Quantum entanglement is one of the most unusual features of modern physics. Two particles can share a quantum state in such a way that measurements of their properties are correlated, even when the particles are separated by vast distances. These correlations cannot always be explained by treating each particle as an independent object carrying its own predetermined properties.
Scientists test this behavior using experiments known as Bell tests, which examine whether the correlations between entangled particles can be explained by theories based on local hidden variables. Repeated experiments have shown that quantum mechanics predicts the observed correlations correctly, while local hidden-variable theories satisfying certain assumptions cannot account for them.
The central idea comes from Bell’s theorem, developed by physicist John Stewart Bell in 1964. His work transformed a philosophical debate about the meaning of quantum mechanics into a question that experiments could answer.
What is quantum entanglement?
In classical physics, objects have properties that can be described independently of how other objects are measured. Two coins, for example, can be prepared so that one is heads and the other is tails. Once the coins are separated, each has a definite face, whether or not anyone looks at it.
Quantum mechanics describes physical systems differently. A particle can exist in a superposition, a combination of possible measurement outcomes, until an interaction with a measuring apparatus produces a definite recorded result. When two particles are entangled, their shared quantum state cannot be fully described by assigning an independent quantum state to each particle.
Consider two particles prepared in a special joint state called a singlet state. If scientists measure a particular property known as spin along the same direction, their results will always be opposite. Spin is an intrinsic quantum property, often compared loosely to angular momentum, although it does not mean that a particle is literally a tiny spinning ball.
The striking feature is that this relationship persists even when the particles travel far apart. Measuring one particle and obtaining a particular result allows scientists to predict the corresponding result for the other particle if both measurements use the same direction.
However, entanglement is more subtle than a simple rule that the particles always carry opposite answers. The correlations also depend on the directions chosen for measurement. When scientists measure spin along different directions, the outcomes follow a precise pattern described by quantum mechanics.
Those correlations, especially when measurements are made along carefully selected directions, are what make Bell tests possible.
The question that led to Bell’s theorem
The unusual implications of quantum mechanics troubled Albert Einstein and other physicists. Einstein objected to the idea that quantum theory might provide only probabilities for individual measurement outcomes rather than a complete description of physical reality.
One concern was whether a measurement on one particle could somehow affect the physical situation of another distant particle. Einstein famously criticized the apparent nonlocal character of quantum mechanics, although his objections did not amount to a denial of every prediction the theory made.
In 1935, Einstein, Boris Podolsky, and Nathan Rosen published an argument now known as the EPR paradox. They reasoned that if quantum mechanics predicted perfectly correlated outcomes for distant particles, and if measuring one particle did not disturb the other, then the distant particle’s properties might already have definite values. They suggested that quantum mechanics could be incomplete, with additional physical information needed to explain the correlations.
Such additional information is often described using the term hidden variables. In a hidden-variable theory, the quantum state would not necessarily provide the whole story. Other variables, perhaps inaccessible to ordinary measurements, would help determine the outcomes of experiments.
Einstein’s preferred picture also emphasized locality: the idea that an event in one place cannot instantly influence a physically separated event in a way that transmits an effect faster than light.
For decades, the dispute could be treated largely as a question about how to interpret quantum mechanics. Bell changed that in 1964 by showing that a broad class of local hidden-variable theories must satisfy mathematical restrictions on measurement correlations.
Quantum mechanics predicts that some entangled systems can violate those restrictions. If experiments observe such a violation, the results cannot be explained by a local hidden-variable model that satisfies the assumptions underlying Bell’s theorem.
The theorem does not prove that every conceivable hidden-variable theory is impossible. Instead, it identifies a specific conflict between quantum predictions and theories that combine locality with a suitable form of realism about measurement outcomes.
What Bell’s theorem actually says
Bell’s theorem concerns the relationship between measurement settings, measurement outcomes, and any underlying physical information that might explain those outcomes.
Imagine two experimenters, Alice and Bob, who receive entangled particles and take them to distant laboratories. Each experimenter can choose among different measurement settings. Alice records an outcome for her particle, while Bob records an outcome for his.
A local hidden-variable theory proposes that the outcomes can be explained by some underlying information, represented by λ, shared by the particles or their common source. The theory may allow each result to depend on the local measurement setting and on λ. However, Alice’s result cannot depend on Bob’s distant setting, and Bob’s result cannot depend on Alice’s distant setting.
The key restriction is that the theory must reproduce the observed results without allowing influences between the distant measurement events and without relying on special coordination between the hidden variables and the experimenters’ measurement choices.
Bell showed that these assumptions impose limits on the correlations the experimenters can observe. Quantum mechanics predicts correlations that can exceed those limits.
A widely used version of the test is the Clauser–Horne–Shimony–Holt inequality, usually called the CHSH inequality. It provides a practical mathematical way to compare measurement results with the predictions of local hidden-variable theories.
For measurement outcomes represented by +1 and −1, the CHSH quantity combines four correlation measurements made using different pairs of settings. Under the standard local hidden-variable assumptions, its absolute value cannot exceed 2:
|S| ≤ 2
Quantum mechanics predicts that appropriately prepared entangled particles can produce a value as large as 2√2, approximately 2.83, when the measurement settings are chosen optimally.
This difference is experimentally meaningful. Scientists do not need to determine the hidden variables or observe the entire quantum state directly. They need to collect measurement data and establish whether the observed correlations violate the relevant inequality.
A violation rules out the class of local hidden-variable explanations covered by the test, provided the experiment’s assumptions and procedures are adequately satisfied.
How scientists perform a Bell test
A Bell test begins with a source that produces pairs of entangled particles. Depending on the experiment, these might be photons, electrons, or other quantum systems. Photons are especially useful because their polarization can be measured and manipulated with optical equipment.
The source sends one particle toward Alice and the other toward Bob. Their detectors are separated so that the measurements can be performed at distant locations.
Each experimenter chooses a measurement setting from a set of available options. For photons, the settings might correspond to different orientations of polarization analyzers. For particles with spin, they might correspond to different directions along which spin is measured.
Each measurement produces an outcome. In a simplified experiment, the result can be coded as +1 or −1. The experimenters repeat the procedure many times, collecting a large sample of paired outcomes.
The important information is not simply how often Alice obtains +1 or how often Bob obtains −1. Instead, researchers compare the results from both sides to determine how strongly the outcomes are correlated for each combination of measurement settings.
For each setting pair, they calculate a correlation value. They then combine the values into a Bell inequality, such as the CHSH inequality, and compare the result with the maximum permitted by local hidden-variable theories.
If the observed value exceeds the local bound by an amount that is statistically convincing, the data demonstrate a Bell inequality violation.
A convincing result requires much more than finding an unusual pattern in a small collection of measurements. Researchers must account for detector performance, statistical uncertainty, background events, and other possible sources of error. They also need to establish that the experimental design does not leave an ordinary local explanation available through an overlooked loophole.
Why measurement settings matter
A Bell test would accomplish little if scientists always measured both particles in the same way. The most revealing correlations emerge when Alice and Bob use different settings chosen to test how the outcomes relate across several measurement directions.
For an entangled pair in a singlet state, measurements along the same direction yield opposite results. Measurements along different directions remain correlated, but the strength of the correlation changes with the angle between the settings.
Quantum mechanics provides a mathematical prediction for this pattern. For spin measurements on a singlet state, the correlation is −cos θ, where θ is the angle between the measurement directions. The minus sign reflects the opposite outcomes obtained when the directions match.
A local hidden-variable model can reproduce some correlations, including perfect anticorrelation for matching settings. The challenge is reproducing the full pattern across multiple settings without exceeding Bell’s limits.
By choosing measurement directions carefully, researchers expose differences between quantum predictions and the restrictions imposed by local hidden-variable theories.
This is why entanglement alone is not sufficient to demonstrate a Bell violation. Scientists must measure the correlations using an appropriate experimental arrangement and show that the resulting statistics cross the relevant inequality’s bound.
What Bell tests have revealed
Experiments have repeatedly demonstrated violations of Bell inequalities, in agreement with quantum mechanics. The results show that entangled quantum systems exhibit correlations that cannot be explained by local hidden-variable theories satisfying the standard assumptions of Bell tests.
Early experiments established the basic effect, but they also faced technical limitations. Critics could ask whether inefficient detectors had missed some outcomes or whether the measurement settings and outcomes had been coordinated in a way that preserved a local explanation.
These concerns became known as loopholes. A loophole does not necessarily mean an experiment was incorrectly performed. It identifies a weakness in the experimental design that might allow a competing explanation to survive.
Two particularly important issues were the detection loophole and the locality loophole.
The detection loophole arises when detectors fail to record every relevant particle. If the recorded events are an unrepresentative sample, the observed correlations might not reflect the behavior of all emitted pairs. A local model could potentially exploit selective detection to imitate a Bell violation.
The locality loophole arises when the two measurement events are not sufficiently separated in space and time. If a signal traveling at or below the speed of light could pass between the relevant events, a local explanation might allow one measurement to influence the other.
Experiments designed to close these loopholes use more efficient detectors, careful timing, and measurement stations separated far enough that the measurement events are spacelike separated. In this context, spacelike separation means that a light-speed signal cannot travel from one event to the other before it occurs.
In 2015, several experimental groups reported Bell tests that addressed the major detection and locality loopholes simultaneously. These loophole-free experiments provided particularly strong evidence against local hidden-variable explanations.
No experiment can eliminate every imaginable alternative theory without assumptions. However, the accumulation of increasingly precise tests has made the central conclusion robust: nature does not behave according to the combination of locality and hidden-variable assumptions tested by Bell inequalities.
Does entanglement allow faster-than-light communication?
Bell inequality violations can sound as though one particle instantly sends information to another. That interpretation is misleading.
When Alice measures her particle, her individual result is random according to quantum mechanics. She cannot choose the result she obtains, even if she can choose the measurement setting. Bob’s individual results are also random.
If Alice and Bob are far apart, Bob cannot determine Alice’s measurement setting or outcome just by examining his own results. His local data do not reveal a controllable message from Alice.
The correlations become visible only when the two experimenters compare their recorded results through an ordinary communication channel. That comparison requires information to travel no faster than light.
Entanglement therefore does not provide a method for sending messages instantaneously. It produces correlations that violate Bell inequalities without enabling faster-than-light signaling.
This distinction matters because quantum mechanics challenges a particular kind of local explanation, not the rule that usable information cannot be transmitted faster than light.
Relativity and quantum mechanics remain compatible in this operational sense: although entangled measurements exhibit nonclassical correlations, they cannot be used to transmit a controllable signal outside the light cone.
What Bell tests do—and do not—prove
Bell tests provide powerful evidence about the structure of physical reality, but their implications must be stated carefully.
First, they do not show that quantum mechanics predicts every possible measurement outcome with certainty. Quantum theory generally predicts probabilities, and Bell tests examine whether the resulting statistical patterns agree with those predictions.
Second, they do not rule out all hidden-variable theories. A hidden-variable model that abandons locality can reproduce quantum predictions in principle. Bohmian mechanics, for example, is a deterministic interpretation of quantum theory that includes nonlocal features.
Third, Bell tests do not establish that distant particles exchange ordinary messages or that consciousness causes quantum measurement outcomes. Neither claim follows from a Bell inequality violation.
The experiments also depend on assumptions about how the test is conducted. In particular, the measurement settings must be sufficiently independent of the hidden variables under consideration, and the data must be analyzed in a way that accounts for relevant experimental imperfections. If a theory rejects these assumptions, it may avoid the usual Bell argument, although it must then explain why the apparent independence of the settings is misleading or why the experimental framework needs revision.
These qualifications do not erase the experimental result. They clarify what the result establishes: quantum correlations cannot be reproduced by theories that satisfy the relevant combination of locality, measurement-setting independence, and other standard assumptions.
The broader question of how to interpret quantum mechanics remains open. Physicists and philosophers continue to debate what the quantum state represents and how measurement should be understood. Bell tests constrain possible answers without selecting a single interpretation as the only correct one.
Why Bell tests matter beyond fundamental physics
Bell tests are important not only because they address a longstanding question about quantum theory but also because they help establish the principles behind emerging quantum technologies.
Quantum cryptography can use quantum systems to help detect eavesdropping. In certain protocols, the security analysis can be tied directly to observed Bell correlations, reducing reliance on assumptions about the internal workings of the devices. Such approaches are known as device-independent methods because security can be established from observed behavior under specified conditions rather than from complete trust in the hardware.
Bell violations also play a role in some approaches to quantum random-number generation. Quantum theory predicts that suitable measurement outcomes contain intrinsic randomness, and sufficiently strong Bell correlations can help certify that unpredictability under explicit assumptions.
More broadly, the ability to create, preserve, and measure entanglement is essential to quantum information science. Quantum computers, quantum communication systems, and quantum networks rely on quantum states that cannot be described as ordinary collections of independent classical bits.
Not every quantum technology requires a Bell test, and not every entangled state violates a Bell inequality under every measurement arrangement. Nevertheless, Bell experiments provide a rigorous way to demonstrate that a system exhibits correlations beyond those allowed by local hidden-variable models.
The lasting significance of Bell’s theorem
Bell’s theorem changed the study of quantum foundations by showing that a debate about completeness and locality could be turned into a measurable experimental question.
The central lesson is not simply that entangled particles behave strangely. It is that their correlations follow rules that cannot be reproduced by a broad and intuitively appealing class of local hidden-variable theories.
Bell tests reveal this difference by preparing entangled particles, measuring them using independently selected settings, comparing the outcomes, and testing the resulting correlations against mathematical inequalities.
The experiments do not answer every question about the meaning of quantum mechanics. They do establish that the classical picture of distant objects carrying local instructions sufficient to explain all their measurement outcomes cannot account for the observed quantum correlations under the standard assumptions.
That result is one of the clearest demonstrations that the quantum world does not fit neatly into the familiar intuitions of everyday experience—and one of the strongest examples of how carefully designed experiments can turn a foundational question in physics into a testable scientific claim.