Decoherence effects in entangled fermion pairs at colliders

This paper calculates the effects of quantum decoherence on maximally entangled fermion pairs at colliders by identifying the relevant Kraus operators with integrated Altarelli-Parisi splitting functions, addressing a factor often neglected in recent LHC measurements of top-quark spin entanglement.

The Gist

The Big Picture: Quantum Entanglement at the Particle Zoo

Imagine two particles, like a top quark and its anti-particle, born together in a high-energy crash at a particle collider (like the Large Hadron Collider). Because they were born from the same event, they are "entangled." In the quantum world, this means they are like a pair of magic dice: if you roll one and it lands on "heads," the other instantly lands on "tails," no matter how far apart they are. They share a single, inseparable quantum state.

Scientists have recently started measuring this "magic connection" (entanglement) in these particles. However, there is a problem: in the real world, these particles don't just sit still. Before they decay (disappear), they often emit tiny bursts of energy, like little sparks of light or gluons.

The Problem: The "Static" on the Radio

The authors of this paper ask a simple question: What happens to that perfect quantum connection when the particles emit these sparks?

Think of the entangled pair as two people trying to have a secret, perfect conversation in a quiet room.

• The Ideal Scenario: The room is silent. They understand each other perfectly. This is what previous experiments assumed: a "closed system" where nothing interferes.
• The Real Scenario: The room suddenly fills with static, wind, and background noise (the radiation). The two people are still talking, but the noise is "leaking" information out of the room. The perfect connection gets fuzzy. In physics, this loss of perfect connection is called decoherence.

For a long time, scientists thought this noise was so quiet it didn't matter. This paper argues that while the noise is small, it is measurable and actually reduces the "quantumness" of the connection.

The Solution: A New Way to Calculate the Noise

The authors developed a new mathematical tool to calculate exactly how much this "noise" ruins the entanglement.

1. The "Magic Filter" (Kraus Operators): In quantum mechanics, we use special math tools called "Kraus operators" to describe how a system gets messy when it interacts with its environment. Think of these as filters that the noise passes through, changing the signal.
2. The "Recipe Book" (Altarelli-Parisi Functions): The authors made a brilliant discovery. They found that these complex quantum filters are mathematically identical to a very famous set of "recipes" used by particle physicists for decades. These recipes, called Altarelli-Parisi splitting functions, describe how a particle splits into smaller pieces (like a parent particle splitting into a child particle and a spark).

The Analogy: Imagine you are trying to figure out how much a cake will shrink when you take a bite.

• Old way: You try to guess the shrinkage by looking at the whole cake and hoping for the best.
• This paper's way: They realized that the "bite" (the radiation) follows a specific, well-known recipe. Instead of guessing, they used the existing recipe book to calculate exactly how much the cake shrinks.

What Did They Find?

They tested this on a specific scenario: a heavy particle decaying into a pair of fermions (like top quarks).

• The Result: The radiation does cause decoherence. The perfect entanglement drops slightly.
• How much? It's a small drop (about 1% for certain types of interactions), but it is there.
• The Cause: The drop happens mostly because of "collinear radiation." Imagine the particles shooting out sparks that travel in almost the exact same direction as the particles themselves. These sparks carry away just enough information to slightly blur the quantum connection.
• The Exception: If the radiation is a specific type of "scalar" (a simple energy burst with no spin), it doesn't mess up the connection at all. It's like the noise being a pure tone that doesn't interfere with the conversation.

The Bottom Line

This paper provides a bridge between two worlds: Quantum Information (the study of entanglement and qubits) and Particle Physics (the study of colliders and radiation).

They showed that the "noise" from particle radiation can be treated as a quantum process that degrades entanglement. By using standard particle physics recipes, they can now predict exactly how much the "magic connection" between particles will weaken. This is a crucial step for future experiments that want to measure quantum entanglement with extreme precision; they can no longer ignore the "static" in the room.

Technical Summary

Technical Summary: Decoherence Effects in Entangled Fermion Pairs at Colliders

Problem Statement
Recent measurements at the Large Hadron Collider (LHC) by the ATLAS and CMS collaborations have successfully observed quantum entanglement in the spins of top-antitop ($t\bar{t}$) pairs. Current experimental analyses typically model the $t\bar{t}$ system as a closed quantum system at the point of decay. However, this approach neglects the effects of radiation (gluons or photons) emitted by the top quarks during the brief interval between production and decay. Such radiation interacts with the system, potentially causing quantum decoherence and a reduction in spin entanglement. While Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO) corrections to spin correlations in Quantum Chromodynamics (QCD) are generally considered small, the specific impact of radiation-induced decoherence on entanglement measures has not been rigorously quantified in the context of high-energy collider physics.

Methodology
The authors formalize the effects of radiation as a decoherence process acting on an open quantum system. They treat a fermion-antifermion pair ($f\bar{f}$), such as $t\bar{t}$ or $\tau^+\tau^-$, initially in a maximally entangled Bell state (specifically the triplet state arising from a scalar boson decay), as a bipartite system of spin qubits. The unobserved radiation is treated as an "environment."

The evolution of the system's density matrix $\rho$ is described using a quantum map $\mathcal{E}$ in the operator-sum representation:
$$\mathcal{E}[\rho] = \sum_j K_j \rho K_j^\dagger$$
where $K_j$ are Kraus operators. The authors calculate these operators by integrating over the phase space of unresolved soft and collinear radiation. They distinguish between:

1. Virtual corrections: Which, for scalar decays, result in an identity map (renormalizing the probability but preserving the state structure).
2. Real emission: Split into soft and hard (collinear) components. Soft emission is shown to preserve the spin structure for scalar and vector couplings but introduces additional Kraus operators for pseudoscalar and axial couplings. Hard (collinear) emission is identified as the primary source of non-trivial decoherence.

The authors explicitly construct the reduced density matrix by tracing over the radiation degrees of freedom. They analyze various interaction types: scalar, pseudoscalar, vector, and axial-vector, characterized by coupling constants $g_S, g_P, g_V, g_A$.

Key Contributions

1. Formalism of Radiation as Decoherence: The paper establishes a framework where radiation effects in particle physics are mapped directly to quantum information processes. It demonstrates how to derive Kraus operators for spin qubits from standard quantum field theory (QFT) calculations involving real and virtual emissions.
2. Kraus Operators and Splitting Functions: A central theoretical contribution is the identification of a direct correspondence between the Kraus operators describing the spin evolution and the integrated Altarelli-Parisi (AP) splitting functions for $q \to qg$ processes. The authors show that the collinear emission limit allows the splitting amplitudes to be interpreted as the operators governing the decoherence of the spin state.
3. Explicit Calculation for Scalar Decays: The authors perform explicit calculations for a scalar boson decaying into a fermion pair. They derive the specific form of the Kraus operators for different Lorentz structures (scalar, pseudoscalar, vector, axial) and calculate the resulting change in concurrence (a measure of entanglement).

Results

• Concurrence Reduction: The study calculates the concurrence $C$ of the $t\bar{t}$ pair as a function of the fermion velocity $\beta$ in the collinear limit. The results show that while scalar emission acts as an identity map (preserving entanglement), vector, axial, and pseudoscalar emissions lead to a reduction in concurrence.
• Magnitude of Effect: The decoherence effects are suppressed by the coupling strength $\alpha_\Gamma$ and the mass of the fermion (specifically suppressed by $1/m_t$ in the collinear limit). For a QCD-type interaction ($\alpha_V \sim 0.1$), the authors estimate a decrease in concurrence of approximately 1% at high $\beta$.
• Coupling Dependence: The nature of the Kraus operators depends on the coupling. For instance, pseudoscalar interactions introduce a phase-flip type operator ($\sigma_3$), while vector interactions introduce bit-flip and bit-phase-flip combinations ($\sigma_\pm$).

Significance and Claims
The paper positions itself as a "proof-of-concept" study. Its primary significance lies in providing a formalism to interpret radiation effects in particle physics through the lens of quantum information theory. The authors claim that:

• This framework is necessary for future precision studies of entanglement at colliders, where higher-order corrections and decoherence effects must be accounted for to accurately compare theory with data.
• The correspondence between Kraus operators and Altarelli-Parisi splitting functions offers a new perspective that could be valuable for "quantum accurate simulations of parton showers."
• The approach is general and can be extended to other entangled systems (e.g., $h \to \tau^+\tau^-$, $e^+e^- \to \tau^+\tau^-$) and higher-order effects, though these specific applications are left for future work.

The authors do not claim to have resolved all phenomenological uncertainties or to have provided a complete experimental proposal; rather, they provide the theoretical machinery to quantify decoherence effects that were previously neglected in entanglement measurements.