Bell's Theorem: The Most Profound Discovery in the History of Physics

In 1935, Einstein, Podolsky, and Rosen published a paper intended to show that quantum mechanics was incomplete.

Their argument was careful, precise, and devastating.

If quantum mechanics is correct, if entangled particles really do exhibit the correlations it predicts, then either:

(a) there is instantaneous action at a distance, violating special relativity, or

(b) quantum mechanics is incomplete, there are hidden variables that determine the outcomes of measurements in advance.

Einstein found (a) unacceptable. He called it spukhafte Fernwirkung, “spooky action at a distance.”

He concluded: quantum mechanics is incomplete. Hidden variables must exist.

For twenty-nine years, this remained a philosophical debate.

Then John Bell did something remarkable.

He turned it into physics.

I. The Setup

Consider a source that emits pairs of particles in an entangled spin state, the singlet state:

\[|\Psi^-\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\rangle_A|\downarrow\rangle_B - |\downarrow\rangle_A|\uparrow\rangle_B\right)\]

Alice receives particle A. Bob receives particle B. They are separated, far apart.

Alice measures the spin of her particle along direction $\hat{a}$. Bob measures the spin of his particle along direction $\hat{b}$.

Each measurement yields $+1$ or $-1$.

Quantum mechanical prediction:

The correlation between their results:

\[E(\hat{a}, \hat{b}) = \langle\Psi^-|\hat{\sigma}_A\cdot\hat{a} \otimes \hat{\sigma}_B\cdot\hat{b}|\Psi^-\rangle = -\hat{a}\cdot\hat{b} = -\cos\theta_{ab}\]

where $\theta_{ab}$ is the angle between the measurement directions.

If $\theta = 0°$: perfect anti-correlation. $E = -1$. If $\theta = 90°$: no correlation. $E = 0$. If $\theta = 180°$: perfect correlation. $E = +1$.

This is the quantum prediction. It has been verified to extraordinary precision.

The question Bell asked: Can any local hidden variable theory reproduce this correlation?

II. Local Hidden Variables: The Precise Definition

A local hidden variable (LHV) theory is any theory satisfying:

1. Realism (hidden variables):

The outcome of every measurement is predetermined by variables $\lambda$, the “hidden variables”, that exist prior to measurement.

The outcome of Alice’s measurement: $A(\hat{a}, \lambda) = \pm 1$

The outcome of Bob’s measurement: $B(\hat{b}, \lambda) = \pm 1$

These are definite functions of the measurement setting and the hidden variable.

2. Locality:

Alice’s outcome does not depend on Bob’s measurement setting, and vice versa:

$A = A(\hat{a}, \lambda) \quad \text{(not } \hat{b}\text{)}$ $B = B(\hat{b}, \lambda) \quad \text{(not } \hat{a}\text{)}$

What happens at Bob’s detector cannot influence Alice’s result, they are spacelike separated.

3. Statistical independence (no conspiracy):

The hidden variable $\lambda$ is distributed according to some probability distribution $\rho(\lambda)$, independent of the measurement settings:

\[\rho(\lambda | \hat{a}, \hat{b}) = \rho(\lambda)\]

The hidden state of the particles is not influenced by the experimenters’ choices of measurement directions.

Given these three assumptions, realism, locality, statistical independence, Bell derived an inequality that any LHV theory must satisfy.

III. Bell’s Inequality: The Derivation

Consider three measurement directions $\hat{a}$, $\hat{b}$, $\hat{c}$.

For any single hidden variable $\lambda$, define:

$A_a = A(\hat{a}, \lambda) = \pm 1$ $A_b = A(\hat{b}, \lambda) = \pm 1$
$A_c = A(\hat{c}, \lambda) = \pm 1$

Note the algebraic identity:

\[A_a A_b - A_a A_c = A_a A_b(1 - A_b A_c)\]

Since $A_b^2 = 1$:

\[|A_a A_b - A_a A_c| = |1 - A_b A_c|\]

Taking the expectation value over $\lambda$, and using the perfect anti-correlation of the singlet ($B(x,\lambda) = -A(x,\lambda)$, so $\langle A_a A_b \rangle = -E(\hat{a},\hat{b})$):

\[|E(\hat{a},\hat{b}) - E(\hat{a},\hat{c})| \leq 1 + E(\hat{b},\hat{c})\]

This is the Bell inequality in its original form.

More commonly used: the CHSH inequality (Clauser, Horne, Shimony, Holt, 1969):

Consider four measurement settings: $\hat{a}$, $\hat{a}'$ for Alice and $\hat{b}$, $\hat{b}'$ for Bob.

Define the CHSH correlator:

\[S = E(\hat{a},\hat{b}) - E(\hat{a},\hat{b}') + E(\hat{a}',\hat{b}) + E(\hat{a}',\hat{b}')\]

For any LHV theory:

\[|S| \leq 2\]

This is the CHSH inequality. Any theory satisfying locality, realism, and statistical independence must satisfy this.

The derivation is elementary. The implication is revolutionary.

IV. Quantum Mechanics Violates the Bell Inequality

Now compute the quantum mechanical prediction for $S$.

Choose measurement directions:

$\hat{a} = 0°, \quad \hat{a}' = 90°$ $\hat{b} = 45°, \quad \hat{b}' = 135°$

The quantum correlations:

\[E(\hat{a},\hat{b}) = -\cos(45°) = -\frac{1}{\sqrt{2}}\] \[E(\hat{a},\hat{b}') = -\cos(135°) = +\frac{1}{\sqrt{2}}\] \[E(\hat{a}',\hat{b}) = -\cos(45°) = -\frac{1}{\sqrt{2}}\] \[E(\hat{a}',\hat{b}') = -\cos(45°) = -\frac{1}{\sqrt{2}}\]

Therefore:

\[S_{QM} = E(\hat{a},\hat{b}) - E(\hat{a},\hat{b}') + E(\hat{a}',\hat{b}) + E(\hat{a}',\hat{b}')\] \[= -\frac{1}{\sqrt{2}} - \left(+\frac{1}{\sqrt{2}}\right) + \left(-\frac{1}{\sqrt{2}}\right) + \left(-\frac{1}{\sqrt{2}}\right)\] \[= -\frac{4}{\sqrt{2}} = -2\sqrt{2} \approx -2.828\] \[|S_{QM}| = 2\sqrt{2} \approx 2.828\]

The quantum mechanical prediction: $|S_{QM}| = 2\sqrt{2} \approx 2.828$

The Bell bound for any LHV theory: $|S_{LHV}| \leq 2$

Quantum mechanics violates the Bell inequality by a factor of $\sqrt{2}$.

The Tsirelson bound, the maximum possible violation in any quantum theory, is exactly $2\sqrt{2}$. Quantum mechanics saturates it.

This is not a small effect, not within experimental error. It is a clean, unambiguous violation of what any local realistic theory can produce.

V. The Experiments: Closing the Loopholes

Bell’s theorem is a mathematical result. Its physical significance depends on experiment.

The loopholes:

Any Bell experiment must contend with loopholes, experimental imperfections that could, in principle, allow an LHV theory to fake a Bell violation.

The detection loophole: If detectors are inefficient, if they miss some particles, an LHV theory could conspire so that the detected particles are a biased sample that violates Bell inequalities even though the full sample does not.

Requires: detector efficiency $> 2(\sqrt{2}-1) \approx 82.8\%$ for the CHSH inequality.

The locality loophole: If Alice’s choice of measurement setting can influence Bob’s outcome, or if there is a common cause that sets both, the Bell inequality derivation is invalid.

Requires: spacelike separation of measurement events, with random setting choices made after the particles have left the source.

The freedom-of-choice loophole: If the experimenters’ choices of settings are correlated with the hidden variable $\lambda$, the statistical independence assumption fails.

Requires: truly random setting choices, independent of the particle source.

Early experiments (1972-1982):

Freedman & Clauser (1972): first Bell test. Aspect, Grangier & Roger (1982): first with time-varying settings.

These closed some loopholes but not all. LHV theories were constrained but not ruled out.

Loophole-free Bell tests (2015):

In 2015, three experiments closed all major loopholes:

Hensen et al. (Delft): Used electron spins in nitrogen-vacancy centers separated by 1.3 km. Detected violation: $S = 2.42 \pm 0.20$. Closed both the detection and locality loopholes.

Giustina et al. (Vienna): Used photons with high-efficiency detectors. Closed detection loophole with $p < 3.74 \times 10^{-31}$.

Shalm et al. (NIST): Used photons, random number generators, and spacelike separation. Violation with $p < 3.6 \times 10^{-9}$.

The Big Bell Test (2018): Used human choices from 100,000 participants worldwide to generate measurement settings, addressing the freedom-of-choice loophole.

Violation confirmed.

The conclusion is definitive:

No local hidden variable theory can reproduce the quantum correlations.

The universe is not locally realistic.

VI. What Must Be Abandoned

Bell’s theorem, combined with the experimental results, forces us to abandon at least one of:

1. Realism: Physical quantities do not have definite values prior to measurement. The moon is not there when nobody looks.

2. Locality: The outcome of a measurement here can be instantaneously influenced by a measurement there, regardless of spatial separation.

3. Statistical independence (Free will / No conspiracy): The measurement settings chosen by experimenters are correlated with the hidden state of the particles, either through a common cause in the past or through cosmic conspiracy.

These are not equivalent choices. Each corresponds to a different interpretation of quantum mechanics.

Abandon realism → Copenhagen, QBism, Relational QM: There are no hidden variables. Quantum states are complete descriptions. Properties do not exist prior to measurement.

Abandon locality → Bohmian mechanics: Keep realism, particles have definite positions. Accept nonlocality, the pilot wave is nonlocal. The guidance equation acts instantaneously across arbitrary distances.

Abandon statistical independence → Superdeterminism: The particles and the detectors share a common past cause that correlates the hidden variable with the measurement settings. No experiment can ever test LHV theories, because the settings are always correlated with what is being measured.

Most physicists find superdeterminism repugnant. It implies that the universe conspires to prevent us from testing it, and it undermines the entire notion of scientific experiment.

But it cannot be experimentally ruled out.

VII. Nonlocality Without Signaling

The experimental violation of Bell inequalities establishes quantum nonlocality.

But, crucially, this nonlocality cannot be used to send signals.

This is not obvious. Let me explain it carefully.

Alice and Bob share an entangled pair. Alice measures along $\hat{a}$ and gets $+1$. She knows instantaneously that Bob’s result (if he measures along the anti-parallel direction) will be $-1$.

But Bob does not know that Alice measured, and he does not know what she got. He just sees $-1$, which he would have gotten with 50% probability anyway.

The nonlocal correlation is only visible when Alice and Bob compare their results, which requires classical communication, limited by the speed of light.

More precisely:

Bob’s marginal distribution, the probability of his outcomes, ignoring Alice’s results, is independent of Alice’s measurement setting:

$P(B = +1 | \hat{b}) = \frac{1}{2}$ regardless of what $\hat{a}$ Alice chose.

The nonlocality is in the joint distribution, in the correlations, not in the marginal distributions.

You cannot signal with entanglement because you cannot control which outcome you get. The randomness of quantum measurement is exactly what prevents faster-than-light signaling.

This is the no-signaling theorem:

Any theory that reproduces quantum predictions must satisfy the no-signaling condition, the marginal distributions of one party are independent of the other’s settings.

Quantum mechanics satisfies this. But it satisfies it in a striking way: the correlations are nonlocal, but the nonlocality is precisely calibrated to prevent signaling.

Why? We don’t know.

VIII. Tsirelson’s Bound and Beyond

The maximum quantum violation of the CHSH inequality is $2\sqrt{2}$, the Tsirelson bound.

Is this the maximum possible? Could a theory be more nonlocal than quantum mechanics?

Popescu-Rohrlich (PR) boxes:

In 1994, Popescu and Rohrlich asked: what is the maximum possible violation of the CHSH inequality in any no-signaling theory?

The answer: $$

\leq 4$$.

A hypothetical “PR box” would achieve $|S| = 4$: maximum possible nonlocality consistent with no-signaling.

The PR box satisfies no-signaling, it cannot be used to send information, but it is more nonlocal than quantum mechanics.

Why doesn’t nature use PR boxes? Why is quantum nonlocality limited to $2\sqrt{2}$ rather than $4$?

Several information-theoretic principles have been proposed as explanations:

Information causality (Pawłowski et al., 2009): The amount of information Bob can access about Alice’s data, using their shared correlations and $m$ bits of classical communication, is at most $m$ bits.

Quantum mechanics satisfies information causality. PR boxes violate it.

Macroscopic locality (Navascués & Wunderlich, 2010): When many measurements are coarse-grained, the statistics should be locally explainable.

Quantum mechanics passes this test. PR boxes do not.

Local orthogonality (Fritz et al., 2012): Events that are locally exclusive should be globally exclusive.

Quantum mechanics has this property. PR boxes lack it.

These principles recover the Tsirelson bound. They explain why quantum mechanics is exactly as nonlocal as it is.

But they do not explain why these principles hold. They replace one mystery with another.

IX. Bell Nonlocality in Quantum Information

Bell nonlocality is not only foundational. It is a resource.

Device-independent quantum cryptography:

In device-independent QKD, Alice and Bob certify the security of their key by observing a Bell inequality violation, without trusting their devices.

If the devices produce a Bell violation, an eavesdropper cannot have full information about the outcomes, because any eavesdropper who did would have to share a local realistic theory with Alice and Bob, which would not produce the violation.

The Bell violation certifies security from first principles, from physics alone.

This is remarkable: the violation of a mathematical inequality guarantees cryptographic security.

Device-independent randomness certification:

A Bell violation certifies genuine randomness.

If Alice’s outcomes were predetermined, if there were a hidden variable $\lambda$ determining her results, the Bell inequality would not be violated.

Therefore: the random outcomes of a Bell experiment are genuinely random, certified by physics, not by the quality of a random number generator.

This is the only known method of certifying true randomness from first principles.

Quantum communication complexity:

Bell nonlocality provides exponential advantages in certain communication complexity tasks, tasks where two parties must compute a function of their combined inputs while minimizing communication.

Nonlocal correlations allow them to do this with exponentially fewer bits than any classical protocol.

X. What Bell’s Theorem Tells Us About Reality

Bell’s theorem is the most important result in the foundations of physics since quantum mechanics itself.

Its implications are not yet fully absorbed, even by physicists who use quantum mechanics daily.

Locality is false.

This is not a matter of interpretation. It is an experimental fact. The universe exhibits nonlocal correlations that cannot be explained by any local theory.

Special relativity tells us that nothing travels faster than light. Bell’s theorem tells us that the universe is nonlocal.

These two facts are compatible, but their compatibility is not obvious. The reconciliation requires that nonlocality not be usable for signaling, a constraint that quantum mechanics satisfies by precise calibration.

Why? We don’t know.

Realism is in trouble.

If you want to maintain locality, if you find nonlocal hidden variables unacceptable, then you must abandon realism: the idea that physical quantities have definite values prior to measurement.

The moon is not there when nobody looks. Not as a philosophical conceit. As a physical statement.

The universe cannot be explained by local realistic physics.

This is what Bell proved. This is what the experiments confirmed.

The structure of reality at the quantum level violates the intuitions built into every macroscopic experience we have ever had.

XI. The Conspiracy That Cannot Be Ruled Out

I want to end with the deepest loophole, the one that cannot be closed.

Superdeterminism.

Bell’s derivation assumes statistical independence: the hidden variable $\lambda$ is uncorrelated with the measurement settings $\hat{a}, \hat{b}$.

If this assumption fails, if the past light cone of the particle source intersects the past light cones of the setting choices in a way that creates correlations, then Bell’s theorem does not apply.

An LHV theory could reproduce all quantum predictions if the universe is so finely tuned that every “random” choice of measurement setting is correlated with the hidden state of the particles.

This cannot be ruled out experimentally, because any experiment we design uses measurement settings from within the universe, which share a common past with everything else.

The Big Bell Test used human choices. But humans are physical systems, their choices are determined by their past, which shares a common origin with the particles.

To truly close this loophole, you would need measurement settings from a source causally disconnected from the particle source, from outside the universe.

This is impossible.

Superdeterminism is the last refuge of local realism.

Most physicists reject it, because it requires a cosmic conspiracy so fine-tuned as to be indistinguishable from magic, and because it undermines the entire enterprise of experimental science.

But it cannot be ruled out.

Gerard ‘t Hooft, Nobel laureate, takes superdeterminism seriously.

He may be right.

Or the universe may simply be nonlocal, deeply, irreducibly, experimentally confirmed nonlocal, and we have not yet found the right framework to make sense of it.

Coda

Bell wrote his theorem in 1964, while working at CERN on theoretical particle physics.

He did it in his spare time. It was not his job.

He submitted it to Physics, a short-lived journal that published only five issues before folding. The paper appeared in Vol. 1, No. 3.

It was rescued and reprinted.

He spent the rest of his life arguing for its importance, arguing that the foundations of quantum mechanics were not a philosophical sideshow but a central unsolved problem in physics.

He was largely ignored.

He died in 1990, of a cerebral hemorrhage, shortly after being nominated for the Nobel Prize.

He never received it.

The Nobel was eventually awarded to Aspect, Clauser, and Zeilinger in 2022, for experimental tests of Bell inequalities.

Bell’s theorem is now recognized as the most important result in the foundations of physics of the 20th century.

“Either the wavefunction, as given by the Schrödinger equation, is not everything, or it is not right.”

— John Bell, Speakable and Unspeakable in Quantum Mechanics (1987)

The universe is nonlocal.

We do not know what this means.

We are certain that it is true.

*Essential reading:

Bell’s original 1964 paper “On the Einstein-Podolsky-Rosen Paradox” (Physics 1, 195) is six pages and worth reading directly.*

Speakable and Unspeakable in Quantum Mechanics *(Cambridge UP, 1987) collects everything Bell wrote on foundations and is essential.

The first experimental confirmation came in Aspect, Grangier & Roger, “Experimental Tests of Bell’s Inequalities” (Phys. Rev. Lett. 49, 91, 1982).

The loophole-free experiment is Hensen et al., “Loophole-free Bell inequality violation” (Nature 526, 682, 2015).

For the comprehensive technical review, see Brunner et al., “Bell nonlocality” (Rev. Mod. Phys. 86, 419, 2014).

For the standard textbook treatment, see Nielsen & Chuang,* Quantum Computation and Quantum Information, Chapter 2.

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