Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. The first such result was introduced by John Stewart Bell in 1964, building upon the Einstein–Podolsky–Rosen paradox, which had called attention to the phenomenon of quantum entanglement.

In the context of Bell's theorem, "local" refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of Bell, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

In his original paper, Bell deduced that if measurements are performed independently on the two separated particles of an entangled pair, then the assumption that the outcomes depend upon hidden variables within each half implies a mathematical constraint on how the outcomes on the two measurements are correlated. Such a constraint would later be named a Bell inequality. Bell then showed that quantum physics predicts correlations that violate this inequality. Multiple variations on Bell's theorem were put forward in the years following his original paper, using different assumptions and obtaining different Bell (or "Bell-type") inequalities.

The first rudimentary experiment designed to test Bell's theorem was performed in 1972 by John Clauser and Stuart Freedman. More advanced experiments, known collectively as Bell tests, have been performed many times since. Often, these experiments have had the goal of "closing loopholes", that is, ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. Bell tests have consistently found that physical systems obey quantum mechanics and violate Bell inequalities; which is to say that the results of these experiments are incompatible with local hidden-variable theories.

The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by philosophers. While the significance of Bell's theorem is not in doubt, different interpretations of quantum mechanics disagree about what exactly it implies.

There are many variations on the basic idea, some employing stronger mathematical assumptions than others. Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical pictures of nature. The original theorem proved by Bell in 1964 is not the most amenable to experiment, and it is convenient to introduce the genre of Bell-type inequalities with a later example.

Hypothetical characters Alice and Bob stand in widely separated locations. Their colleague Victor prepares a pair of particles and sends one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements (perhaps by flipping a coin to decide which). Denote these measurements by ${\displaystyle A_{0}}$ and ${\displaystyle A_{1}}$. Both ${\displaystyle A_{0}}$ and ${\displaystyle A_{1}}$ are binary measurements: the result of ${\displaystyle A_{0}}$ is either ${\displaystyle +1}$ or ${\displaystyle -1}$, and likewise for ${\displaystyle A_{1}}$. When Bob receives his particle, he chooses one of two measurements, ${\displaystyle B_{0}}$ and ${\displaystyle B_{1}}$, which are also both binary.

Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure ${\displaystyle A_{0}}$ and obtains the result ${\displaystyle +1}$, then the particle she received carried a value of ${\displaystyle +1}$ for a property ${\displaystyle a_{0}}$. Consider the combination$${\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}=(a_{0}+a_{1})b_{0}+(a_{0}-a_{1})b_{1}\,.}$$Because both ${\displaystyle a_{0}}$ and ${\displaystyle a_{1}}$ take the values ${\displaystyle \pm 1}$, then either ${\displaystyle a_{0}=a_{1}}$ or ${\displaystyle a_{0}=-a_{1}}$. In the former case, the quantity ${\displaystyle (a_{0}-a_{1})b_{1}}$ must equal 0, while in the latter case, ${\displaystyle (a_{0}+a_{1})b_{0}=0}$. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal ${\displaystyle \pm 2}$. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination ${\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}}$ across all the trials will be less than or equal to 2. No single trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages$${\displaystyle |\langle A_{0}B_{0}\rangle +\langle A_{0}B_{1}\rangle +\langle A_{1}B_{0}\rangle -\langle A_{1}B_{1}\rangle |\leq 2\,.}$$ This is a Bell inequality, specifically, the CHSH inequality. Its derivation here depends upon two assumptions: first, that the underlying physical properties ${\displaystyle a_{0},a_{1},b_{0},}$ and ${\displaystyle b_{1}}$ exist independently of being observed or measured (sometimes called the assumption of realism); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of locality).