Scattering entanglement entropy between one particle species and the others

arXiv:2610.10704 · hep-th, hep-ph, quant-ph · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Scattering entanglement entropy between one particle species and the others".

Mira: The gist The authors construct an entanglement entropy of a general scattering process between one particle species and the rest of the final state,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we're looking at this paper, "Scattering entanglement entropy between one particle species and the others," and what it says about constructing an entanglement entropy for a general scattering process between one particle species A and the rest of the final state.

Mira: The authors show they can write this entropy using things like scattering probabilities and differential cross sections. It’s essentially taking a complex quantum interaction and turning it into something calculable with standard physics tools.

Lev: But what does that mean for us on hardware? If we try to run this, do we even have the setup to measure those differential cross sections mentioned?

Kai: The paper shows that the reduced density matrix in particle A’s Fock space is block-diagonal in zero, one, and multi-particle states. This allows them to write down the Von Neumann entropy of particle A as a function of scattering probabilities and differential cross sections, provided we're only producing at most one particle A in the final state.

Mira: They break it down into terms: S A is written as S A = - p zero p zero - T

rho 1A (rho 1A): - T

p mul rho mul (p mul rho mul): , and that simplifies to S A = - p zero p zero - p one p one + p one S one - p mul p mul + p mul S mul.

Lev: That structure is interesting, but the paper immediately cuts down the multi-particle term, S mul, saying it can always be restored later. So for now, what's the focus on those probabilities and cross sections?

Kai: The authors focus on that one-particle entropy part, S one which they write as S one = -, alpha p, alpha p, alpha. They explain that is the three-momentum of particle A and alpha labels any additional discrete degrees of freedom it has.

Mira: Now, when they take the continuum limit for these momenta, they get a formula for S A: S A = - p zero p zero - p one p one - p one alpha Z d, p alpha P (p alpha P), where d is the phase space element and is a regulator.

Lev: That regulator term,, that's what worries me for real experiments. They say the von Neumann entropy typically scales with N, where N is the dimension of the Hilbert space, so we need some way to handle that infinite-dimensional momentum Hilbert space when we push it to zero resolution.

Kai: Exactly. The paper points out that the term S one contains information on both total scattering probabilities and differential distributions—that differential part depends on a phase-space resolution factor because of that infinite-dimensional momentum Hilbert space, which has been seen in previous studies.

Mira: They also looked at Tsallis and Rényi entropies, which are different measures of entanglement, and found that at the leading order, they are proportional to the total scattering probability and carry no differential information. Specifically for Tsallis entropy T n(rho A) for n two assuming a small p one they get T n(rho A) = n/(n-one)p one plus higher order terms.

Lev: If the leading order is just proportional to the total probability, then the differential details in the cross section aren't that important for those measures, right?

Kai: That’s what they found. They also did something similar for Rényi entropy R n(rho A) for n two and it gives them R n(rho A) = n/(n-one)p one at the leading order, which is the same as the Tsallis result.

Mira: This suggests that for many simple cases, knowing just how often particle A is produced in a certain way is more important than knowing the fine details of its momentum distribution in those scattering events.

Lev: So, if we were designing a real experiment to probe this, would we focus our measurement on the total rates first before worrying about the detailed momentum distributions?

Kai: That seems like what they are suggesting for these measures. But they did mention that restoring that multi-A contribution, p mul S mul, is hard because there isn't a simple expression for it in their current framework.

Mira: And the authors themselves admit that if we consider multiple particle A produced, like m > one whether we can still isolate the subsystem with just one particle A and get a meaningful entropy expression is a challenge they haven't fully solved yet.

Lev: So, to summarize this paper on "Scattering entanglement entropy between one particle species and the others," it constructs an entropy formula using scattering probabilities and cross sections, showing that the one-particle part S one holds differential information about the channel A plus X, where X is any number of other particles.

Kai: And while they found that Tsallis and Rényi entropies are insensitive to differential distribution at leading order, the core difficulty remains in handling those multi-particle contributions when you try to get the full picture of entanglement in these scattering processes.

Mira: The implication here is a clear separation: for the one-particle sector, we can isolate information about particle A itself even when everything else is interacting. But that doesn't solve the problem for states where multiple A particles are produced.

Lev: It sounds like the next step for this research would be developing a way to tackle that multi-particle entropy term, S mul, which they noted wasn't addressed in their framework.

Kai: Precisely. The authors suggest that perhaps tackling those higher particle numbers might give us a better way to understand the full entanglement picture in these processes.

Conclusion: Kai: So we've looked at how they built this entanglement entropy for scattering between one particle A and everything else in the final state, and now we're wrapping up with what this actually means for us on the radio.

Mira: The paper is titled "Scattering entanglement entropy between one particle species and the others," and it shows they can write that entropy using scattering probabilities and differential cross sections. It’s basically taking a complex quantum interaction and turning it into something we can calculate with standard physics tools.

Kai: Right, so the core idea is this entropy breaks down into pieces related to the different ways particles scatter. It separates the information about particle A itself from how it interacts with everything else in that final state.

Mira: What’s interesting is their result for that one-particle part of the entropy, which they call S1, because it turns out that term doesn't actually depend on the total scattering probability at all.

Kai: That’s a big deal because it means you can isolate information about just particle A’s state even when everything else is interacting wildly in the final state. It points to some fundamental separation of information in these kinds of processes.

Mira: They also looked at those Tsallis and Rényi entropies, which are different ways to measure entanglement, and found that at the leading order, they behave similarly—they both depend mainly on that one-particle probability term.

Kai: It suggests that for many simple cases, the differential information in the scattering cross section is less important than just knowing how often particle A is produced in a certain way.

Mira: But they also pointed out a limitation, which is they can't easily find a simple expression for the multi-particle contribution to that entropy. That part gets really messy.

Kai: So while this gives us a clear picture of the one-particle information, we still have that multi-particle term to tackle if we want the full story on entanglement in these processes. We need to figure out how to handle those higher particle numbers in our framework next.

Yuan-Hao Zheng, *, Jiayin Gu, †

Department of Physics and Center for Field Theory and Particle Physics, Fudan University · Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan University

hep-th, hep-ph, quant-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 9 pages, 0 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 78/100

The gist: The gist The authors construct an entanglement entropy of a general scattering process between one particle species and the rest of the final state, showing that this entropy can be written in terms

Key concepts

Entanglement Entropy (SA)
This measures the quantum entanglement between a specific particle species (A) and the rest of the final state. It quantifies how much information about particle A is mixed with the other particles, expressed through its Von Neumann entropy.
Fock Space Decomposition
The total Hilbert space describing all possible final states is decomposed into subspaces based on how many particles of species A are present (zero, one, or multi-particle). This decomposition simplifies calculating the reduced density matrix for particle A.
Continuum Limit and Differential Information
Taking the continuous limit of momentum variables reveals that the entropy expression includes a term dependent on phase-space resolution ($\Delta\Phi$). This term shows that $S_1$ carries information about how scattering probabilities are distributed across different momentum configurations.

Terminology

Summary

The gist The authors construct an entanglement entropy of a general scattering process between one particle species and the rest of the final state, showing that this entropy can be written in terms of scattering probabilities and differential cross sections

Construction of Entanglement Entropy

The reduced density matrix in particle A’s Fock space is block-diagonal in zero-, one-, and multi-particle states, and is diagonal in the momentum basis for the one-particle state. This allows the Von Neumann entropy of particle A, SA, to be written down in terms of scattering probabilities and differential cross sections assuming no more than one particle A is produced in the final state. The decomposition of the Fock space is given by F = FA ⊗ Fothers. The reduced density matrix of A, ρA, can be written as a block matrix where ρ1A (ρmulA) is the normalized reduced density matrix in the one- (multi-) particle subspace of FA.

Entanglement Entropy Formula

The entanglement entropy SA is expressed as SA = − p0 log p0 − Tr[p1ρ1A log(p1ρ1A)] − Tr[pmulρmulA log(pmulρmulA)]. This simplifies to SA = − p0 log p0 − p1 log p1 + p1S1 − pmul log pmul + pmulSmul, where S1 ≡ − Tr(ρ1A log ρ1A) and Smul ≡ − Tr(ρmulA log ρmulA). The one-particle entropy S1 can be written in the form S1 = − X⃗k,α p⃗k,α log p⃗k,α, where ⃗k is the 3-momentum of particle A and α labels collectively any additional discrete degrees of freedom it has.

Continuum Limit and Differential Information

Taking the continuum limit for ⃗k, with p⃗k,α = p(α⃗k)p(⃗k) → pα(Φ)℘(Φ)dΦ, we obtain SA = −p0 log p0−p1 log p1−p1 X α Z dΦ pα℘ log(pα℘∆Φ), where ∆Φ is a regulator representing the phase-space resolution. As ∆Φ → 0, SA is clearly ill-defined without a regulator because the von Neumann entropy typically scales with log N where N is the dimension of the Hilbert space. The term S1 contains information on both total scattering probabilities and differential distributions — the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space.

Tsallis and R´enyi Entropies

The authors also obtain the n ≥ 2 Tsallis and R´enyi entropies in their framework, finding that at the leading order, they are proportional to the total scattering probability and carry no differential information. For Tsallis entropy Tn(ρA) for n ≥ 2, assuming a small p1, we have Tn(ρA) = n/(n − 1)p1 + O(p21), which is in agreement with the results in Refs. [13, 14]. Similarly, for R´enyi entropy Rn(ρA) for n ≥ 2, we obtain Rn(ρA) = n/(n − 1)p1 + O(p21), which is the same as the Tsallis entropy at the leading order in p1.

Summary and Outlook

The entanglement entropy SA factorizes into several pieces, depending on scattering probabilities and differential cross sections. The normalized entropy associated with the one-particle-A subspace, S1, is independent of the total scattering probability and contains the differential information in the channel A + X (where X contains any number of other particles). This dependence on phase-space resolution or other regulators was also observed in previous studies. The authors note that they can restore the multi-A contribution, −pmul log pmul + pmulSmul, in Eq. (23), but there is no simple expression for the multi-A entropy Smul that can be obtained within their framework. The smallness of the entanglement entropy is merely a reflection that most initial state particles pass through each other without any interactions. The normalized entropy associated with the one-particle-A subspace, S1, is independent of the total scattering probability and contains the differential information in the channel A + X (where X contains any number of other particles). This dependence on phase-space resolution or other regulators was also observed in previous studies. The authors also calculated the n ≥ 2 Tsallis and R´enyi entropies in their framework and found out that they are insensitive to the differential distribution at the leading order in the scattering probabilities. The field is bounded by how different definitions of entropy carry different weights for the total rate and the differential information. This work is desirable to generalize their framework to consider any arbitrary bipartition of the final state. The immediate challenge is that when multiple particle A are produced, whether one could still consider the subsystem with just one particle A and obtain a meaningful expression for the entropy. This might provide a way to tackle the multi-particle entropy Smul (defined in Eq. (9)), which is not addressed in their framework.

How it works

The construction begins by assuming a pure initial momentum eigenstate i⟩, and the final state is described by a unitary time evolution Si⟩ = Xa fa⟩⟨faSi⟩. The entanglement entropy SA is constructed as the von Neumann entropy of the subsystem of A. Due to energy and momentum conservation, the selection rule ⟨Kµ; α S i⟩ = 0 unless Kµ = Pµ. The final state is written as S i⟩ = Xkµ,α,β c kµ; α⟩A Pµ − k µ; β⟩O.

Diagonalization Properties

The total density matrix of the final state is ρ = Xckµ,α,βc∗ k′µ,α′,β′ k µ; α⟩A Pµ − k µ; β⟩O ⟨k′µ; α′A ⟨Pµ − k′µ; β'O. Tracing out Fothers imposes the condition k µ = k′µ, leading to ρA = Xkµ,α,α' ρA(k µ, α, α') k µ; α⟩A ⟨k µ; α'A. This diagonalization in kµ automatically separates zero-, one and multi-particle states. The zero-particle state is the only state with kµ = 0, a one-particle state always has k2 = m2 for particle A, and a multi-particle state always has k2 ≥ 4m2.

Explicit Expression in Continuum Limit

The explicit expression for SA in the continuous ⃗k3 A exception of this diagonalization is SA = −p0 log p0 − p1 log p1 − p1 X⃗k,α p⃗k,α log p⃗k,α. Taking the continuum limit yields SA = −p0 log p0−p1 log p1−p1 X α Z dΦ ρ α℘ log(p α℘∆Φ), where ℘(Φ) is the probability density in the Φ space. This expression shows that S1 contains information on both total scattering probabilities and differential distributions.

Improvements for AI systems

  1. textbfEmphasis on Differential Information Extraction from Scattering Probabilities: The improved AI system can calculate differential cross sections as a direct input to determine entanglement entropy, using equation (18) which shows how SA depends on the phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space.

  2. textbfQuantification of Entanglement in Multi-Particle Production: The system can incorporate the multi-particle contribution, specifically the term −pmul log pmul + pmulSmul, to provide a more complete picture of entanglement when multiple particle A are produced, even though an explicit expression for Smul is not provided.

  3. textbfHandling Non-Unitary/Non-Pure States: The system can utilize the Tsallis and R´enyi entropies (equations 29 and 32) to analyze non-pure states or systems where standard von Neumann entropy is insufficient, finding that at the leading order, they are proportional to the total scattering probability and carry no differential information.

  4. textbfModeling Entanglement via Phase Space Resolution: The AI can incorporate a regulator term log ∆Φ into the calculation of S1 (equation 24), demonstrating how entanglement is sensitive to phase-space resolution and suggesting that for sufficiently small ∆Φ, the contribution from momentum entanglement to S1 could be much larger than the ones from any discrete degrees of freedom.

Abstract

We construct the entanglement entropy of a general scattering process between one particle species (denoted as A) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle A 's Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle A, S A, in terms of scattering probabilities and differential cross sections, assuming no more than one particle A is produced in the final state. S A contains the information of both the total scattering probabilities (or cross sections) and the differential distributions --- the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution (Si) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle A is subleading. We also obtain the n at least2 Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.

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