Simulating Bell inequalities with Qibo

This paper presents educational material and Qibo-based software tools organized into three modules of increasing difficulty to help students simulate Bell inequality violations, thereby exploring fundamental quantum concepts like entanglement and non-locality while gaining practical experience with statistical analysis and hardware noise.

The Gist

Imagine you are trying to teach a class of students about the weirdest, most mind-bending rules of the universe: Quantum Mechanics. Specifically, you want to show them how two tiny particles can be so deeply connected that what happens to one instantly affects the other, even if they are on opposite sides of the galaxy. This is called "entanglement."

For decades, scientists have debated whether this connection is real or if the particles are just carrying secret "instruction manuals" (hidden variables) that tell them what to do before they even separate. In 1964, a physicist named John Bell came up with a mathematical test—a set of rules called Bell Inequalities—to settle the argument. If the particles follow the "instruction manual" theory, they must obey these rules. If they follow the weird rules of quantum mechanics, they will break them.

This paper is essentially a teacher's toolkit designed to help students run these tests themselves using a computer program called Qibo. Instead of just reading about the math, students can build a virtual lab, run simulations, and see the "magic" happen on their screens.

Here is how the paper breaks it down, using simple analogies:

1. The Three-Module Lesson Plan

The authors organized their teaching material into three steps, getting harder as you go, like levels in a video game.

• Level 1: The Bell-Wigner Inequality (The "Sock" Analogy)
  Imagine you have a pair of socks. If you put a left sock in one box and a right sock in another, and you send them to different cities, you know exactly what's in the other box once you open one. This is the "local hidden variable" idea: the socks were always left or right; you just didn't know yet.

  The paper starts here because the math is simple. It asks: "If the particles are like these pre-determined socks, what are the odds of them matching in specific ways?" The simulation shows that if the particles were just "socks," they would follow a strict rule. But when the students run the simulation with quantum particles, the particles break the rule. They act as if they are talking to each other instantly, rather than just following a pre-written list.

• Level 2: The Original Bell Inequality (The "Perfect Mirror")
  This level gets a bit more complex. It looks at how the particles correlate. Imagine two dancers who are perfect mirrors of each other. If one spins left, the other spins right.

  The paper explains that in a "normal" world, there's a limit to how perfectly they can mirror each other across different angles. But in the quantum world, the dancers are so perfectly synchronized that they exceed this limit. The simulation lets students tweak the angles of the "dance floor" and watch the numbers jump over the limit, proving that the "mirror" isn't just a reflection of a pre-set plan, but something more dynamic.

• Level 3: The CHSH Inequality (The "Four-Direction" Challenge)
  This is the most famous and robust version of the test. Imagine the dancers are now being watched by four different judges standing in different directions. The judges ask the dancers to perform specific moves based on where they are standing.

  The paper shows that if the dancers are following a "script" (hidden variables), their combined scores can never exceed a certain number (2). But when the students run the quantum simulation, the score jumps to 2.82 (which is $2\sqrt{2}$). This is the "smoking gun" that proves the universe is not local; the particles are truly connected in a way that defies our everyday logic.

2. The Virtual Lab (Qibo)

The paper emphasizes that you don't need a real physics lab with lasers and vacuum chambers to see this. The authors used Qibo, which is like a "flight simulator" for quantum computers.

• The Code: They provide Python code (a type of computer language) that students can copy and paste.
• The Process: The code creates two "qubits" (quantum bits, which are like super-powered coins that can be heads, tails, or both at once). It entangles them, spins them in different directions (simulating measurements), and counts the results.
• The Result: The students see graphs where the quantum results clearly violate the "classical" limits.

3. The Real-World Messiness (Noise and Statistics)

The paper also teaches a very practical lesson: Real life is messy.
In a perfect computer simulation, the results are smooth. But if you run this on a real quantum computer (like the ones at CERN or in labs), the results get "noisy."

• The Analogy: Imagine trying to hear a whisper in a quiet room (perfect simulation) versus trying to hear it at a rock concert (real hardware). The "noise" from the hardware can hide the signal.
• The Lesson: The authors show students how to calculate how many times they need to run the experiment (shots) to get a clear answer. If they run it too few times, the random "static" makes it look like the rule wasn't broken. If they run it enough times, the true quantum nature shines through.

4. Why This Matters for Education

The authors argue that this tool is a game-changer for teaching.

• For Physics Students: It turns abstract, scary math into something they can touch and see. They can "play" with the angles and see the violation happen in real-time.
• For Computer Science Students: It gives them a chance to learn deep physics concepts without needing a PhD in theoretical physics first. They can focus on the code and the logic.

Summary

In short, this paper presents a digital playground where students can prove that the universe is stranger than we think. By using the Qibo software, they can simulate the famous Bell tests, watch the "classical rules" break, and understand that quantum entanglement is a real, measurable phenomenon—not just a theory. It bridges the gap between "reading about magic" and "performing the magic trick" on a computer screen.

Technical Summary

Technical Summary: Simulating Bell Inequalities with Qibo

Problem Statement
The paper addresses the pedagogical challenge of introducing complex quantum mechanical concepts—specifically entanglement, spin correlations, hidden variables, and non-locality—to undergraduate and graduate students. While theoretical arguments regarding Bell inequalities are mathematically and conceptually demanding, there is a need for practical educational tools that allow students to visualize these phenomena and understand the impact of statistical fluctuations and hardware noise. Although various quantum computing frameworks exist (e.g., Qiskit, Cirq), the authors note a specific gap in educational tools for simulating the Bell-Wigner inequality and the original Bell inequality, with existing resources often focusing solely on the more complex CHSH-type inequalities.

Methodology
The authors developed a modular educational package consisting of three Python-based Jupyter notebooks implemented using Qibo, an open-source quantum computing framework capable of interfacing with both classical simulators and real quantum hardware. The material is structured into three modules of increasing difficulty:

1. Module 1: The Bell-Wigner Inequality: This module introduces the theoretical framework based on local hidden (LH) variable theories and Wigner's argument regarding eight distinct particle populations. It constructs quantum circuits to measure probabilities $P(\hat{a}+, \hat{b}+)$, $P(\hat{a}+, \hat{c}+)$, and $P(\hat{c}+, \hat{b}+)$ for three coplanar directions ($\hat{a}, \hat{b}, \hat{c}$). The circuits utilize Hadamard and CNOT gates to generate the singlet Bell state $|\Psi^-\rangle$, followed by rotation gates ($R_z, R_y$) to align measurement axes.
2. Module 2: The Original Bell Inequality: This section shifts focus from probabilities to spin correlations $C(\hat{a}, \hat{b})$. It implements the same quantum circuits as Module 1 but calculates the correlation function $C = \langle \alpha \beta \rangle$ using the frequencies of all four possible measurement outcomes ($00, 01, 10, 11$). The module tests the inequality $|C(\hat{a}, \hat{b}) - C(\hat{a}, \hat{c})| \leq 1 + C(\hat{c}, \hat{b})$.
3. Module 3: The CHSH Inequality: The final module addresses the Clauser-Horne-Shimony-Holt (CHSH) inequality, which avoids the strict assumption of perfect anti-correlation required by the original Bell inequality. It simulates measurements along four directions ($\hat{a}, \hat{b}$ for Alice; $\hat{c}, \hat{d}$ for Bob) to compute the quantity $S = |C(\hat{a}, \hat{c}) - C(\hat{a}, \hat{d}) + C(\hat{b}, \hat{c}) + C(\hat{b}, \hat{d})|$.

The simulations employ a frequentist approach, running circuits for a specified number of shots ($N_{shots}$) to reconstruct probability distributions and calculate the relevant inequality quantities ($Q_W, Q_B, Q_S$). The code allows for the variation of angular parameters ($\theta_{ab}, \theta_{ac}, \phi$) to explore different geometric configurations.

Key Results
The simulations successfully reproduce the theoretical predictions for quantum mechanics, demonstrating violations of the inequalities under specific geometric configurations:

• Bell-Wigner: The quantity $Q_W$ becomes positive (violating the LH bound of $\leq 0$) when the measurement directions are coplanar with specific angular separations (e.g., $\theta_{ab} = \pi/2, \theta_{ac} = \pi/4, \phi=0$).
• Original Bell: The quantity $Q_B$ exceeds the classical bound of 1, with maximal violation occurring at $\theta_{ac} = \pi/4$ for $\theta_{ab} = \pi/2$.
• CHSH: The quantity $Q_S$ exceeds the classical bound of 2, reaching the quantum mechanical maximum of $2\sqrt{2}$ in the optimal coplanar configuration.

The paper also analyzes the impact of statistics and noise. It derives the relationship between the number of shots and statistical error ($\sigma_{stat} \propto 1/\sqrt{N_{shots}}$) and contrasts this with intrinsic hardware noise ($\sigma_{noise}$), which is independent of shot count. The authors provide a method to estimate the optimal number of shots required to make statistical error comparable to device noise.

Significance and Claims
The authors position this work primarily as an educational tool rather than a new physical discovery. The significance lies in:

1. Bridging Theory and Practice: The modules allow students to move from abstract theoretical derivations to concrete implementation, visualizing the transition between classical (local hidden variable) and quantum regimes.
2. Addressing Pedagogical Gaps: By including the Bell-Wigner and original Bell inequalities, the material offers a more accessible entry point for students than the mathematically heavier CHSH inequality alone.
3. Practical Awareness: The inclusion of statistical analysis and noise modeling prepares students for the realities of current quantum hardware (NISQ era), where imperfect state preparation and readout errors affect results.
4. Accessibility: The use of Qibo and standard Python notebooks makes the material flexible for both physics students (focusing on theoretical concepts) and computer science students (focusing on algorithmic implementation).

The authors note that the material has been tested in a Master's level Quantum Computing course at the University of Ferrara, receiving positive feedback from students with diverse backgrounds. They emphasize that the tool is designed to facilitate classroom discussion on fundamental issues like non-locality and the limitations of local realism, without claiming to solve open problems in quantum foundations or propose new experimental setups beyond the educational scope.