Qubit entanglement from forward scattering

This paper analytically derives the concurrence for mixed qubit states in relativistic $2\to 2$ scattering, demonstrating that the leading-order entanglement depends on the real part of the inelastic forward amplitude and the initial state, while also revealing a connection between this amplitude and the reduction of linearized entropy.

The Gist

The Big Picture: A Cosmic Dance of Particles

Imagine two dancers (particles) entering a ballroom. They have two things about them:

1. Where they are going: Their momentum (speed and direction).
2. Who they are: Their "identity" or "flavor" (like their spin, charge, or a specific type of particle). In quantum mechanics, this identity acts like a qubit (a quantum bit of information, like a coin that can be heads, tails, or both at once).

When these dancers collide, they interact. Sometimes they bounce off each other, sometimes they change partners, and sometimes they just graze past one another. This interaction is described by something called the S-matrix, which is essentially the "rulebook" of how they interact.

The big question the authors ask is: Does this collision make their "identities" (qubits) become "entangled"?

Entanglement is a spooky connection where two particles become so linked that you can't describe one without the other. If you measure one, you instantly know something about the other, no matter how far apart they are.

The Problem: Too Much Noise

In a real experiment, we can't easily separate the dancers' "identities" from their "movement." When they collide, they get tangled up in a mess of directions and speeds. It's like trying to hear a specific conversation in a crowded, noisy room.

To study the entanglement of their identities (the qubits), physicists usually have to do one of two things:

1. The "Freeze Frame" Approach: Pick a specific angle where they fly out and ignore everything else. This is easy to calculate but ignores the fact that energy and probability must be conserved (the "conservation of dance moves").
2. The "Blur" Approach (This Paper): Instead of picking one angle, we look at all possible angles and average them out. We "trace out" the movement to see what's left of the identity. This is more realistic but mathematically messy because the resulting state is "mixed" (a bit fuzzy).

The Big Discovery: The "Ghost" of the Forward Path

The authors found a surprisingly simple rule for how much entanglement is created, specifically when the particles start off as separate, unentangled partners (a "product state").

They discovered that the main source of entanglement comes from a very specific, weird scenario: The Forward Scattering Amplitude.

The Analogy:
Imagine two cars driving toward each other on a highway.

• The "Inelastic" Crash: Most of the time, they crash, spin out, and go in random directions. This creates a lot of chaos (entropy) and links their movement to their identity.
• The "Ghost" Pass-Through: Imagine a scenario where the cars drive so close they barely touch, but their "radio signals" (quantum numbers) swap or get confused, yet they keep driving in almost the exact same direction they came from.

The paper proves that this "Ghost Pass-Through" (Forward Scattering) is actually the dominant reason the particles become entangled in their identities.

• The Imaginary Part (The Crash): Usually, physicists think the "crash" (the total cross-section, or the imaginary part of the math) creates the mess. This creates a link between where they go and who they are.
• The Real Part (The Ghost): The authors found that the Real Part of the forward amplitude is what actually creates the entanglement between the two particles' identities. It's a subtle, "real" connection that happens even when they don't scatter wildly.

The "Concurrence" Meter

To measure this entanglement, they use a tool called Concurrence. Think of this as a "Spookiness Meter."

• 0: The particles are independent (not entangled).
• 1: They are perfectly entangled (maximally spooky).

The paper gives a formula showing that for most collisions, the "Spookiness Meter" reads a value directly proportional to the Real part of the Forward Amplitude.

Why is this cool?
It means you don't need to simulate the entire chaotic collision to know how entangled the particles will be. You just need to look at the math describing the "almost-nothing-happened" scenario (forward scattering).

Two Real-World Examples

The authors tested their theory on two scenarios:

1. The Two-Higgs Doublet Model (2HDM):

   • The Scenario: A model of physics with extra types of Higgs bosons (particles that give mass to others).
   • The Result: Here, the particles interact like billiard balls hitting each other directly (contact interactions). The "Forward" and "Sideways" scattering are similar. The math showed that the entanglement is generated by the "Real" part of the interaction, confirming their rule.

2. Electron-Positron Annihilation (QED):

   • The Scenario: An electron and a positron (anti-electron) smash together to create muons.
   • The Twist: In this specific case, due to the laws of angular momentum, the "Forward Scattering" (where they keep going straight) is forbidden. The "Ghost Pass-Through" is zero.
   • The Result: Because the "Forward" part is zero, the Concurrence (Spookiness) is also zero at the leading order. The particles don't get entangled in their identities in the way the first example did. This proves the rule: No forward amplitude = No leading-order identity entanglement.

The "Entropy" Side Note

The paper also looked at Linearized Entropy, which measures how "mixed up" the system is.

• They found that the "Forward Scattering" (the Real part) actually acts like a cleaner. It slightly reduces the messiness (entropy) of the system.
• They linked this reduction to a concept called Coherence, which is basically how "quantum" a state is. The forward scattering preserves a bit of that quantum "purity."

Summary: What Does This Mean for Us?

1. Simplicity in Chaos: Even in the complex world of high-energy particle collisions, the amount of entanglement created between particle identities is governed by a simple, specific part of the math: the Real part of the forward scattering amplitude.
2. The "Almost" Collision: The most significant quantum link between two particles often comes from the interaction where they almost didn't interact at all (they kept going straight), rather than the big, messy crash.
3. New Tools for New Physics: This gives physicists a new, easier way to calculate entanglement without doing heavy simulations. It also suggests that if we want to find new symmetries in the universe (rules that make physics look the same), we should look at how they affect this specific "Forward Scattering" entanglement.

In a nutshell: The universe creates quantum connections between particles not just when they crash, but most strongly when they "almost" pass through each other without changing direction, swapping their quantum "souls" in the process.

Technical Summary

1. Problem Statement

The paper addresses the challenge of quantifying quantum entanglement generated between discrete internal degrees of freedom (qubits, such as spin, polarization, or flavor) of particles in relativistic $2 \to 2$ scattering processes.

While entanglement between momentum modes is well-understood (proportional to the total cross-section), quantifying entanglement between internal quantum numbers in a mixed final state is more complex. Previous approaches often projected the final state onto a specific momentum vector, yielding a pure state but violating probability conservation (unitarity) between initial and final states. Alternatively, tracing out momentum degrees of freedom preserves unitarity but results in a mixed state, requiring specialized entanglement measures like concurrence rather than simple entropy of a pure state.

The authors aim to:

1. Derive an analytic expression for the concurrence of the momentum-reduced density matrix in terms of the scattering amplitude and initial state.
2. Identify the specific components of the scattering amplitude (real vs. imaginary parts, forward vs. non-forward) that drive qubit entanglement at leading order.
3. Investigate the relationship between qubit entanglement and the "entanglement flow" between momentum and internal degrees of freedom.

2. Methodology

The authors employ a perturbative expansion of the S-matrix within the framework of Quantum Field Theory (QFT).

• Formalism: They define the total Hilbert space as $\mathcal{H}_{tot} = \mathcal{H}_{mom} \otimes \mathcal{H}_{qd}$, where $\mathcal{H}_{qd}$ represents the discrete quantum numbers (qubits). The initial state is a product of a momentum wave packet and a qubit state $|A\rangle$.
• Momentum Tracing: Instead of projecting onto a specific angle, they trace out the momentum degrees of freedom to obtain the reduced density matrix $\rho_Q = \text{Tr}_{mom}(|out\rangle\langle out|)$. This ensures unitarity is preserved via the optical theorem.
• Perturbative Expansion: The calculation is performed up to order $\mathcal{O}(M^2)$ (where $M$ is the scattering amplitude). They introduce a small parameter $\Delta$, related to the overlap of initial and final momentum wave packets in solid angle, to organize the expansion.
• Entanglement Measures:
  • Linearized Entropy ($E$): Used to quantify the entanglement between the momentum and qubit partitions.
  • Concurrence ($C$): Used to quantify the entanglement between the two qubits themselves in the mixed state.
• Derivation: They derive a general formula for the concurrence $C(\rho_Q)$ involving the initial state $|A\rangle$, the spin-flipped state $|B\rangle = \Sigma |A^*\rangle$, and various integrals of the scattering amplitude $M$ (both forward and integrated over phase space).

3. Key Contributions and Results

A. Analytic Formula for Concurrence

The authors derive a general expression for the concurrence of the reduced density matrix. For an initial product state (where the two qubits are initially unentangled), the formula simplifies significantly at the leading order:
$$C(\rho_Q) \simeq 2\Delta |\langle B | \text{Re} M_{fw} | A \rangle|$$
where:

• $M_{fw}$ is the forward scattering amplitude ($p_{in} \to p_{in}$).
• $\text{Re} M_{fw}$ is the real part of the forward amplitude.
• $\Delta$ is a normalization factor related to the wave packet overlap.

Crucial Finding: Unlike the linearized entropy (which depends on the imaginary part of the forward amplitude or the total cross-section), the leading-order qubit entanglement is generated exclusively by the real part of the forward scattering amplitude.

B. Subleading Correction to Linearized Entropy

The paper analyzes the linearized entropy $E = 1 - \text{Tr}(\rho_Q^2)$. They show that while the leading term is proportional to the inelastic cross-section (imaginary part of $M_{fw}$), the real part of the forward amplitude provides a subleading correction ($\mathcal{O}(\Delta^2)$).

• This correction reduces the linearized entropy.
• For computational basis states, this reduction is mathematically equivalent to the relative entropy of coherence.
• This implies that the real part of the amplitude acts to preserve the separability between momentum and qubit degrees of freedom, effectively "reducing" the entanglement flow from qubits to momentum.

C. Phenomenological Applications

The authors apply their formulas to two specific models:

1. Two-Higgs Doublet Model (2HDM):

   • Analyzed high-energy scattering of scalar fields ($h^0 h^0 \to h^0 h^0$).
   • Since the interaction is a contact term, the amplitude is isotropic.
   • Result: The concurrence is non-zero and directly proportional to the coupling $\lambda_5$ (via the real part of the forward amplitude).
   • Bell State Analysis: For an initial maximally entangled Bell state, they find a quantitative relation $C_{out} \approx 1 - E_{out}$. This suggests an "entanglement flow" where entanglement initially stored in the qubits is transferred to the momentum-flavor partition during scattering.

2. Electron-Positron Annihilation ($e^+e^- \to \mu^+\mu^-$) in QED:

   • Analyzed at high energies ($s \gg m_\mu^2$).
   • Due to angular momentum conservation and helicity selection rules, the inelastic forward amplitude vanishes (the matrix element is zero at $\theta=0$).
   • Result: The leading-order concurrence vanishes ($C \approx 0$). Entanglement is only generated at higher orders or via non-forward scattering, which is perturbatively suppressed. This confirms the theoretical prediction that if $\text{Re} M_{fw} = 0$, no leading-order qubit entanglement is generated.

4. Significance and Implications

• Theoretical Insight: The work establishes a direct link between the real part of the forward scattering amplitude and the generation of quantum correlations between internal degrees of freedom. This distinguishes it from the generation of momentum-qubit entanglement, which is driven by the imaginary part (total cross-section).
• Symmetry and Entanglement: The results offer a new perspective on the "entanglement minimization" principle. If symmetries of the Lagrangian force the real part of the forward amplitude to vanish (or satisfy specific conditions), they would suppress the generation of qubit entanglement. This provides a potential mechanism to test emergent symmetries in high-energy physics.
• Experimental Relevance: While measuring the full density matrix is difficult, the derived dependence on the forward amplitude suggests that specific kinematic configurations or interference effects in forward scattering could be sensitive probes of qubit entanglement.
• Methodological Advancement: The paper provides a robust, unitary-preserving framework for calculating entanglement in mixed states using perturbative S-matrix theory, moving beyond the limitations of fixed-angle projections.

5. Conclusion

Kowalska and Sessolo demonstrate that in relativistic scattering, the real part of the forward scattering amplitude is the primary driver of entanglement between qubits in the final state, provided the initial state is a product state. This finding contrasts with the imaginary part's role in momentum-qubit entanglement. The study bridges quantum information theory and high-energy phenomenology, offering analytic tools to predict entanglement generation in models like the 2HDM and QED, and suggesting that the suppression of qubit entanglement could serve as a signature for underlying symmetries.