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Thermal screening and critical scaling of quantum energy teleportation in a harmonic chain

This paper establishes a finite-temperature Gaussian-state framework for quantum energy teleportation in a one-dimensional harmonic chain, demonstrating that while extracted energy is exponentially screened by thermal correlation lengths at finite temperatures, it follows a distinct $d^{-4}$ power-law scaling in the zero-temperature critical limit, with squeezed measurements modifying the extraction efficiency without altering the large-distance scaling behavior.

Original authors: Taisanul Haque

Published 2026-09-02

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CC BY 4.0

Original authors: Taisanul Haque

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 quantum world, energy does not always have to travel through space like a ball rolling down a hill. Sometimes, it can appear at a distant location without ever crossing the gap between them. This counterintuitive process, known as quantum energy teleportation, relies on a strange feature of nature called entanglement, where particles remain linked regardless of distance. The protocol works like this: a scientist at one spot measures a local part of a system, which injects a tiny amount of energy. They then send the result of that measurement via a standard radio signal to a partner far away. Using that information, the partner performs a specific local action that extracts energy from their own spot. Crucially, no usable energy travels during the radio transmission; instead, the partner uses the pre-existing quantum connection between the two spots to pull energy out of the vacuum of their own location. While this phenomenon has been studied in simple systems of just a few particles, scientists have long wondered how it behaves in large, continuous materials, especially when the material is warm rather than frozen at absolute zero.

A new study by Taisanul Haque at the University of Göttingen brings this concept into the realm of large, warm materials. The researcher modeled a long chain of connected springs and masses, a system that behaves like a continuous line of vibrating atoms or a chain of linked microwave circuits. By treating this chain as a collection of quantum states that exist at a specific temperature, the team developed a precise mathematical framework to see how much energy could be teleported from one end of the chain to another. The work reveals that temperature acts as a powerful filter. When the chain is kept at a fixed, warm temperature, the ability to extract energy drops off incredibly fast as the distance between the two scientists increases. The energy is effectively screened out by the heat, disappearing exponentially over a distance determined by the temperature. This means that in a warm environment, you cannot teleport energy across a large room; the connection is too weak to carry the signal.

However, the story changes dramatically when the temperature is lowered toward absolute zero and the system is tuned to a critical point, a state where the material is on the verge of a major change in its physical properties. In this specific, delicate limit, the rapid drop-off caused by heat vanishes. Instead, the energy extraction follows a much slower, steady decline based on distance. The researchers derived an exact rule showing that the extractable energy decreases as the inverse of the distance raised to the fourth power. This means that even across a long chain, a small but measurable amount of energy can still be teleported, provided the system is cold enough and tuned just right. The study confirms that while heat destroys long-range quantum energy transfer, the special geometry of a critical system allows it to survive, albeit with a predictable and mathematically precise weakness.

The paper also investigates whether the way the first scientist measures the system can be improved to get more energy out. They tested a technique called squeezing, which alters the precision of the measurement in one direction at the cost of another. The results show that this technique acts like a volume knob: it can increase or decrease the total amount of energy extracted, but it cannot change the fundamental rule of how that energy fades with distance. Whether the measurement is squeezed or not, the energy still follows the same exponential drop in warm conditions or the same power-law drop in the cold, critical limit. This finding suggests that the limits of quantum energy teleportation are set by the natural state of the material itself, not by the cleverness of the measurement tool.

These findings provide a clear, controlled map of how quantum energy teleportation works in a realistic, extended system. The study moves beyond simple, isolated particles to show how heat and criticality shape the flow of energy in a continuous medium. It establishes that while thermal noise is a formidable barrier that screens out long-distance quantum effects, the unique structure of a critical system can overcome this barrier to allow energy to be teleported over significant distances. The work offers a solid theoretical benchmark for future experiments, particularly those involving chains of superconducting circuits, where these effects might one day be observed and utilized.

Technical Summary: Thermal Screening and Critical Scaling of Quantum Energy Teleportation in a Harmonic Chain

Problem Statement
Quantum Energy Teleportation (QET) is a protocol where a local measurement injects energy into a correlated many-body system, and a distant conditional operation extracts positive energy using the measurement outcome and pre-existing correlations. While extensive literature exists for ground-state protocols and few-body systems (e.g., spin chains), the behavior of QET in spatially extended bosonic systems at finite temperature remains less explored. Specifically, there is a need for an analytically tractable framework to understand how thermal fluctuations screen energy extraction and how critical behavior influences long-distance scaling in the thermodynamic limit. Previous numerical studies on harmonic chains suggested a critical decay exponent but lacked an analytic derivation.

Methodology
The author develops a finite-temperature Gaussian-state formulation of QET for a one-dimensional harmonic chain. The system is modeled by a pinned spring-mass chain (or equivalently, coupled linear microwave resonators) with a dimensionless Hamiltonian $H = \frac{1}{2} \sum (p_j^2 + q_j^2 - \alpha q_j q_{j+1})$ , where $\alpha$ controls the coupling strength and the gap.

State and Protocol: The initial state is the Gibbs state $\rho_\beta$ . The protocol involves Alice performing a Gaussian measurement (coherent-state POVM or squeezed Gaussian POVM) on a block of sites and Bob applying a local displacement based on the classical communication of the outcome.
Reduction to Correlators: For translationally invariant Gibbs states, the author shows that the optimized Gaussian QET protocol in a single-site setting reduces entirely to thermal two-point correlation functions. Specifically, the extracted energy depends on the position and momentum covariance functions, $g_n(\beta, \alpha)$ and $h_n(\beta, \alpha)$ .
Analytic Derivation: The author derives exact expressions for the extracted energy $E_{ext}$ using Matsubara frequency expansions for finite temperature and closed-form integrals for the zero-temperature critical limit. They enforce an ordered limit: first the thermodynamic limit ( $N \to \infty$ ), then the temperature regime, and finally the critical limit ( $\alpha \uparrow 1$ ).
Numerical Diagnostics: The analytical results are benchmarked against numerical simulations of finite chains ( $N=215$ and $N=2000$ ) to resolve crossovers between algebraic and exponential decay regimes.

Key Contributions and Results

Reduction to Thermal Correlators: The primary theoretical contribution is the demonstration that for single-site protocols, the optimized extracted energy is governed by a single thermal coupling function, $h_d(\beta, \alpha)$ . This simplifies the problem to the asymptotic analysis of this specific correlator.
Thermal Screening (Finite Temperature): At fixed finite temperature, the extracted energy is exponentially suppressed with distance $d$ . The decay is controlled by a thermal correlation length $\xi_\beta$ . The author derives the exact asymptotic form $E_{ext} \sim e^{-4d/\xi_E}$ , where the energy decay length is half the correlation length of the momentum coupling.
Critical Scaling (Zero Temperature): In the zero-temperature critical limit ( $\beta \to \infty$ followed by $\alpha \uparrow 1$ ), the thermal screening vanishes, and the system exhibits algebraic decay. The author derives the exact asymptotic law:
$E_{ext} \sim d^{-4}$
This resolves a previous numerical observation of an apparent decay close to $d^{-3.6}$ , providing a rigorous analytic proof of the $d^{-4}$ exponent.
Crossover Analysis: The paper analyzes the crossover between the algebraic regime (near-critical, low temperature) and the exponential regime (finite temperature or gapped ground state). Numerical diagnostics of the effective logarithmic slope and local attenuation rate confirm that the apparent drift in decay exponents at intermediate distances is a crossover effect caused by finite correlation lengths (either thermal or gap-induced).
Squeezed Measurements: The author analyzes squeezed Gaussian measurements. They find that $p$ -squeezing (reducing momentum noise) enhances the extracted energy prefactor, while $q$ -squeezing suppresses it. Crucially, squeezing modifies the extraction efficiency but does not alter the large-distance scaling class (the exponent remains $-4$ in the critical limit).

Significance and Claims
The paper positions the harmonic chain as an analytically controlled platform for studying QET in correlated quantum matter. The author claims that their work clarifies the interplay between thermal screening and criticality:

Finite temperature acts as a screen that destroys long-range energy extraction resources, limiting them to a thermal correlation length.
Proximity to a critical point restores long-range tails, allowing for algebraic decay even in the presence of thermal effects if the temperature is sufficiently low relative to the correlation length.
The protocol's performance is dictated more by the correlation structure of the underlying Gaussian state than by specific protocol details (like squeezing), which only affect prefactors.

The results serve as a benchmark for bosonic QET implementations, particularly in circuit-QED architectures where coupled microwave resonators map directly to the harmonic chain model. The work does not propose new experimental setups but provides the theoretical foundation and exact scaling laws necessary to interpret such experiments.

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