foundations intermediate · 16 min read · By LIPAI WANG · April 29, 2026

Bell's Theorem and the CHSH Inequality: How We Know the Universe Isn't Locally Realistic

Bell's theorem (1964) is the most-cited result in the foundations of quantum mechanics. It proves that no theory based on local hidden variables can reproduce all the predictions of quantum mechanics — and the CHSH form gives a quantitative inequality that experiments can test. Every loophole-free Bell test since 2015 has confirmed quantum mechanics. This tutorial derives the CHSH inequality, explains the Tsirelson bound, and covers what 'loophole-free' actually means.

Prerequisites: Tutorial 3: Multi-Qubit Entanglement

In 1935, Einstein, Podolsky, and Rosen argued that quantum mechanics must be incomplete: it predicts strong correlations between distant entangled particles, and any reasonable physical theory should explain those correlations through pre-existing properties of the particles, not through some “spooky action at a distance.” Their argument was philosophical — quantum mechanics either fails to be a complete description of reality, or there are hidden variables we haven’t yet identified.

For 30 years EPR’s argument lived as a philosophical curiosity. Then in 1964, John Bell proved a stunning theorem: no theory in which particles carry pre-determined values for all possible measurements can reproduce the correlations that quantum mechanics predicts. This is not a philosophy statement — it is a quantitative inequality that any local-hidden-variables theory must satisfy, and that quantum mechanics provably violates.

The CHSH inequality (Clauser-Horne-Shimony-Holt 1969) put Bell’s theorem into experimentally testable form. Every loophole-free Bell test conducted since 2015 has confirmed that the CHSH inequality is violated by exactly the amount quantum mechanics predicts. The universe is provably not locally realistic. This is one of the deepest empirical results in physics.

This tutorial derives the CHSH inequality from first principles, explains the Tsirelson bound that gives quantum mechanics its specific advantage, and covers what “loophole-free” actually means in modern Bell tests.

The setup: two parties, two settings, two outcomes

The standard Bell-test scenario: two distant parties, Alice and Bob, each receiving one particle from an entangled pair. Each party can choose between two measurement settings. Each measurement returns one of two outcomes, $\pm 1$.

• Alice picks setting $A$ or $A'$, gets outcome $a \in \{+1, -1\}$.
• Bob picks setting $B$ or $B'$, gets outcome $b \in \{+1, -1\}$.

Define the correlation for a pair of settings:

$$E(A, B) \;=\; \langle a \cdot b \rangle_{A, B}$$

where the average is over many runs with the same setting choices. Correlation ranges from $-1$ (perfectly anti-correlated) to $+1$ (perfectly correlated).

The CHSH quantity is the specific combination

$$S \;=\; E(A, B) + E(A, B') + E(A', B) - E(A', B').$$

This particular combination, with one minus sign, is what Bell’s argument exploits.

The local-hidden-variables prediction

Suppose a hidden-variable theory says: each entangled pair carries hidden information $\lambda$ that pre-determines the outcomes for every possible measurement setting. When Alice measures $A$, her outcome is $a(A, \lambda) \in \{\pm 1\}$, depending on her setting and the hidden variable. Similarly for Bob.

The “local” assumption: Alice’s outcome $a(A, \lambda)$ depends only on her setting $A$ and the shared hidden variable $\lambda$, not on Bob’s setting choice. Likewise Bob’s outcome depends only on $B$ and $\lambda$, not on Alice’s choice. This rules out faster-than-light communication of setting choices.

The “realism” assumption: the values $a(A, \lambda), a(A', \lambda), b(B, \lambda), b(B', \lambda)$ all exist as definite values, even before measurement. The measurement just reveals them.

Under these two assumptions, the CHSH quantity is bounded:

$$|S_\text{LHV}| \;\leq\; 2.$$

Here is the elementary proof. For any single hidden-variable value $\lambda$, the four possible outcomes are $\pm 1$. The combination

$$a(A, \lambda) \cdot b(B, \lambda) + a(A, \lambda) \cdot b(B', \lambda) + a(A', \lambda) \cdot b(B, \lambda) - a(A', \lambda) \cdot b(B', \lambda)$$

can be rewritten as

$$a(A, \lambda) \bigl[b(B, \lambda) + b(B', \lambda)\bigr] + a(A', \lambda) \bigl[b(B, \lambda) - b(B', \lambda)\bigr].$$

For any choice of $\pm 1$ outcomes, exactly one of the two square brackets is $\pm 2$ and the other is $0$. So the whole expression has absolute value exactly $2$ for every $\lambda$. Averaging over $\lambda$ gives $|S_\text{LHV}| \leq 2$.

This is the CHSH inequality: any local-hidden-variables theory satisfies $|S| \leq 2$.

The quantum-mechanical prediction

Quantum mechanics predicts a different answer. Consider the maximally entangled Bell state

$$|\Phi^+\rangle \;=\; \tfrac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr).$$

Choose Alice’s settings as measurements along axes at angles $0$ and $\pi/2$ (i.e., $\sigma_z$ and $\sigma_x$). Choose Bob’s settings as axes at angles $\pi/4$ and $-\pi/4$ — rotated by $45°$ relative to Alice’s. The quantum-mechanical correlations are

$$E_\text{QM}(A, B) \;=\; \cos(\theta_A - \theta_B),$$

where $\theta_A, \theta_B$ are the measurement angles.

Plugging in the four specific angle pairs:

$$S_\text{QM} \;=\; \cos(\pi/4) + \cos(\pi/4) + \cos(\pi/4) - \cos(3\pi/4) \;=\; 4 \cdot \tfrac{1}{\sqrt{2}} \;=\; 2\sqrt{2} \;\approx\; 2.828.$$

Quantum mechanics predicts $|S| = 2\sqrt{2}$ for the optimal measurement settings on a Bell state. This violates the CHSH inequality. Therefore quantum mechanics cannot be a local-hidden-variables theory.

The maximum quantum-mechanical CHSH value is $2\sqrt{2}$, the Tsirelson bound (Cirel’son 1980). It is the largest value any quantum theory can produce, and quantum mechanics achieves it for maximally entangled states with optimal measurements.

The CHSH game perspective

There is a clean game-theoretic interpretation. Imagine Alice and Bob play a cooperative game where they each receive a random bit ($A \in \{0, 1\}$ for Alice, $B \in \{0, 1\}$ for Bob), and they each output a bit ($a, b \in \{0, 1\}$). They win if $a \oplus b = A \cdot B$ (XOR equals AND of inputs). They cannot communicate during the game; they can pre-share any strategy or shared randomness.

• Best classical (LHV) strategy: wins with probability $0.75$. Equivalent to $|S| = 2$.
• Best quantum strategy: wins with probability $\cos^2(\pi/8) \approx 0.854$. Equivalent to $|S| = 2\sqrt{2}$.

The quantum advantage is $\sim 10\%$ — small but provably impossible classically. The CHSH game is the operational version of Bell’s theorem.

This game framing matters in modern applications: device-independent quantum cryptography (BB84-style protocols whose security rests on observed CHSH violation), randomness amplification, and certification of quantum-mechanical behavior in untrusted devices.

A small numerical experiment

Concrete code computing the CHSH value on a quantum simulator:

import numpy as np
import pennylane as qml

dev = qml.device("default.qubit", wires=2, shots=10_000)

def bell_state(circuit_fn, settings):
    """Prepare |Phi+> and measure with specified angle settings."""
    @qml.qnode(dev)
    def circuit():
        qml.Hadamard(wires=0)
        qml.CNOT(wires=[0, 1])
        # Measure Alice in basis rotated by alice_angle.
        qml.RY(-2 * settings["alice"], wires=0)
        # Measure Bob in basis rotated by bob_angle.
        qml.RY(-2 * settings["bob"], wires=1)
        return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1))
    return circuit()


# Optimal CHSH angles.
alice_settings = [0.0, np.pi / 4]
bob_settings = [np.pi / 8, -np.pi / 8]

correlations = {}
for i, a_angle in enumerate(alice_settings):
    for j, b_angle in enumerate(bob_settings):
        E = bell_state(None, {"alice": a_angle, "bob": b_angle})
        correlations[(i, j)] = E
        print(f"E(A{i}, B{j}) = {E:+.4f}")

# CHSH combination: E(A0, B0) + E(A0, B1) + E(A1, B0) - E(A1, B1).
S = correlations[(0, 0)] + correlations[(0, 1)] + correlations[(1, 0)] - correlations[(1, 1)]
print(f"\nCHSH value: S = {S:.4f}")
print(f"Classical bound: 2.0000")
print(f"Tsirelson bound: {2 * np.sqrt(2):.4f}")

Sample output:

E(A0, B0) = +0.7064
E(A0, B1) = +0.7102
E(A1, B0) = +0.7048
E(A1, B1) = -0.7088

CHSH value: S = 2.8302
Classical bound: 2.0000
Tsirelson bound: 2.8284

The simulated CHSH value reaches $\sim 2.83$, matching the Tsirelson bound to within shot-noise statistics. This is the quantitative confirmation that quantum mechanics violates Bell’s inequality. Real laboratory experiments (next section) achieve similar violations with statistical confidence levels exceeding $10^{30}$ standard deviations.

Loopholes and the 2015 loophole-free experiments

The CHSH inequality is mathematically clean, but real experiments have to close several loopholes — alternative explanations that could rescue local hidden variables:

Locality loophole

If Alice’s setting choice could somehow influence Bob’s outcome (e.g., via signals slower than light), the local-realism assumption is violated experimentally rather than by quantum mechanics. To close: ensure Alice’s and Bob’s measurements are space-like separated — light cannot travel between them in the time it takes to perform the measurement.

This was first closed in 1982 by Aspect-Dalibard-Roger, using fast switching of measurement settings on entangled photons.

Detection loophole

If the detector efficiency is low, the local-hidden-variables theory could exploit selective non-detection. Imagine the hidden variables knew which pairs would be detected and arranged outcomes to violate Bell only on the detected pairs. To close: detector efficiency must exceed a threshold (~$2/3$ for the CHSH setup).

Fully closed in 2013 by Giustina et al. and Christensen et al. using high-efficiency superconducting nanowire detectors and ion-trap-based experiments.

Memory loophole

If individual measurement results are correlated across runs, simple statistical analyses can fail. To close: use protocols where the Bell test is robust to inter-run correlations, or randomize setting choices independently per run.

Free-choice (measurement independence) loophole

If Alice’s and Bob’s setting choices are not truly independent of the hidden variables, the inequality can be violated trivially. To close: use unpredictable measurement settings — typically driven by quantum random number generators or astronomical sources of randomness.

The 2015 loophole-free experiments

In 2015, three groups simultaneously closed all loopholes:

• Hensen et al. (Delft, October 2015): NV centers in diamond, 1.3 km separation. CHSH violation $S = 2.42 \pm 0.20$, ~$2$-sigma significance. Closed locality, detection, free-choice loopholes.
• Giustina et al. (Vienna, December 2015): entangled photons. Higher statistical power but slightly less complete loophole closure.
• Shalm et al. (NIST, December 2015): entangled photons with high-efficiency detectors.

Subsequent experiments have improved statistical confidence. By 2018-2019, loophole-free Bell violations had been demonstrated at thousands of standard deviations.

The empirical picture as of 2026: local realism is ruled out at confidence levels far beyond any physics result of comparable foundational importance. Bell’s theorem combined with experiment is one of the most robustly established results in modern physics.

What this means for quantum computing

Bell’s theorem and CHSH are not just foundations curiosities. They have practical implications:

1. Device-independent cryptography. Protocols that demonstrate CHSH violation can be used to bound an eavesdropper’s information. Tutorial 21’s threat-model discussion of post-quantum cryptography contrasts with this older line of work that uses Bell’s theorem itself for security.
2. Randomness certification. A demonstrated CHSH violation certifies that the underlying process is genuinely random (cannot be predicted by any local-hidden-variables theory). Used in commercial randomness-as-a-service products.
3. Entanglement verification. Bell tests are the gold standard for confirming entanglement in laboratory systems. Any quantum computing platform claiming to produce entangled states should be able to demonstrate CHSH violation.
4. Foundations of quantum mechanics. Many interpretations of QM (many-worlds, QBism, relational, Bohmian) all have to grapple with Bell’s theorem. The result narrows the space of viable interpretations.

For working quantum-computing developers, the practical takeaway: your hardware should be able to demonstrate Bell-state CHSH violations close to $2\sqrt{2}$ as a sanity check. Devices that can’t are either not entangled or have severe noise — both reasons to suspect the device is not actually quantum.

Common misconceptions

“Bell’s theorem proves quantum mechanics is right.” It proves local hidden variables cannot reproduce QM predictions. Quantum mechanics is right because experiments confirm its predictions; Bell’s theorem is the rigorous tool that lets experiments distinguish QM from any local-hidden-variables alternative.

“Bell’s theorem proves faster-than-light communication.” No. The correlations Bell’s theorem predicts cannot be used to send signals — both Alice and Bob each see locally random outcomes. The “non-locality” is in the correlations, not in any signal.

“Bell’s theorem rules out all hidden variables.” Only local hidden variables. Non-local hidden-variables theories (like Bohmian mechanics) are consistent with quantum mechanics’ predictions, including CHSH violation. They just abandon locality.

“You need a perfect quantum state to violate CHSH.” Far from it. CHSH violation persists down to fairly noisy entangled states. The state needs to have entanglement-of-formation above a threshold that depends on the measurement setup; for typical Bell tests, fidelities of $\sim 80\%$ with the ideal Bell state are enough to see violation.

“The CHSH bound of $2\sqrt{2}$ is just an arbitrary mathematical fact.” The Tsirelson bound has a deep operational meaning — it is the maximum of the relevant operator norm in any quantum theory. Other principles (information causality, no-trivial-communication-complexity) give the Tsirelson bound from independent axioms. There is genuinely something special about $2\sqrt{2}$.

Decision rule

Bell-test reasoning is most useful when:

1. You’re verifying a hardware claim. Does the system genuinely produce entanglement? Demand a CHSH test.
2. You’re designing a security-critical quantum protocol. Device-independent protocols rest on observed CHSH violation. Without it, security guarantees weaken.
3. You’re explaining to a skeptic that quantum mechanics is real. Bell tests are the cleanest argument: no local theory matches the experimental data, and the experiments are now loophole-free.
4. You’re building intuition for entanglement. CHSH gives a concrete operational signature of entanglement — much more useful than abstract definitions for first-time learners.

In production quantum-computing work, CHSH measurements are standard sanity checks for any new entanglement-generating subsystem.

Exercises

1. The CHSH game

Two players (Alice and Bob) play the CHSH game. They are each given a random bit ($A$ for Alice, $B$ for Bob), and must output bits $a, b$. They win if $a \oplus b = A \cdot B$. They can coordinate strategy in advance but not communicate during the game. What is the best classical winning probability, and how does the quantum strategy beat it?

Show answer

Best classical strategy: always output $a = b = 0$. Then $a \oplus b = 0$, which equals $A \cdot B$ when $A = 0$ or $B = 0$ — three out of four input combinations. Win probability $= 3/4 = 0.75$. The quantum strategy uses a shared Bell pair: Alice measures in one of two bases depending on her input; Bob measures in one of two bases (rotated 45°) depending on his. Output the measurement results. The CHSH violation translates to win probability $\cos^2(\pi/8) \approx 0.854$. Quantum beats classical by ~10 percentage points. This is the operational form of CHSH violation — the quantum advantage in a simple cooperative game.

2. Why the Tsirelson bound is $2\sqrt{2}$

The CHSH operator on a maximally entangled state has expectation $2\sqrt{2}$ at the optimal angles. Why specifically $2\sqrt{2}$ and not something larger?

Show answer

The CHSH operator $C = A \otimes B + A \otimes B' + A' \otimes B - A' \otimes B'$ on two qubits with all $A, A', B, B'$ unit Hermitian operators has spectral norm $\|C\| \leq 2\sqrt{2}$. The bound follows from the operator inequality $C^2 \leq 4 + (A A' - A' A) (B B' - B' B) \leq 8$ (using $\|A A' - A' A\| \leq 2$ for $\pm 1$-eigenvalue operators, and similarly for $B$). So $|C| \leq 2\sqrt{2}$. No quantum theory with two-qubit measurement operators can exceed this bound. The bound is achieved exactly by Bell states with optimal measurement angles — the quantum-mechanical limit is sharp. Theories beyond quantum mechanics (e.g., the “PR-box” hypothetical) could in principle violate $2\sqrt{2}$ but would violate other principles like information causality. $2\sqrt{2}$ marks the boundary of what quantum mechanics permits.

3. Why the detection loophole matters

A Bell test has detector efficiency $50\%$. A local-hidden-variables theory that knew which trials would be detected could in principle arrange outcomes to violate CHSH on the detected subset. What detector efficiency do you need to close the loophole?

Show answer

For a fair-sampling closure with maximally entangled states and optimal CHSH measurements, detector efficiency must exceed approximately $2/3$ (specifically, $2(\sqrt{2}-1) \approx 82.8\%$ for the strictest CHSH-style bounds with no-fair-sampling assumption). Detector efficiencies under this threshold leave room for a hidden-variables theory to engineer the detected subset to look like CHSH violation while the full distribution satisfies the inequality. Modern photonic and ion-trap experiments use detectors with efficiency $> 90\%$, comfortably above the loophole threshold. The detection loophole was the last to be fully closed (2013-2015) because high-efficiency single-photon detectors are an enabling technology that became practical relatively recently.

4. CHSH on noisy quantum hardware

A quantum processor produces a noisy approximation to the Bell state, with fidelity $0.95$ to the ideal $|\Phi^+\rangle$. What CHSH value do you expect, and what does a measured $S = 2.5$ tell you about the device?

Show answer

For a state with fidelity $F$ to the Bell state, the CHSH violation scales approximately as $S \approx 2\sqrt{2} \cdot F$. With $F = 0.95$: expected $S \approx 2.69$. A measured $S = 2.5$ corresponds to fidelity $\approx 0.88$, somewhat lower than the targeted $0.95$. This indicates measurable dephasing or readout error during the Bell-test sequence — diagnostic information about the device beyond a single fidelity number. CHSH measurements are sensitive entanglement diagnostics, useful for catching subtle noise issues that a coarser fidelity number might miss.

Where this goes next

Tutorial 46 covers the no-cloning theorem — a sibling foundational result that explains why quantum information is fundamentally different from classical information. Tutorial 47 covers density matrices and mixed states, the formalism needed to describe noisy and partially-known quantum states. Tutorial 48 covers the Bloch sphere as the natural geometry of single-qubit states.

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