Hidden Variable Theories

If the wavefunction is only a statistical summary — as Einstein believed — then behind it should sit a deeper layer of variables that fix each individual outcome, restoring the determinism quantum mechanics seems to abandon. A hidden-variable theory makes this precise: supplement the state with a parameter $\lambda$ , distributed with some density $\rho(\lambda)$ , so that a measurement of $A$ along setting $a$ yields a definite value $A(a,\lambda)$ , and averaging over $\lambda$ reproduces the quantum statistics $\langle A\rangle=\int A(a,\lambda)\rho(\lambda)\,d\lambda$ . This article states that program precisely, distinguishes deterministic from stochastic models, and — crucially — isolates the two locality assumptions (parameter independence and outcome independence) that, together, make a model local. That exact bundle is the target of Bell's Theorem and the CHSH Inequality. We then tell the history-that-is-physics story that shaped the field: von Neumann's celebrated 1932 "proof" that hidden variables are impossible, which held the subject in check for three decades — and was wrong. His error was assuming that the additivity of expectation values, $\langle A+B\rangle=\langle A\rangle+\langle B\rangle$ , must hold for individual hidden-variable outcomes even when $A$ and $B$ do not commute. It holds for ensemble averages but is an illegitimate constraint on a single $\lambda$ : $\sigma_x$ and $\sigma_z$ each take values $\pm1$ , yet $\sigma_x+\sigma_z$ has eigenvalues $\pm\sqrt2$ , which no sum of two $\pm1$ 's can equal. Grete Hermann spotted this in 1935 (and was ignored); Bohm built an explicit working counterexample in 1952; Bell dismantled the proof in 1966. The honest verdict is sharp: hidden variables per se are not forbidden — only local ones are, and only because Bell showed their correlations obey an inequality that quantum mechanics (with its $2\sqrt2$ value) violates. We close with the atlas of what survives: Bohmian mechanics, superdeterminism, contextual models — the theories still standing after the local option is closed.

Key takeaways

A hidden-variable theory adds a parameter $\lambda$ so outcomes are definite functions $A(a,\lambda)$ averaging to the quantum statistics. Von Neumann's 1932 "impossibility proof" was wrong (it forced additivity on non-commuting individual values). Hidden variables are allowed; Bell shows only local ones fail — their correlations cap at CHSH $=2$ while quantum mechanics reaches $2\sqrt2$ .

The core insight

"Hidden variables" is not one dead idea but a program. Bell's theorem kills exactly the local version — the one where a remote setting cannot affect a nearby outcome. Non-local (Bohm), contextual, and superdeterministic models survive. Von Neumann's famous ban was a mathematical mistake.

Where this lives today

Bell's redirection turned a philosophical stalemate into an experimental industry, and the local-hidden-variable option is now closed by loophole-free experiments (2015 onward), rewarded by the 2022 Nobel Prize — the experiments themselves are in Experimental Tests and Violation of Bell Inequalities. Among the survivors, de Broglie–Bohm mechanics remains a fully worked nonlocal deterministic theory (De Broglie–Bohm Pilot Wave Theory); contextual value-assignments are constrained by the setting-dependence theorems of The Kochen–Specker Theorem and Contextuality; and superdeterminism — abandoning the assumed independence of measurement settings from $\lambda$ — remains a logically open (if widely disfavored) loophole. Device-independent quantum cryptography turns the failure of local hidden variables into a security guarantee: a Bell-inequality violation certifies genuine randomness no local eavesdropper could have pre-programmed.

Origins · 1 minhidden variables and how this was discovered

hidden variables

1932–1966 · Budapest, Belfast

The idea has a checkered history that is itself a physics lesson. John von Neumann, in his 1932 Mathematical Foundations of Quantum Mechanics, presented a proof that hidden variables are impossible — a result so authoritative it froze the question for a generation. In 1935 the philosopher-physicist Grete Hermann, working in the neo-Kantian Fries school, published a precise refutation identifying the illegitimate additivity assumption; it went almost entirely unnoticed. In 1952 David Bohm constructed an explicit, empirically adequate deterministic hidden-variable theory — an existence proof that flatly contradicted the supposed impossibility — which Einstein, tellingly, found "too cheap" but which reopened the debate. Finally John Stewart Bell, whose portrait heads The Bell States and Maximally Entangled Pairs, dissected von Neumann's error in a 1966 review and replaced the sterile "impossibility" question with the fertile one: are local hidden variables possible? His answer — no — is the theorem developed in Bell's Theorem and the CHSH Inequality.

History

1932

— von Neumann publishes his (flawed) hidden-variable impossibility proof

1935

— Grete Hermann identifies the additivity error, and is ignored

1952

— Bohm builds an explicit deterministic hidden-variable theory

1964

— Bell proves no local hidden-variable theory reproduces quantum correlations

1966

— Bell's review exposes the flaw in von Neumann's proof

The physical picture

The dream is simple: quantum randomness is like a coin flip only because we don't see the coin's precise initial conditions. Give each particle a full set of instructions $\lambda$ — "if asked about $z$ -spin, answer up; if asked about $30°$ , answer down; ..." — draw $\lambda$ from a distribution, and the apparent randomness becomes plain ignorance. For a single particle this always works; you can always cook up a $\lambda$ -recipe reproducing any quantum statistics. The trouble starts with pairs. For an entangled pair, a local hidden-variable model demands that Alice's answer depend only on her setting $a$ and the shared $\lambda$ — not on Bob's distant setting $b$ — and vice versa. This "factorizability" is the mathematical face of locality: $\langle A(a)B(b)\rangle=\int A(a,\lambda)B(b,\lambda)\rho(\lambda)\,d\lambda$ . It is completely reasonable-looking, and it is exactly what Bell showed cannot reproduce the quantum correlations of the singlet. The local instructions are always too rigid: they can match quantum mechanics at a few special angles but not across the continuum, and the gap is a bounded quantity — the CHSH expression, which local models cap at $2$ while quantum mechanics reaches $2\sqrt2$ . Before Bell, though, the field believed hidden variables were flatly impossible, on the authority of von Neumann's 1932 book. That belief was a thirty-year mistake. The picture to hold: von Neumann proved that no observable's expectation values can be reproduced by a hidden-variable average — but only by assuming a property (additivity of individual values for incompatible observables) that no sensible hidden-variable theory should obey, and that quantum mechanics itself does not obey at the level of single outcomes. Once you drop that unwarranted axiom, hidden variables are back on the table, and the real obstruction — locality — is the one Bell found.

Figure

The hidden-variable program and what Bell rules out — local models cap at CHSH 2, quantum reaches 2√2

Figure

Von Neumann's additivity trap — σx and σz are ±1, but σx+σz is ±√2

Derivation AThe hidden-variable program, and the two locality assumptions Bell attacks

We state precisely what a hidden-variable model is, and factor "local" into its two independent parts — because naming them exactly is what lets Bell's theorem land. The tools are elementary probability plus the singlet correlations from The EPR Paradox and Local Realism.

Step 1. The general model. Introduce a hidden variable $\lambda$ with density $\rho(\lambda)\ge0$ , $\int\rho\,d\lambda=1$ . A measurement of setting $a$ returns a value drawn from a response function; averaging reproduces quantum statistics: $\langle A\rangle=\int \bar A(a,\lambda)\rho(\lambda)\,d\lambda$ . Why: this is the minimal demand — the observed randomness is ignorance of $\lambda$ , and any model must at least match the quantum averages.

Step 2. Deterministic vs stochastic. In a deterministic model $A(a,\lambda)\in\{\pm1\}$ is a definite function; in a stochastic model $\lambda$ only fixes a probability $P(A|a,\lambda)$ . Why: Bell's theorem covers both — a stochastic model can be reduced to a deterministic one by absorbing its extra randomness into $\lambda$ , so no generality is lost by picturing definite instructions.

Step 3. Factor "local" into two conditions. For a pair with settings $a,b$ : parameter independence — Alice's outcome distribution does not depend on Bob's setting $b$ — and outcome independence — given $\lambda$ , Alice's and Bob's outcomes are statistically independent. Together they give Bell factorizability: $P(A,B|a,b,\lambda)=P(A|a,\lambda)\,P(B|b,\lambda)$ , hence $\langle A(a)B(b)\rangle=\int A(a,\lambda)B(b,\lambda)\rho(\lambda)\,d\lambda$ . Why: this factorized correlation is the precise mathematical meaning of "no non-local influence and no conspiracy" — and it is exactly the assumption from which Bell derives his inequality in Bell's Theorem and the CHSH Inequality.

Step 4. Set up the target. A factorizable model of the singlet must reproduce $\langle\sigma_a\sigma_b\rangle=-\cos(a-b)$ (from The EPR Paradox and Local Realism). The question Bell answers: can any choice of $A(a,\lambda),B(b,\lambda),\rho(\lambda)$ do it for all angles at once? Why: the correlation is fixed by quantum mechanics; the game is whether local instructions can fake it. We will see in Example 2 that they cannot — the CHSH combination they allow tops out at $2$ , below the quantum $2\sqrt2$ . *The response functions and correlation forms are pinned in `verify_aq-qm-hidden-variable-theories.py`: the explicit local model $A(a,\lambda)=\mathrm{sign}(\hat a\cdot\hat\lambda)$ is built and its correlation computed.*

Go deeper · Derivation B · 2 minVon Neumann's flawed impossibility proof — the additivity trap

Derivation BVon Neumann's flawed impossibility proof — the additivity trap

Here is the history-that-is-physics crown: exactly why the most famous no-go theorem in the subject was wrong, and how to see the flaw in one line. Getting this right corrects thirty years of received wisdom. The tool is the elementary spectrum of Pauli operators.

Step 1. What von Neumann assumed. His 1932 proof required that any valid state assignment obey $\langle A+B\rangle=\langle A\rangle+\langle B\rangle$ for all observables $A,B$ — including non-commuting ones — and demanded this hold for the hypothetical hidden-variable "dispersion-free" states (each individual $\lambda$ ) as well as for ensembles. Why: additivity of expectation values is true and unremarkable for ensemble averages in quantum mechanics; von Neumann's slip was to impose it on the value assignment of a single hidden state.

Step 2. Why that is illegitimate for individual outcomes. Consider spin- $\tfrac12$ : $\sigma_x$ and $\sigma_z$ each have eigenvalues $\pm1$ , so any dispersion-free state must assign each of them a value in $\{\pm1\}$ . But $\sigma_x+\sigma_z$ has eigenvalues $\pm\sqrt2$ . Why: a genuine measurement of the observable $\sigma_x+\sigma_z$ returns $\pm\sqrt2$ — but no sum of two individually-assigned $\pm1$ values can equal $\pm\sqrt2$ (such a sum lands in $\{-2,0,+2\}$ ). Additivity of individual values is simply false for incompatible observables, because $\sigma_x+\sigma_z$ is measured by a different apparatus than either alone.

Step 3. The verdict. Von Neumann's axiom is satisfied by quantum ensembles but is an unphysical requirement on hidden states; dropping it dissolves the "impossibility." Why: a hidden-variable theory is free to make $A(a,\lambda)$ non-additive across incompatible settings — indeed it must, since those settings correspond to incompatible experiments. Von Neumann proved the impossibility of a class of theories nobody sensible would propose.

Step 4. The rehabilitation. Grete Hermann identified this exact gap in 1935 (buried in a philosophy journal, unnoticed for decades); David Bohm constructed an explicit, empirically adequate hidden-variable theory in 1952 — a living counterexample to a "proof" of impossibility; John Bell, in a 1966 review, spelled the flaw out and redirected the question to locality, where the real obstruction lives. Why: this is the correction to the historical record — hidden variables were declared dead on a technicality, and the field lost thirty years before Bell asked the productive question. The surviving no-go is not von Neumann's but Bell's, and it forbids only local hidden variables. *Verified: $\mathrm{eig}(\sigma_x+\sigma_z)=\{\pm\sqrt2\}$ , not $\{\pm2\}$ — the additivity of individual $\pm1$ assignments is arithmetically impossible.*

Limiting cases

Model class	Local?	Deterministic?	Killed by Bell?
Local hidden variables (Bell factorizable)	yes	either	yes
de Broglie–Bohm pilot wave	no	yes	no (nonlocal)
Superdeterministic	(violates measurement independence)	yes	no (loophole)
Contextual (Kochen–Specker)	value depends on context	either	different no-go
Orthodox QM (no hidden variables)	—	no	not applicable

Only the top row is refuted by Bell. The lesson is that "hidden variables are impossible" is false; "local hidden variables are impossible" is the theorem.

Common trap“Von Neumann proved hidden variables are impossible.”

He did not — and treating his 1932 result as a valid no-go is the classic mistake this article repairs. His proof assumed the additivity $\langle A+B\rangle=\langle A\rangle+\langle B\rangle$ holds for individual hidden states even when $A,B$ don't commute. That is false: $\sigma_x,\sigma_z$ take values $\pm1$ but $\sigma_x+\sigma_z$ takes values $\pm\sqrt2$ , which no sum of $\pm1$ 's can produce, because measuring $\sigma_x+\sigma_z$ is a physically different experiment. Bohm's 1952 theory is an explicit working hidden-variable model, so the impossibility claim is simply refuted by example. The genuine no-go is Bell's (1964), and it targets only local hidden variables. Cite von Neumann's theorem as a cautionary tale, never as a proof.

Worked examples

1 · An explicit local model and where it breaks (confidence builder)

Build the simplest local hidden-variable model for the singlet. Enumerate: let $\lambda$ be a random unit vector $\hat\lambda$ (the shared "instruction"), and let Alice answer $A(\hat a,\lambda)=\mathrm{sign}(\hat a\cdot\hat\lambda)$ , Bob answer $B(\hat b,\lambda)=-\mathrm{sign}(\hat b\cdot\hat\lambda)$ (opposite, to build in anti-correlation), with $\hat\lambda$ uniform. Apply: averaging over $\hat\lambda$ gives the model correlation $E_{\rm LHV}(\hat a,\hat b)=-1+\tfrac{2}{\pi}\,\theta_{ab}$ (a straight-line "triangle" in the angle $\theta_{ab}$ ), which we confirmed by Monte Carlo against the analytic form. Boundary number: compare with quantum mechanics, $E_{\rm QM}=-\cos\theta_{ab}$ . At $\theta=0$ both give $-1$ ; at $\theta=\pi$ both give $+1$ — the model matches at the special angles. But at $\theta=\pi/4$ the model gives $E_{\rm LHV}=-1+\tfrac{2}{\pi}\cdot\tfrac{\pi}{4}=-0.5$ , while quantum mechanics gives $E_{\rm QM}=-\cos(\pi/4)=-0.707$ . The local instructions are not curved enough — they interpolate linearly where quantum mechanics dips as a cosine, and that gap between $-0.5$ and $-0.707$ is precisely the room a Bell inequality exploits. No cleverer local model closes it; the linearity is forced by locality.

2 · The CHSH gap: local ceiling 2 versus quantum 2√2 (exam level)

Evaluate the CHSH combination $S=E(a,b)-E(a,b')+E(a',b)+E(a',b')$ . Enumerate the local bound first: for any deterministic local assignment, each $E$ is a product of predetermined $\pm1$ values, so $S=A_a(B_b-B_{b'})+A_{a'}(B_b+B_{b'})$ ; since $B_b\pm B_{b'}$ is $0$ or $\pm2$ (and the two are mutually exclusive), $|S|\le2$ — checked by brute enumeration of all $2^4=16$ sign assignments, whose maximum is exactly $2$ . Stochastic and mixed local models are convex combinations, so they too obey $|S|\le2$ . Apply the quantum value: with the singlet at the optimal angles $a=0,\ a'=\tfrac{\pi}{2},\ b=\tfrac{\pi}{4},\ b'=\tfrac{3\pi}{4}$ (a $22.5°$ spacing), each $E=-\cos(\text{angle})$ gives $|S|=2\sqrt2\approx2.828$ . Boundary number: $2\sqrt2>2$ — quantum mechanics overshoots the local ceiling by $41\%$ , and $2\sqrt2$ is the maximum any quantum model can reach (Tsirelson's bound). We verified both numbers directly: the LHV enumeration caps at $2$ ; the singlet expectation at those angles equals $2\sqrt2$ to machine precision. This gap is the content of Bell's theorem — the derivation of the inequality itself, and the proof that $2\sqrt2$ is optimal, belong to Bell's Theorem and the CHSH Inequality; here the point is that the local hidden-variable program has a hard, computable ceiling that reality exceeds.

Going further

The hidden-variable program set the stage; the decisive verdict is Bell's Theorem and the CHSH Inequality, which derives the inequality whose local ceiling ( $2$ ) and quantum value ( $2\sqrt2$ , Tsirelson) we verified here, and the experiments that settled it are in Experimental Tests and Violation of Bell Inequalities. The premises Bell attacks were laid out in The EPR Paradox and Local Realism. Among survivors: the fully nonlocal deterministic theory is De Broglie–Bohm Pilot Wave Theory; the value-assignment obstruction that holds even without locality is The Kochen–Specker Theorem and Contextuality; the single-run refutation is The GHZ State and All-or-Nothing Nonlocality; and the broader interpretational stances are The Copenhagen Interpretation, The Many-Worlds Interpretation, and Relational and QBist Interpretations. The singlet correlations used throughout are from The EPR Paradox and Local Realism and The Bell States and Maximally Entangled Pairs.

If hidden variables aren't impossible, why don't physicists use them?

Because the surviving versions pay a steep price. Bohmian mechanics works but is explicitly nonlocal — the guiding wave depends instantaneously on all particles' positions — which most physicists find no more comfortable than the orthodox picture, and it adds unobservable trajectory structure with no new predictions. Superdeterminism must correlate the hidden variable with every future measurement choice, which many regard as abandoning experimental science. So hidden variables are permitted but not compelling: they buy determinism at the cost of locality or of measurement independence, and give no empirical advantage. Orthodox quantum mechanics, taking the correlations as fundamental, is simpler.

Is the local model's failure a matter of not being clever enough?

No — it is a theorem, not a failure of imagination. Bell proved that every local factorizable model, deterministic or stochastic, obeys the same inequality ( $S\le2$ ), regardless of how the response functions $A(a,\lambda)$ and distribution $\rho(\lambda)$ are chosen. The linear "triangle" correlation of the simplest model is not a fixable quirk; the boundedness of the CHSH combination is forced by factorizability alone. Quantum mechanics reaches $2\sqrt2$ , so no local model — however ingenious — can reproduce it. The derivation is in Bell's Theorem and the CHSH Inequality.

Didn't Kochen–Specker also rule out hidden variables?

It rules out a different class: non-contextual hidden variables, where each observable has a fixed value independent of which compatible observables are measured alongside it. Kochen–Specker (1967) shows no such consistent value-assignment exists for systems of dimension three or more — a no-go that needs no locality assumption, only non-contextuality. It complements Bell: Bell attacks locality, Kochen–Specker attacks context-independence. Contextual hidden-variable theories survive both, at the cost of making an observable's value depend on the measurement context. See The Kochen–Specker Theorem and Contextuality.

What is the difference between parameter independence and outcome independence?

They are the two halves of "locality" in a Bell model. Parameter independence: the probability distribution of Alice's outcome does not depend on Bob's setting $b$ — this is the piece whose violation would allow signalling. Outcome independence: given the hidden variable $\lambda$ , Alice's and Bob's outcomes are statistically independent — the correlations are entirely mediated by the shared $\lambda$ . Their conjunction is Bell factorizability. Quantum mechanics respects parameter independence (no signalling) but violates outcome independence — the outcomes stay correlated even after conditioning on any $\lambda$ — which is why it can exceed the local bound without allowing faster-than-light communication.

Was Grete Hermann's contribution really ignored?

Yes, for decades. Hermann published her critique of von Neumann's additivity assumption in 1935, in a philosophy journal associated with the Fries school, and identified essentially the point Bell would make thirty years later. Working outside the physics mainstream and in a philosophical idiom, her work went unread by physicists, and von Neumann's authority carried the day. Her priority was recognized only in the 1970s and more fully in recent scholarship. It is a cautionary tale about how the authority of a great mathematician can suppress a correct objection, and it is why the honest history matters here.

Does the failure of local hidden variables mean the universe is nonlocal?

Not necessarily — it means you cannot keep both locality and pre-existing definite values (local realism). You may keep locality by abandoning realism (as orthodox quantum mechanics does — outcomes are not predetermined), or keep realism by abandoning locality (as Bohm does). Quantum mechanics itself is no-signalling (Bob's marginal is untouched, per The EPR Paradox and Local Realism), so it never contradicts relativity, but its correlations cannot be explained by any local story of pre-set values. "Nonlocal correlations without nonlocal signalling" is the precise, subtle situation — which is exactly why it took a theorem to pin down.