⚛️ high-energy theory

Momentum-space entanglement and mutual information in Kaluza-Klein reduction

This paper demonstrates that in Kaluza-Klein reductions of interacting scalar fields, the four-dimensional effective theory accurately describes low-energy observables via local operators but fails to capture the non-pure nature of the reduced quantum state, as the momentum-space entanglement and mutual information between zero modes and the massive tower arise from time-nonlocal effects and grow logarithmically with the number of levels rather than being dominated by the lightest modes.

Original authors: Udaykrishna Thattarampilly, Yunlong Zheng

Published 2026-10-28

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CC BY 4.0

Original authors: Udaykrishna Thattarampilly, Yunlong Zheng

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

The universe we experience is built on four dimensions: three of space and one of time. Yet, for decades, physicists have suspected that reality might be far more crowded. Theories attempting to unify the fundamental forces of nature often require extra dimensions that are curled up so tightly they remain invisible to our instruments. Imagine a garden hose viewed from miles away; it looks like a one-dimensional line, but if you were an ant walking on its surface, you would find a second dimension wrapping around it. In the same way, our familiar four-dimensional world might be a slice of a larger, five-dimensional reality, with the extra dimension compacted into a tiny circle.

When scientists try to describe physics in this higher-dimensional world from the perspective of someone living in the four-dimensional slice, they use a method called Kaluza-Klein reduction. This process treats the hidden dimension as a set of vibrating modes, much like the different notes a guitar string can produce. The lowest note, or "zero mode," corresponds to the massless particles we see every day, such as photons. The higher notes form a "tower" of massive particles that are too heavy to be created at current energy levels. For a long time, the standard view was that these heavy particles simply fade away into the background, leaving the low-energy world completely pure and predictable. However, a new study challenges this assumption by asking a different question: does the hidden tower of particles leave a trace in the very state of the universe, even if it doesn't change the numbers we measure in experiments?

Researchers Udaykrishna Thattarampilly and Yunlong Zheng set out to investigate this by looking at how information is shared between the visible world and the hidden tower. They focused on a simple model: a single type of particle moving in a five-dimensional space where the extra dimension is a circle. They examined two scenarios, one where particles interact in groups of three and another where they interact in groups of four. Their goal was to determine if the act of ignoring the heavy, hidden particles to focus only on the light ones actually destroys information, leaving the visible world in a "mixed" or uncertain state. In quantum physics, a "pure" state is one where everything is perfectly known and connected, while a "mixed" state implies that some information has been lost to the environment.

The team found that for free particles that do not interact, the reduction is perfect; the visible world remains pure, and no information is lost. However, the moment interactions are turned on, the situation changes dramatically. The heavy tower of particles begins to entangle with the light ones. This entanglement means that the state of the visible particles is no longer independent; it is inextricably linked to the hidden tower. To measure this connection, the researchers calculated a quantity called mutual information, which acts like a thermometer for how much two systems know about each other. They discovered that in the scenario where particles interact in groups of three, the connection between the visible world and a specific heavy particle level drops off slowly as the particle gets heavier. Specifically, the connection weakens in proportion to the inverse of the particle's mass.

This slow drop-off has a profound consequence. Because the connection fades so gradually, every single level in the tower, from the lightest to the heaviest, contributes a small but significant amount of information. When the researchers added up the contributions from all the levels in the tower, the total amount of missing information grew logarithmically with the number of levels. This means that the more levels you include in your calculation, the more information you realize is missing from the four-dimensional description. The visible world is not just a clean, isolated system; it is a mixed state, constantly entangled with the hidden dimensions.

The findings were different for the scenario where particles interact in groups of four. In this case, the connection to the heavy tower was dominated by the highest energy levels, near the theoretical limit of the theory. Here, the mutual information depended heavily on the cutoff point of the calculation, suggesting that the connection is set by the physics at the very edge of our understanding rather than by the specific properties of the hidden particles. This contrast highlights a crucial distinction: while the heavy particles do not significantly alter the forces or energies we measure at low levels—those effects are dominated by the lightest particles—they do fundamentally alter the quantum state of the universe. The local laws of physics that describe our everyday world are sufficient to predict what happens in experiments, but they are insufficient to describe the actual state of the system.

The study also addressed the role of supersymmetry, a theoretical framework that pairs every known particle with a heavier partner. While supersymmetry is famous for canceling out the vacuum energy of the universe, the researchers found that it does not cancel out this entanglement. Even in a perfectly supersymmetric universe with zero vacuum energy, the four-dimensional observer would still be left with a mixed state, entangled with the hidden tower. The information loss is a sum of squared probabilities, and adding more particles simply adds more positive terms to this sum, never canceling them out.

Ultimately, this work reveals a subtle but important limit to our description of reality. The local effective theories that physicists use to describe the four-dimensional world are excellent at predicting the outcomes of measurements, such as how particles scatter or how forces behave. However, these theories describe only the observables, not the full state of the system. The missing information, which grows with the number of hidden levels, resides in the non-local parts of the physics that are discarded when simplifying the theory. The universe, it seems, retains a memory of its hidden dimensions in the very fabric of its quantum state, a memory that persists even when the heavy particles are too massive to be seen. This suggests that a complete understanding of the universe requires more than just the local laws of physics; it demands a recognition of the deep, invisible connections that bind our visible world to the hidden dimensions beyond.

Technical Summary: Momentum-space entanglement and mutual information in Kaluza–Klein reduction

Problem Statement
In theories with compact extra dimensions, the standard four-dimensional (4D) effective description is derived via Kaluza–Klein (KK) reduction, where heavy massive modes are integrated out. The Appelquist–Carazzone theorem guarantees that low-energy observables (scattering amplitudes, correlation functions) are reproduced by a local 4D effective Lagrangian, with corrections suppressed by powers of the heavy mass scale. However, this decoupling applies to observables, not necessarily to the quantum state itself.

This paper investigates the quantum state of a 4D observer who has access only to the massless zero modes (below a resolution scale $\mu$ ) but lacks access to the massive KK tower. The central question is whether tracing out the tower leaves the zero-mode sector in a pure state or a mixed state, and how the entanglement between the accessible sector and the tower scales with the KK level mass. While momentum-space entanglement has been studied in 4D field theories (e.g., by Balasubramanian, McDermott, and Van Raamsdonk), its behavior in the context of dimensional reduction—where entire KK levels (relativistic species) are discarded rather than just momentum shells—remains unexplored.

Methodology
The authors analyze a real scalar field on a five-dimensional spacetime $M_4 \times S^1$ with a compactification radius $R$ . They consider two interaction models: a cubic ( $\Phi^3$ ) and a quartic ( $\Phi^4$ ) self-interaction.

Setup: The 5D field is expanded into Fourier modes $\phi_n(x)$ . The Hilbert space is partitioned into an accessible sector $A$ (zero modes with momentum $|k| \le \mu$ ) and an environment $B$ (the entire KK tower $n \neq 0$ and zero modes with $|k| > \mu$ ).
Perturbative Expansion: The interacting vacuum $|\Omega\rangle$ is computed to leading order in the coupling $\lambda$ . The reduced density matrix $\rho_A = \text{Tr}_B |\Omega\rangle\langle\Omega|$ is derived.
Entanglement Measures:
- Entropy: The von Neumann entropy $S_A$ and purity deficit are calculated. The authors note that $S_A$ diverges in the quartic theory due to power-law divergences from hard zero-mode configurations.
- Mutual Information: To isolate the contribution of specific KK levels and avoid UV-sensitive divergences, the authors compute the mutual information $I_n \equiv I(A : B_n)$ between the accessible sector and a single KK level $n$ . This quantity measures the total correlation (classical and quantum) between the zero modes and the specific tower level.
Channel Decomposition: The calculation involves decomposing interaction vertices into "channels" based on how many soft (accessible) and hard (environment) quanta they produce. Key channels include:
- $(2,2)$ : Two soft zero modes and a tower pair ( $\phi_0 \phi_0 \phi_n \phi_{-n}$ ).
- $(1,3)$ : One soft zero mode and three tower quanta.
- Mixed: One soft zero mode, one hard zero mode, and a tower pair.
Numerical Verification: The analytical asymptotic behaviors are verified using Monte Carlo integration of the phase-space kernels.

Key Contributions and Results

Free Theory: For free fields, the reduction is lossless; the zero-mode vacuum remains pure, and entanglement with the tower vanishes. Entanglement is strictly an interaction effect.
Cubic Theory ( $\Phi^3$ ):
- The mutual information $I_n$ with a single KK level $n$ falls off as $1/m_n$ (where $m_n = n/R$ ).
- The coefficient of this tail is independent of the ultraviolet (UV) cutoff.
- Summing over all levels $N$ , the total mutual information grows logarithmically: $I_{\text{total}} \propto \ln N$ .
- This implies that the entanglement is not dominated by the lightest levels; every "octave" of the tower contributes equally.
Quartic Theory ( $\Phi^4$ ):
- The behavior differs significantly. While the $(2,2)$ channel contributes a $1/m_n$ tail, the $(1,3)$ and mixed channels introduce states with additional hard zero-mode quanta.
- These states leave a second hard momentum unconstrained, causing the integral to be dominated by momenta near the UV cutoff $\Lambda_{UV}$ .
- Consequently, $I_n$ depends on the cutoff and does not exhibit a clean $1/m_n$ tail independent of UV physics. The total mutual information is dominated by the cutoff scale rather than the tower structure.
Supersymmetry: The authors demonstrate that unbroken supersymmetry cancels the vacuum energy (Casimir energy) of the compactification but does not cancel the mutual information. Since entanglement at leading order is a sum of squared amplitudes (positive definite), adding fermionic superpartners only increases the total correlation. Thus, a supersymmetric compactification with vanishing vacuum energy still leaves the 4D observer in a mixed state.

Significance and Claims
The paper claims that the local 4D effective theory, derived via a derivative expansion, successfully describes low-energy observables but fails to describe the state of the reduced system.

Decoupling of Observables vs. State: While low-energy amplitudes are dominated by the lightest KK levels and captured by local operators (decoupling), the mutual information with the tower receives equal contributions from all levels (in the cubic case). This "missing information" grows with the number of integrated-out levels.
Non-locality: The entanglement arises from the time-nonlocal terms of the effective action (terms non-local in time on the scale $\Delta t \sim 1/m_n$ ). Standard derivative expansions discard these non-local terms, thereby rendering the reduced state pure in the effective theory, even though the true state is mixed.
Operational Meaning: The mutual information quantifies the difficulty an observer with access to a specific KK level would have in distinguishing the correlated state from a decoupled product state. The $1/m_n$ scaling implies that distinguishing the state becomes harder as the level mass increases, but the cumulative effect of the entire tower is significant.
Limitations: The authors note that for realistic compactifications involving gravitons and gauge fields, the couplings to the tower involve derivatives. Power counting suggests these derivative couplings may shift the mutual information dependence to the UV cutoff, similar to the quartic scalar case. A complete gauge-invariant calculation is left as an open problem.

In conclusion, the paper establishes that dimensional reduction generates intrinsic quantum entanglement between the observable 4D sector and the hidden extra dimensions. This entanglement is a fundamental property of the vacuum state that is invisible to the local effective Lagrangian used to describe low-energy physics.

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