The EPR Paradox

Bohr's reply, and why it did not settle matters

Bohr responded four months later in Physical Review, under the same title. His answer denies EPR's criterion of reality: the phrase "without in any way disturbing a system" is ambiguous, he argued, because choosing what to measure on one particle fixes "the very conditions which define the possible types of predictions regarding the future behaviour of the system." There is no mechanical disturbance at a distance — but there is also no meaningful attribution of properties independent of the whole experimental arrangement.

The reply was accepted at the time as decisive and is now widely regarded as obscure. What matters historically is that it left the question unresolved as a matter of physics. It took Bell to make it decidable.

Einstein's own version was cleaner

Einstein was dissatisfied with the published paper. He wrote to Schrödinger in June 1935 that the essential point had been "smothered by the formalism" — he had not written the text; Podolsky had. His own later statements dropped the position–momentum machinery and pressed a simpler claim: either the state of Bob's particle depends on what Alice does, which he found unacceptable, or the description was incomplete.

That framing is the one Bell tested, and Bell was clear that Einstein's reasoning was sound as far as it went. What Bell added was that the conclusion Einstein drew — completability by local hidden variables — is refutable, and is refuted.

The loopholes, and why they mattered

For decades a determined local realist could resist the experiments by pointing at gaps. These were not crankery; the experimenters took them seriously and spent forty years closing them.

The detection (fair-sampling) loophole. Early photon detectors caught only a few percent of the pairs. If the undetected pairs were systematically different, the observed sample could violate the inequality while the full ensemble did not. Closing it requires detection efficiency above a threshold — around 2/3 for the standard CHSH setup, lower with optimised states — which needed transition-edge-sensor detectors.

The locality (communication) loophole. If a signal could travel from Alice's analyser to Bob's detector before the result was registered, local influence is not excluded. Aspect's 1982 switching experiment attacked this; closing it fully requires the setting choice and the distant outcome to be spacelike separated, which means fast random switching over long baselines.

The freedom-of-choice loophole. The setting choices must be independent of the hidden variables. Later experiments have driven the choices with distant quasar light (Handsteiner and colleagues, 2017) and with human button-presses from around the world (the "BIG Bell Test", 2018), pushing any conspiracy back billions of years or into human volition.

The 2015 experiments closed detection and locality at the same time, which no earlier experiment had. This is why they are the ones that count.

What cannot be closed. Superdeterminism — the position that the source and the settings are correlated by common past causes — is not refutable by any experiment, because it denies the assumption that lets experiments mean anything. Gerard 't Hooft has defended it seriously. Most physicists regard the price as prohibitive: it undermines the possibility of controlled experiment generally.

So what is the status?

The status of this entry is resolved, and that is a claim about the question EPR actually asked. Is quantum mechanics completable by a local hidden-variable theory? No. That is settled, by experiment, and it is one of the few metaphysical-sounding questions ever decided in a laboratory.

What is not settled is which assumption to drop instead. That is the measurement problem, and it belongs to Schrödinger's cat.

1927. At the fifth Solvay Conference, Einstein raises an objection involving a single particle diffracting through a slit and hitting a screen: if the particle is detected at one place, the wave function elsewhere must vanish instantly. It is the germ of the EPR argument, applied to one particle.

1933–1935. Einstein, now at the Institute for Advanced Study in Princeton, works with Podolsky and Rosen. The paper is submitted in March 1935 and appears on 15 May.

Summer 1935. The response is immediate. The New York Times carries the story on 4 May, before publication, under the headline "Einstein Attacks Quantum Theory" — Einstein was annoyed at the leak. Bohr drops his current work to reply; his paper appears in October. Schrödinger writes his own response, publishing "Die gegenwärtige Situation in der Quantenmechanik" later that year, in which he names entanglement and introduces the cat. Einstein complains to Schrödinger in a letter of 19 June that Podolsky's drafting had buried the point.

1936–1951. Little progress. The consensus is that Bohr won; foundational work is professionally unrewarding.

1932/1935. John von Neumann's 1932 book contains a proof that hidden-variable theories are impossible. Grete Hermann publishes a correct criticism of it in 1935, identifying the unjustified linearity assumption. Her work goes almost entirely unnoticed for decades, and von Neumann's proof deters research on hidden variables for a generation.

1952. David Bohm constructs an explicit hidden-variable theory reproducing all quantum predictions, demonstrating by counterexample that von Neumann's proof cannot be doing what it was thought to do. Bohm's theory is manifestly non-local.

1964. Bell publishes "On the Einstein Podolsky Rosen Paradox" in the first issue of a short-lived journal, Physics. It is barely noticed for years, in part because the journal folds. Bell's stated motivation is Bohm's theory: given that a hidden-variable theory exists but is non-local, he asks whether the non-locality is avoidable. It is not.

1966. Bell publishes his critique of von Neumann's impossibility proof — written earlier than the 1964 paper but delayed in publication — vindicating Hermann's point.

1969. CHSH give the testable inequality.

1972. Freedman and Clauser perform the first test. Clauser had hoped for the opposite result and later described his disappointment.

1982. Aspect's Orsay experiments.

1998. Weihs and colleagues in Innsbruck achieve strict spacelike separation of the setting choices with fast random switching over 400 metres.

2015. The three loophole-free experiments.

2017–2018. Cosmic Bell tests with quasar light; the BIG Bell Test with 100,000 human participants generating setting choices.

2022. The Nobel Prize in Physics to Aspect, Clauser and Zeilinger, "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science."

A metaphysical question was decided by experiment. "Are the properties of things fixed independently of measurement, in a world where influences travel no faster than light?" sounds like something for philosophers to argue over indefinitely. Bell converted it into an inequality and experimentalists measured it. There are very few episodes like this in the history of thought, and it is the strongest available argument that foundational questions can be worth physicists' time.

Bell rescued a discredited research programme. Von Neumann's flawed proof had closed off hidden variables for thirty years. Bell's work reopened foundations as a legitimate field and directly produced the tools of quantum information.

Entanglement became a resource. EPR treated the correlation as a reductio — evidence that something was wrong. It is now a quantity you measure, purify, distil, transmit and budget for. Quantum key distribution, teleportation, superdense coding, device-independent randomness and measurement-based quantum computing all consume entanglement the way a classical computer consumes energy. The 2022 Nobel citation explicitly names "pioneering quantum information science" alongside the Bell tests, because the two are the same story.

It sharpened what relativity forbids. Before Bell, "relativity forbids faster-than-light influence" was a vague slogan. Afterwards the statement is precise: relativity forbids faster-than-light signalling, a constraint quantum mechanics obeys exactly, while permitting correlations that no local model can reproduce. That distinction is now standard equipment in physics.

Device-independent cryptography. This is EPR's argument turned into a security guarantee. If two parties observe a CHSH value above 2, they can certify — from the statistics alone, without trusting their own hardware — that their outcomes are not predetermined and cannot be known to an eavesdropper. The protocols of Artur Ekert (1991) and their device-independent descendants derive their security from the impossibility of local hidden variables. A vendor's backdoored device cannot fake a Bell violation.

Certified randomness. Bell violation implies genuine unpredictability, not merely apparent unpredictability. NIST began publishing a public randomness beacon seeded by loophole-free Bell tests in 2018. The claim being made is exactly Bell's: no pre-existing record could have determined these bits.

Quantum networking. The Delft experiment's entanglement-swapping technique is the basic operation of a quantum repeater, the component that would allow entanglement distribution over continental distances. Bell violation is the standard acceptance test for a link.

Persistent public misunderstanding. Entanglement is routinely described in popular accounts as instantaneous communication, and occasionally sold as such. The no-communication theorem is not a subtlety; it is a two-line calculation, and it holds for every operation Alice could perform. Any claim of faster-than-light messaging via entanglement is wrong, and can be recognised as wrong without examining the details of the proposal.

Benchmarking quantum hardware. Bell inequality violation is a routine diagnostic in laboratories building superconducting and trapped-ion processors — a fast, assumption-light check that two qubits are genuinely entangled rather than classically correlated by noise. Einstein's objection has become a piece of test equipment.

Setting up

Alice measures observable $A(a)$ with setting $a$, Bob measures $B(b)$, each returning $\pm 1$. Write the correlation

$$E(a,b) = \langle A(a) B(b) \rangle.$$

A local hidden-variable model asserts that there is a variable $\lambda$ with distribution $\rho(\lambda)$ such that

$$E(a,b) = \int A(a,\lambda), B(b,\lambda), \rho(\lambda), d\lambda,$$

where $A(a,\lambda) = \pm 1$ depends only on Alice's setting and $\lambda$, and $B(b,\lambda) = \pm 1$ only on Bob's setting and $\lambda$. Three assumptions are packed in here:

• Outcome independence — Alice's result does not depend on Bob's result.
• Parameter independence — Alice's result does not depend on Bob's setting.
• Measurement independence — $\rho(\lambda)$ does not depend on $a$ or $b$.

The conjunction of the first two is Bell locality. The third is what superdeterminism denies.

The CHSH inequality

Clauser, Horne, Shimony and Holt (1969) gave the form used in every modern experiment. Alice chooses between settings $a, a'$; Bob between $b, b'$. Define

$$S = E(a,b) - E(a,b') + E(a',b) + E(a',b').$$

For any local hidden-variable model, note that for fixed $\lambda$,

$$A(a,\lambda)\big[B(b,\lambda) - B(b',\lambda)\big] + A(a',\lambda)\big[B(b,\lambda) + B(b',\lambda)\big] = \pm 2,$$

because $B(b,\lambda)$ and $B(b',\lambda)$ are each $\pm 1$, so one bracket is $0$ and the other is $\pm 2$. Averaging over $\rho(\lambda)$ preserves the bound:

$$|S| \le 2.$$

This is the CHSH inequality. It assumes nothing about the physics of the particles — only that the outcomes are functions of local settings and a shared variable.

The quantum prediction

For the singlet state

$$|\psi^-\rangle = \tfrac{1}{\sqrt{2}}\big(|{\uparrow}\rangle_A|{\downarrow}\rangle_B - |{\downarrow}\rangle_A|{\uparrow}\rangle_B\big),$$

measuring spin along unit vectors $\hat{a}$ and $\hat{b}$ gives

$$E(a,b) = -\hat{a}\cdot\hat{b} = -\cos\theta_{ab}.$$

Choose the settings at relative angles of 45°: $\hat a$ and $\hat b$ at 22.5°, $\hat a$ and $\hat b'$ at 67.5°, and so on. Then

$$|S| = 2\sqrt{2} \approx 2.828.$$

This is Tsirelson's bound, the maximum any quantum state can achieve. It sits comfortably above the classical $2$ and well below the algebraic maximum of $4$ that a general no-signalling theory would permit — the value attained by the hypothetical "PR box" of Popescu and Rohrlich. Quantum mechanics is non-local, but not maximally so, and why nature stops at $2\sqrt{2}$ is itself an active research question.

The no-communication theorem

Let the joint state be $\rho_{AB}$ on $\mathcal{H}_A \otimes \mathcal{H}_B$. Alice performs any operation she likes: a measurement, a unitary, a general quantum channel. All such operations are described by a completely positive trace-preserving map $\mathcal{E}_A \otimes \mathcal{I}_B$ — the identity on Bob's factor, because her apparatus acts only on her subsystem.

Bob's accessible state is the reduced density matrix $\rho_B = \operatorname{Tr}A(\rho{AB})$. After Alice acts,

$$\rho_B' = \operatorname{Tr}_A!\big[(\mathcal{E}_A \otimes \mathcal{I}B)(\rho{AB})\big] = \operatorname{Tr}A(\rho{AB}) = \rho_B,$$

where the second equality uses trace preservation of $\mathcal{E}_A$. For a measurement with Kraus operators ${M_i}$ satisfying $\sum_i M_i^\dagger M_i = \mathbb{I}$, the same computation reads

$$\rho_B' = \sum_i \operatorname{Tr}A!\big[(M_i \otimes \mathbb{I})\rho{AB}(M_i^\dagger \otimes \mathbb{I})\big] = \operatorname{Tr}A!\Big[\Big(\sum_i M_i^\dagger M_i \otimes \mathbb{I}\Big)\rho{AB}\Big] = \rho_B .$$

Bob's density matrix is completely unchanged. Since $\rho_B$ determines the statistics of every measurement Bob can make, no experiment on his side can detect that Alice did anything at all, let alone what she chose. Signalling is impossible.

Note where the tension is resolved. The correlations violate local causality; the marginals do not move. Non-locality lives entirely in the joint statistics, which require both records to see. Reconciling these two facts — non-local correlation with strict no-signalling — is sometimes called peaceful coexistence with relativity, a phrase of Abner Shimony's.

Why quantum teleportation is not a counterexample

Teleportation transfers an unknown state $|\psi\rangle$ from Alice to Bob using a shared entangled pair. Alice performs a Bell-basis measurement on $|\psi\rangle$ and her half, obtaining two classical bits, and Bob applies one of four Pauli corrections. Without those two bits Bob's state is $\tfrac{1}{2}\mathbb{I}$ — maximally mixed, carrying nothing. The classical bits travel at or below light speed. The protocol consumes entanglement and a classical channel and beats neither of their limits.

Experimental numbers

Experiment	Year	System	Result	Loopholes closed
Freedman & Clauser	1972	Ca cascade photons	~6σ violation	none simultaneously
Aspect, Grangier, Roger	1982	Ca cascade photons	$S = 2.697 \pm 0.015$	—
Aspect, Dalibard, Roger	1982	time-varying analysers	violation	locality (partially)
Weihs et al.	1998	fibre-separated photons, 400 m	violation	locality (strictly)
Rowe et al.	2001	trapped ions	violation	detection
Hensen et al.	2015	NV centres, 1.3 km, Delft	$S = 2.42 \pm 0.20$	detection + locality
Giustina et al.	2015	photons, Vienna	$p < 3.7\times10^{-31}$	detection + locality
Shalm et al.	2015	photons, NIST	$p < 2.3\times10^{-7}$	detection + locality
Handsteiner et al.	2017	quasar-seeded settings	violation	freedom of choice (to ~600 yr)

The Delft result has a large error bar because entanglement swapping through NV centres is slow — a few hundred trials. Its significance is not the precision but that it is simultaneously airtight on both major loopholes, the first experiment of which that was true.

Which assumption does each interpretation give up?

Bell's theorem is a disjunction, and the honest statement of the conclusion names the disjuncts.

• Copenhagen and standard textbook quantum mechanics reject the reality criterion. An unmeasured spin component has no value to be pre-set, so there is nothing for a hidden variable to record. Locality of signals is preserved by the no-communication theorem.
• Bohmian mechanics keeps definite values and rejects parameter independence. The guiding equation for particle $i$ depends on the instantaneous configuration of all particles; Alice's setting really does affect Bob's particle's trajectory. Non-signalling survives because the distribution of initial positions is in quantum equilibrium, $\rho = |\psi|^2$, which hides the non-locality statistically.
• Everettian / many-worlds accounts reject the assumption that measurements have single outcomes. The dynamics remain strictly local and the correlations only appear when the two branches are compared — which requires a light-speed channel. Advocates such as David Deutsch and David Wallace argue this makes Everett the only fully local interpretation, at the cost of a many-branched ontology and a hard problem about probability.
• Objective collapse (GRW, CSL) keeps single definite outcomes and gives up exact locality: the collapse of an entangled state is a non-local physical event. Relativistic versions exist (Tumulka's rGRWf) but the programme is not complete.
• QBism and pragmatist views deny that measurement outcomes are agent-independent facts, so there are no distant matters of fact for anything to influence. Locality is preserved by relocating the outcomes into an agent's experience.
• Superdeterminism rejects measurement independence: $\rho(\lambda)$ depends on $a$ and $b$. Locality and determinism both survive, but the inference from experiment to theory is compromised in general.
• Retrocausal accounts (Huw Price, Ken Wharton, and time-symmetric approaches) allow the later setting choice to influence the earlier state, preserving a form of locality at the cost of a fixed causal arrow.

No experiment distinguishes among these. This is the live frontier; the death of local realism is not.