⚛️ quantum physics

How thermal is a filtered state?

This paper establishes that in both Floquet and conventional Hamiltonian systems, the trace distance between energy-filtered states and thermal states is bounded by the square root of the filter width, thereby quantifying the precision required for filtered states to accurately approximate thermal behavior.

Original authors: Yilun Yang, J. Ignacio Cirac, Mari Carmen Bañuls

Published 2026-07-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yilun Yang, J. Ignacio Cirac, Mari Carmen Bañuls

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 Big Picture: Cooking a "Thermal" Meal from Scratch

Imagine you have a chaotic kitchen (a quantum system) filled with thousands of ingredients (particles) moving around wildly. You want to cook a specific dish: a Thermal State. In physics, a thermal state is like a perfectly balanced soup where the temperature is uniform, and the ingredients are mixed just right. It represents the state of equilibrium that nature naturally settles into.

The problem is, preparing this "soup" directly in a quantum computer is incredibly hard. It's like trying to mix a giant pot of soup by hand without spilling a drop.

The Solution: The Energy Filter
Instead of mixing the soup directly, the authors propose a clever trick: Energy Filtering.
Imagine you have a giant sieve (a filter) that only lets through ingredients with a specific weight (energy). If you start with a simple, ordered pile of ingredients (a "product state") and shake them through this sieve, you get a new pile that is supposed to look and taste like the thermal soup.

The big question the paper answers is: How fine does the mesh of the sieve need to be?

If the holes are huge, you get a mess.
If the holes are microscopic (exponentially small), you get a perfect thermal state, but it takes forever to make.
The Goal: Find the "sweet spot" where the holes are small enough to make the soup taste thermal, but not so small that the process is impossible.

The New Tool: The "Stroboscopic" Camera (Floquet Dynamics)

To figure this out, the authors didn't look at the soup being stirred continuously. Instead, they used a stroboscopic camera.

Imagine taking a photo of a spinning fan once every second. You don't see the blur; you see the fan in specific, frozen positions. In physics, this is called Floquet dynamics. Instead of watching the system evolve smoothly over time, they look at it in "snapshots" at regular intervals.

They found that if you apply their energy filter in this "snapshot" mode, the resulting state is mathematically equivalent to averaging all the snapshots together.

Analogy: Imagine taking a photo of a dancer every second for an hour. If you overlay all those photos, you get a "ghostly" image showing every position the dancer ever took. The paper proves that the "filtered state" is essentially this ghostly average.

The Main Discovery: How "Thermal" is it?

The authors ran the numbers to see how close this "ghostly average" gets to a real thermal state. They measured the distance between the filtered state and the perfect thermal state using a metric called Trace Distance.

The Result:
The difference between the filtered state and the perfect thermal state is roughly the square root of the filter width.

The Metaphor: Think of the filter width ( $\delta$ $δ$ ) as the "blur" in your photo.
- If you make the filter twice as precise (halving the width), the error doesn't get cut in half. Instead, it gets cut by the square root (about 30%).
- The Takeaway: You don't need a microscopic, perfect filter to get a very good thermal state. A moderately precise filter is enough to get you "close enough" for most practical purposes. The error shrinks predictably as you tighten the filter.

The Entanglement Puzzle: How "Messy" is the Soup?

In quantum physics, "thermal" states are usually very "messy" or "entangled." This means the ingredients are so mixed that you can't describe one part without describing the whole pot. This messiness is measured by Entropy.

Previous studies suggested that filtered states might not be messy enough to be truly thermal. The authors dug deeper and found that the "messiness" depends on how you measure it (using a parameter called $\alpha$ ):

If you look at the "average" messiness ( $\alpha > 1$ ): The filtered state is surprisingly not as messy as a perfect thermal state. It grows very slowly (logarithmically) as you tighten the filter. It's like a soup that is mixed, but you can still see a few distinct clumps of ingredients.
If you look at the "standard" messiness ( $\alpha = 1$ ): The messiness grows linearly. This is closer to what we expect in a thermal state.
If you look at the "extreme" messiness ( $\alpha < 1$ ): The messiness can grow very fast (quadratically), depending on how the filter is cut off.

Why does this matter?
It shows that while filtered states aren't perfectly thermal in every single mathematical sense (especially for certain types of measurements), they are locally thermal. This means if you only look at a small spoonful of the soup (a local part of the system), it looks and behaves exactly like a thermal state.

The Bridge: From Snapshots to Real Time

The paper started by using the "snapshot" (Floquet) method because it's easier to calculate. But real quantum systems usually evolve continuously (Hamiltonian dynamics).

The authors showed that you can translate their "snapshot" results back to the "continuous" world.

Analogy: They proved that the recipe for the "snapshot soup" works just as well for the "continuous stirring soup," provided you adjust the timing correctly.
Result: The same rule applies: The error is proportional to the square root of the filter width, even in the real, continuous world.

Summary of Claims

The Filter Width Rule: To get a state that behaves thermally, you don't need a filter that is impossibly precise. The error in the state scales with the square root of the filter width.
The Equivalence: A filtered state is mathematically the same as a weighted average of the system's history (time average).
Local vs. Global: While the filtered state might not be "perfectly" thermal in every deep mathematical way (specifically regarding how entangled the whole system is), it is locally thermal. If you measure any small part of it, it looks exactly like a thermal state.
Practicality: You can start with a very simple, ordered state (like a product state) and use this filtering method to create a thermal-like state, which is much easier to prepare than trying to create a thermal state from scratch.

In a nutshell: The paper proves that using an energy filter is a reliable way to "fake" a thermal state. You don't need a perfect filter; a reasonably good one gets you 99% of the way there, and the math tells you exactly how close you are.

Technical Summary: "How thermal is a filtered state?"

Problem Statement
Quantum simulation offers a pathway to probe thermal equilibrium properties of many-body systems. However, directly preparing thermal states on quantum devices is challenging. A promising alternative involves using energy filters to approximate thermal ensembles from simple, easily preparable pure states (e.g., product states). While it is established that an exponentially narrow filter (resolving individual eigenstates) yields thermal behavior under the Eigenstate Thermalization Hypothesis (ETH), the precise requirements for the filter width $\delta$ to guarantee thermal behavior remain an open question. Previous works have explored this via entanglement entropy scaling and canonical universality, yielding conflicting or incomplete results regarding the necessary scaling of $\delta$ with system size $N$ and the order of Rényi entropy $\alpha$ . This work aims to rigorously determine how narrow the filter must be to ensure a filtered state behaves thermally.

Methodology
The authors analyze the problem by shifting from continuous-time Hamiltonian evolution to a discrete-time Floquet regime. They utilize a cosine filter (specifically a truncated version) applied to a local Hamiltonian or Floquet operator.

Floquet Mapping: The filter is defined as a linear combination of time-evolution operators at discrete steps. The authors distinguish between the "Hamiltonian regime" (where the time step $T$ scales with system size to resolve energy) and the "fully Floquet regime" (where $T$ is constant, and the spectrum is folded).
Reduced Density Matrix (RDM) Structure: A central technical step involves analyzing the RDM of the filtered state. The authors demonstrate that for generic initial product states, the Loschmidt echo (overlap between time-evolved states) decays exponentially with system size. This decay implies that off-diagonal blocks in the RDM vanish, rendering the RDM effectively a weighted direct sum of RDMs from individual time-evolved steps.
Entropy Calculation: Leveraging this block-diagonal structure, the authors compute Rényi- $\alpha$ entanglement entropies by summing contributions from the diagonal blocks and the distribution of filter coefficients.
Thermal Convergence: To quantify "thermalness," the authors calculate the trace distance (or difference in expectation values of local observables) between the filtered state and the thermal (diagonal) ensemble. They employ the Floquet ETH, which posits that matrix elements of observables in the eigenbasis follow random matrix theory statistics, to bound the contributions of off-diagonal terms.
Extension to Hamiltonian Regime: Finally, the authors map Hamiltonian-filtered states to effective Floquet-filtered states by grouping time-evolution terms, allowing the Floquet results to be applied to the standard Hamiltonian setting.

Key Results

Rényi Entropy Scaling: The authors reproduce and explain distinct scaling behaviors for the entanglement entropy $S_\alpha$ $S_{α}$ of filtered states depending on $\alpha$ $α$ :
- $\alpha > 1$ : The entropy scales logarithmically with the inverse filter width ( $\log(1/\delta)$ ). This contrasts with the volume law expected for fully thermal states, suggesting that for polynomially small $\delta$ , filtered states are not fully thermal in terms of high-order entanglement.
- $\alpha = 1$ (Von Neumann): The entropy scales linearly with $1/\delta$ (plus a logarithmic correction), similar to the entropy of a state evolved for time $t \sim 1/\delta$ .
- $\alpha < 1$ : The scaling depends on the filter type. For a truncated filter, it scales linearly with $1/\delta$ . For an untruncated (exact cosine) filter, it scales quadratically with $1/\delta$ (or $1/\delta^2$ ).
Convergence to Thermal States: Under the assumption of Floquet ETH, the authors prove that the trace distance between a Floquet-filtered state and the thermal state is upper bounded by $O(\sqrt{\delta})$ . Specifically, the difference in expectation values for any local observable is bounded by $O(\sqrt{\delta})$ .
Hamiltonian Regime Equivalence: The $O(\sqrt{\delta})$ bound extends to Hamiltonian-filtered states. By mapping a Hamiltonian-filtered state to an effective Floquet state with a modified time step, the authors show that the convergence rate remains $O(\sqrt{\delta_H})$ .
Numerical Verification: The theoretical predictions are validated using Matrix Product State (MPS) simulations for non-integrable Ising models and exact diagonalization for small systems, confirming the $O(\sqrt{\delta})$ scaling for local observables and the specific entropy scalings.

Significance and Claims
The paper claims to provide a quantitative characterization of the convergence of filtered states to thermal ensembles.

Resolution of Open Questions: It clarifies the relationship between filter width and thermal behavior, showing that while filtered states may not exhibit volume-law entanglement for all $\alpha$ (specifically $\alpha > 1$ ) when $\delta$ is polynomially small, they do converge to thermal expectation values for local observables.
Derivation of Canonical Universality: The work demonstrates that the "canonical universality" property (where expectation values in a narrow energy window approximate thermal values) is not an independent hypothesis but can be derived directly from the Floquet ETH. The authors show that even with correlated off-diagonal matrix elements (beyond simple independence assumptions), the error scales as $\sqrt{\delta}$ .
Algorithmic Efficiency: The authors note that Floquet filters can replace Hamiltonian filters in thermal state preparation. Since the Floquet time step can be larger ( $O(\sqrt{N})$ ) than the Hamiltonian time step required to resolve energy, this approach may reduce the number of measurements required in quantum algorithms.
Simplicity of Initial States: Unlike other methods for preparing Thermal Pure Quantum States (TPQS) that require Haar-random initial states, this approach works with simple product states, making it more feasible for both classical and quantum implementations.

The authors remain modest regarding the limitations, acknowledging that while local observables converge, the entanglement structure (particularly for $\alpha > 1$ ) does not fully mimic a thermal state unless the filter width is exponentially small. They also note that the extension of entanglement entropy results from the Floquet to the Hamiltonian regime is not theoretically guaranteed to be identical due to the entanglement introduced in the effective initial state of the mapping, though numerical evidence supports the similarity.