⚛️ high-energy experiments

Nonlocal advantage of quantum coherence in tau-lepton pairs from electron--positron collisions

This paper proposes a dedicated experimental scheme to observe the Nonlocal Advantage of Quantum Coherence (NAQC) in tau-lepton pairs produced at Belle~II and FCC-$ee$ colliders, demonstrating that high-precision measurements can certify this strongest form of quantum correlation for the first time in a high-energy collider setting.

Original authors: Yoav Afik, Juan Ramón Muñoz de Nova, Sarthak Sharma, Surya Sundar Raman

Published 2026-09-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yoav Afik, Juan Ramón Muñoz de Nova, Sarthak Sharma, Surya Sundar Raman

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 subatomic world, particles do not merely exist; they exist in a state of potential, holding multiple possibilities at once until they are observed. This property, known as quantum coherence, is the engine behind the strange behaviors that distinguish the quantum realm from our everyday experience. When two particles become linked, they can share a connection so deep that measuring one instantly influences the other, regardless of the distance between them. Scientists have long studied different strengths of this connection, ranging from simple entanglement to a more rigorous form called steering, where one observer can effectively force the other into a specific state. At the very top of this hierarchy sits a phenomenon known as the nonlocal advantage of quantum coherence. It represents the strongest possible form of quantum correlation, a state where the connection between particles is so intense that it allows one to generate a level of order and predictability that would be impossible if the particles were acting alone.

For decades, physicists have searched for ways to observe these delicate quantum effects in high-energy environments, such as the massive particle colliders that smash atoms together at near-light speeds. While evidence of simpler quantum links has been found in heavy particles like top quarks, the strongest form of this connection has remained elusive in these chaotic, high-speed collisions. The challenge has been that these powerful quantum effects usually vanish almost instantly or exist only in tiny, hard-to-reach pockets of the collision data. Now, a team of researchers has identified a unique setting where this elusive connection not only survives but thrives, offering a clear path to observing the most robust quantum correlation ever measured in a collider.

The researchers focused their attention on the tau lepton, the heaviest member of the electron family. Unlike other heavy particles that immediately break apart into a spray of other particles, tau leptons have a unique property: they preserve their internal spin information until they decay. This spin acts like a tiny compass needle, and because the tau lepton does not interact with the strong nuclear force, this needle remains intact long enough to be measured. When an electron and a positron collide, they can annihilate each other to create a pair of these tau leptons. The team calculated that the quantum link between these two newly born particles is exceptionally strong, particularly when the collision happens at specific energy levels.

By analyzing the physics of these collisions, the team discovered that the quantum connection between the tau pair is not a fleeting anomaly but a persistent feature across a large portion of the possible outcomes. They found that in roughly 18 percent of the events at one major experiment and 31 percent at a future facility, the particles exhibit this maximum-strength quantum correlation. This is a significant finding because it means the signal is not hidden in a needle-in-a-haystack scenario; instead, it is present in a substantial fraction of the data, making it accessible to current and upcoming detectors. The researchers developed a specific test, or "witness," that allows scientists to confirm the presence of this strong connection by measuring just a single value, rather than needing to reconstruct the entire complex state of the particles.

The study points to two specific locations where this discovery can be made. The first is the Belle II experiment in Japan, which operates at a collision energy of 10.58 GeV. The second is the proposed Future Circular Collider (FCC-ee) in Europe, which would operate at a much higher energy of 91.19 GeV. The team estimated that with the precision expected at these facilities, a definitive observation of this effect is within reach. At Belle II, a measurement precision of about one percent would be sufficient to confirm the effect with high statistical certainty, while the FCC-ee would require a precision of a few percent. The sheer number of tau pairs expected to be produced at these machines means that the data required to make this observation is already being collected or will be soon.

Beyond simply detecting the phenomenon, the researchers proposed a way to actively demonstrate the nature of this connection using a "steering game." In this scenario, one particle is measured first, and that measurement choice determines the state of the other particle in a way that cannot be explained by classical physics. This would mark the first time such a game has been played in a high-energy collider, moving beyond passive observation to an active demonstration of quantum control. The ability to perform this test in a particle collider would represent a major milestone, proving that the strongest form of quantum correlation exists not just in controlled laboratory settings with photons, but in the violent, high-energy collisions that recreate the conditions of the early universe.

The implications of this work extend beyond the immediate detection of a new quantum effect. By establishing that the nonlocal advantage of quantum coherence is present and measurable in tau lepton pairs, the study provides a new tool for probing the fundamental laws of physics. It offers a way to test the limits of quantum mechanics in a regime where it has rarely been examined. Furthermore, because the connection between the particles is asymmetric, meaning it behaves differently depending on which particle is measured first, it could serve as a sensitive probe for new physics that violates the symmetry between matter and antimatter. The researchers have laid out a clear roadmap for experimentalists to follow, transforming a theoretical concept into a concrete target for the next generation of particle physics experiments.

Technical Summary: Nonlocal Advantage of Quantum Coherence in Tau-Lepton Pairs from Electron–Positron Collisions

Problem Statement
Quantum correlations in high-energy collider physics have recently expanded beyond entanglement and Bell nonlocality to include concepts like quantum steering and the Nonlocal Advantage of Quantum Coherence (NAQC). NAQC represents a particularly strong form of quantum correlation, defined as the ability of one party to remotely prepare conditional states for another party such that the average quantum coherence exceeds the maximum achievable by any single local quantum state. While NAQC has been observed in photonic setups and studied in neutrino oscillations, its detection in high-energy colliders has been challenging. In systems like top-antitop quark pairs, NAQC is confined to highly restricted regions of phase space, rendering experimental observation unrealistic with current data. The paper identifies a need to find a high-energy system where NAQC persists over a large portion of phase space, allowing for feasible experimental certification.

Methodology
The authors investigate the production of tau-lepton pairs ( $\tau^+\tau^-$ ) in electron–positron ( $e^+e^-$ ) collisions. The methodology involves:

Theoretical Framework: Modeling the $\tau^+\tau^-$ pair as a two-qubit system using the electroweak theory. The spin quantum state is described by a density matrix $\rho$ characterized by spin polarization vectors ( $B^\pm$ ) and a spin-correlation matrix ( $C$ ).
Basis Selection: The analysis utilizes the "diagonal basis," which diagonalizes the spin-correlation matrix. This basis is chosen because it maximizes the NAQC witness. The authors also consider the helicity basis and the beam basis, noting that the diagonal basis optimizes sensitivity to quantum observables.
NAQC Measures: The study employs two coherence measures: the $l_1$ -norm of coherence ( $C_{l_1}$ ) and the relative entropy of coherence ( $C_{re}$ ). NAQC is established if the nonlocal coherence measure $C_{NA}$ exceeds the maximum average coherence achievable by a single qubit ( $\bar{C}_{max}$ ).
Witness Construction: The authors derive a specific NAQC witness, $W_{NA}(\rho) = \sum |C_{ii}|$ , which provides a sufficient condition for NAQC. If $W_{NA} \geq \sqrt{6}$ , the state exhibits NAQC.
Experimental Scenarios: The analysis focuses on two benchmark center-of-mass (COM) energies:
- $\sqrt{s} = 10.58$ GeV (Belle II), where the photon channel dominates.
- $\sqrt{s} = 91.19$ GeV (FCC-ee, Z-peak), where the Z-boson channel dominates.
Angular Averaging: Recognizing that experiments measure over finite angular regions, the authors calculate angular-averaged density matrices ( $\rho_\Sigma$ ) over a selection window $|\cos\Theta| < x_c$ . They demonstrate that if the averaged state exhibits NAQC, the underlying individual states must also possess it.
Simulation and Validation: Analytical results are validated using Monte Carlo simulations (MadGraph v3.5.7 and TauDecay library) for the process $e^+e^- \to \tau^+\tau^- \to \pi^+\bar{\nu}_\tau \pi^-\nu_\tau$ .

Key Contributions

Identification of an Ideal System: The paper establishes that $\tau^+\tau^-$ production in $e^+e^-$ collisions is an ideal testing ground for NAQC because, unlike top-quark pairs, it exhibits strong quantum correlations and NAQC over a broad region of phase space.
Development of a Single-Magnitude Witness: The authors construct a dedicated NAQC witness ( $W_{NA}$ ) that allows for the certification of NAQC by measuring a single magnitude derived from the spin-correlation matrix, avoiding the need for full quantum tomography in initial assessments.
Experimental Steering Game Design: The paper proposes the first experimental scheme to implement a "steering game" in a high-energy collider. This scheme involves reconstructing conditional quantum states based on measurement outcomes to directly test the definition of NAQC.
Quantitative Feasibility Study: The study provides concrete estimates for the fraction of events contributing to the NAQC signal and the precision required for observation at specific facilities.

Results

Phase Space Structure: NAQC is present in three distinct bands in the $(\cos\Theta, \sqrt{s})$ plane: a low-energy strip, a narrow band around the Z resonance, and a high-energy strip. The $l_1$ -norm measure captures NAQC in broader regions than the relative entropy measure.
Event Fractions: A significant fraction of total events exhibits NAQC:
- Belle II: Approximately 18% ( $f_{NAQC} \approx 0.18$ ) of events in the optimal angular window.
- FCC-ee: Approximately 31% ( $f_{NAQC} \approx 0.31$ ) of events.
Precision Requirements:
- Belle II: A 5 $\sigma$ observation of NAQC requires a measurement precision of approximately 1%.
- FCC-ee: A precision of a few percent is sufficient for a 5 $\sigma$ observation.
Witness Performance: The proposed observable $D_i$ (related to the trace of the correlation matrix) serves as an effective experimental witness. For Belle II, the $D_n$ observable is proposed; for FCC-ee, the $D_r$ observable is proposed.
Statistical Potential: Based on projected integrated luminosities (50 ab $^{-1}$ for Belle II and 125 ab $^{-1}$ for FCC-ee), the expected number of $\tau^+\tau^- \to \pi^+\bar{\nu}_\tau \pi^-\nu_\tau$ events exhibiting NAQC is estimated at $8 \times 10^7$ for Belle II and $8 \times 10^8$ for FCC-ee.

Significance
The paper argues that the observation of NAQC in this context would constitute the strongest form of quantum correlation measured in a collider to date. The significance rests on several points:

Hierarchy of Correlations: The results provide a theoretical layer of confirmation for the conjectured hierarchy of quantum correlations in high-energy systems: NAQC $\subset$ Bell Nonlocality $\subset$ Steering $\subset$ Entanglement $\subset$ Discord. While a general proof that NAQC implies Bell nonlocality is still missing, the paper cites strong numerical evidence supporting this direction. This work extends previous experimental characterizations in top-quark systems by proposing a test in a new high-energy qubit system.
Operational Advantage: It demonstrates that NAQC, which implies an operational advantage in coherence generation, is accessible in high-energy physics, bridging the gap between quantum information theory and collider phenomenology.
First Steering Game in Colliders: The proposed experimental scheme represents the first implementation of a steering game in a high-energy environment, paving the way for future investigations into asymmetric quantum correlations and potential CP-violating physics beyond the Standard Model.
Feasibility: The study demonstrates that NAQC is not merely a theoretical curiosity confined to narrow kinematic regions but is robust enough to be observed with current and near-future experimental capabilities, specifically at Belle II and FCC-ee.

More like this