.

Results

Evolution of Amplitudes: Numerical results show that deep inside the horizon, pair amplitudes remain small. As modes approach horizon crossing, the diagonal amplitudes grow significantly, while the cross-field amplitude ( $Z_{12,k}$ ) exhibits a transient enhancement before decreasing. However, the coupling leaves a lasting imprint on the state's evolution.
Squeezing Strengths: Both principal squeezing strengths ( $r_{1k}$ and $r_{2k}$ ) are small in the subhorizon regime and grow after horizon crossing, reflecting the amplification of inflationary perturbations. The two strengths separate at late times, indicating that the coupled sectors are not amplified equally.
Complexity Comparison: The calculated circuit complexity grows after horizon crossing for both coupled and uncoupled models. Crucially, the difference in complexity between the coupled and uncoupled cases is found to be very small for the parameters considered. The signed difference ( $\Delta C_{2,k}$ ) is resolved only on a much narrower scale than the total complexity.
Symmetry: The complexity is found to be symmetric with respect to the sign of the coupling parameter ( $\lambda$ ), i.e., $C_{2,k}(\lambda) = C_{2,k}(-\lambda)$ , causing the linear correction to vanish near $\lambda=0$ .

Significance and Claims
The paper claims that while interfield coupling fundamentally alters the structure of the four-mode squeezed state by generating cross-field pair amplitudes and modifying the pair-amplitude matrix, its effect on the equal-weight circuit complexity is minimal for the tested parameters.

The authors emphasize that this result highlights the dependence of complexity comparisons on the chosen reference state, gate set, and cost function. Specifically, the equal-weight Nielsen distance is insensitive to the orientation of the squeezing channels when the principal strengths are fixed. Consequently, different four-mode pair-amplitude matrices (arising from coupled vs. uncoupled dynamics) can yield nearly identical complexity values.

The study concludes that this specific complexity measure does not resolve every change in the quantum state or serve as a direct probe of interfield correlations in this context. It serves as a reference for studying Gaussian circuit complexity in more general two-field models but does not, in its current form, determine Lyapunov exponents or scrambling times, nor does it directly link to primordial observables without further analysis. The authors suggest that unequal gate penalties might be necessary to make the complexity sensitive to the angular evolution and orientation of the squeezing channels.

Technical Summary: Circuit Complexity of Coupled Four-Mode Squeezed States in Two-Field Inflation

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