This episode is the second part of Tim Maudlin’s lecture on Bell’s theorem and quantum nonlocality, explaining how Bell used Einstein’s own EPR argument to prove that locality — the idea that distant events cannot instantaneously influence each other — is mathematically incompatible with the predictions of quantum mechanics, forcing the conclusion that the physical world is fundamentally nonlocal.
EPR Completeness Criterion and Einstein’s Reality Criterion
The EPR paper asks whether the quantum mechanical wave function provides a complete description of physical reality, defining completeness as: every element of physical reality must have a counterpart in the physical theory.
Einstein’s reality criterion: if, without disturbing a system, we can predict with certainty (probability = 1) the outcome of a measurement, then there exists an element of physical reality corresponding to that quantity.
This criterion is a sufficient condition for reality, not a necessary one, and requires both perfect predictability and no disturbance of the system.
Einstein’s objection to quantum mechanics was not about indeterminism (“God playing dice”) or superluminal signaling, but about “spooky action at a distance” — the idea that a measurement in one location could instantaneously change the physical state of a distant system, which he saw as a threat to relativity’s rejection of absolute simultaneity.
Locality Implies Determinism
In the EPR argument, locality (no action at a distance) plays a crucial logical role: it justifies the claim that separating Alice and Bob’s labs at spacelike separation ensures neither disturbs the other’s system.
If locality holds, the perfect correlations predicted by quantum mechanics (EPR correlations) force the conclusion that the outcomes must be predetermined — determinism is inferred from locality, not assumed.
Bell emphasized: “What is held sacred is the principle of local causality… determinism is not a presupposition of the analysis.”
Bohm’s Spin Reformulation
David Bohm’s 1951 textbook reformulated the EPR argument using spin measurements instead of position/momentum, which simplified the mathematics and introduced a continuous infinity of measurable spin directions (via Stern-Gerlach magnet orientations).
This reformulation was essential for Bell’s theorem: with spin, Alice and Bob can measure at arbitrary relative angles, producing imperfect but statistically predictable correlations that become the basis for Bell’s inequality.
Bohm himself was initially a Copenhagen adherent; after discussing his book with Einstein in 1951, he abandoned Copenhagen and developed the pilot wave (hidden variables) theory in 1952.
Stern-Gerlach Experiments and Spin Quantization
A Stern-Gerlach apparatus uses an inhomogeneous magnetic field to measure spin: classically, a spinning charged particle would be deflected continuously depending on orientation; quantum mechanically, spin-½ particles show only two discrete outcomes (up/down) regardless of magnet orientation.
The quantum prediction is binary: each run yields either “spin up” or “spin down” along the chosen axis, with probabilities determined by the quantum state.
Actual Stern-Gerlach data (sent to Bohr on a postcard) shows the discrete splitting with a gap in the middle, confirming quantization.
Entangled Spin States and the Singlet State
Two-particle spin states include product states (independent spins) and entangled states (no independent spin state for either particle).
The singlet state — the spin analog of the EPR state — is (1/√2)(|↑⟩ₐ|↓⟩ᵦ − |↓⟩ₐ|↑⟩ᵦ), with a crucial minus sign giving it rotational symmetry: perfect anti-correlation holds for any shared measurement direction.
If Alice and Bob measure along the same axis, outcomes are perfectly anti-correlated (100% opposite), enabling the EPR reality criterion: Alice can predict Bob’s outcome with certainty without disturbing his particle (assuming locality).
Pilot Wave Theory and Its Nonlocality
Bohm’s 1952 pilot wave theory adds particle positions as “additional variables” guided by the wave function via a deterministic guidance equation; the wave function never collapses.
It reproduces all non-relativistic quantum predictions but is manifestly nonlocal: the motion of Bob’s particle can depend instantaneously on Alice’s magnet orientation, even at arbitrary distance.
Einstein rejected it precisely because it retained the nonlocality he sought to eliminate; determinism alone was not his goal.
Configuration Space vs. Physical Space
The pilot wave theory’s dynamics can be written in configuration space (3N dimensions for N particles), where the entire system is a single point.
Maudlin argues this is a mathematical trick, not physical space: locality in configuration space does not correspond to locality in the 3D space where labs, magnets, and detectors exist.
A theory with only one particle in high-dimensional configuration space (“the marvelous point theory”) cannot be empirically connected to our experience without cheating.
Bell’s Theorem: The 1964 Paper
Bell picked up where EPR left off: EPR proved that if locality holds, the wave function is incomplete and any completion must be deterministic. Bell asked: can a local deterministic theory reproduce all quantum predictions, including imperfect correlations at offset angles?
Von Neumann’s 1932 “no hidden variables” proof was widely cited but flawed (Grete Hermann and Einstein identified the faulty assumption); Bell called it “foolish.”
Kochen-Specker contextuality was also examined by Bell and found to involve only “tame” contextuality (different apparatuses give different results), not the spooky action at a distance that Bell targeted.
The Core Logic: Contrapositive of EPR
EPR: (Quantum predictions accurate ∧ Locality) → Wave function incomplete.
Contrapositive: (Quantum predictions accurate ∧ Wave function complete) → Locality violated.
Bell’s strategy: assume locality and quantum predictions are correct, then show the required deterministic local hidden variables cannot reproduce the quantum statistics for offset measurement angles.
Offset Angles and the Cosine-Squared Prediction
For the singlet state, quantum mechanics predicts the probability of disagreement between Alice and Bob’s outcomes is cos²(θ/2), where θ is the angle between their magnet orientations.
At θ = 0°: 100% disagreement (perfect anti-correlation).
At θ = 60°: 75% disagreement.
At θ = 120°: 25% disagreement.
At θ = 90°: 50% disagreement (no correlation).
The 75%/25% Impossibility Proof (Nick Herbert’s Version)
Restrict Alice to {0°, +60°} and Bob to {0°, −60°}. Four experimental conditions: (0,0), (0,−60), (+60,0), (+60,−60).
Locality + perfect anti-correlation at 0° forces predetermined “instruction sets” for each particle at each angle.
To get 75% disagreement at 60° offset, the +60° outcomes must differ from 0° outcomes 25% of the time (similarly for −60°).
But then at 120° offset (+60° vs −60°), the outcomes must disagree at least 50% of the time — contradicting the quantum prediction of 25% disagreement.
No assignment of predetermined outcomes can satisfy all three statistical constraints simultaneously. Locality is mathematically impossible.
Assumptions: Only Locality and Statistical Independence
Bell’s proof requires only two assumptions:
Einstein locality: no causal influence between spacelike separated events.
Statistical independence: the choice of measurement settings is statistically independent of the particle states at the source (ensured by physical randomizers like coin flips, digits of π, or chaotic devices — not free will).
“Free will” is a red herring; Conway’s “free will theorem” merely reproves Bell’s theorem with misleading language.
Denying statistical independence (superdeterminism) undermines all experimental science and is an unreasonable loophole.
Uses three particles in a GHZ state sent to Alice, Bob, Charlie, each choosing between X and Z spin measurements.
Quantum mechanics makes certain (100%) predictions for four experimental conditions:
All three measure X → odd number of “up” outcomes.
Two measure Z, one measures X → even number of “up” outcomes.
Mermin’s “cocktail napkin” proof: assume predetermined outcomes (U/D) for each of the six settings. The parity constraints (odd, even, even, even) require an odd total number of U’s across all four conditions, but each setting appears in exactly two conditions, forcing an even total — contradiction.
This proves nonlocality without statistics, using only perfect correlations.
Loopholes and Unreasonable Doubts
Matrix/superdeterminism loophole: claim all experimental results are illusory (e.g., simulated by aliens) while the underlying physics is local. This abandons empirical science entirely — “the cure is worse than the disease.”
Statistical independence denial: claiming no physical randomizer can achieve true independence. This rejects the methodology of randomized experiments across all science.
These are logically possible but epistemically irrational; they protect locality by destroying the basis for believing any physical theory.
Conclusion
Bell’s theorem proves that any theory reproducing quantum predictions must be nonlocal. The experimental violations of Bell inequalities (Clauser, Aspect, Zeilinger — Nobel 2022) confirm the quantum predictions.
Nonlocality is not superluminal signaling; it is a fundamental physical dependency between spacelike separated events.
Maudlin’s view: the proof of nonlocality is “the most astonishing proof of any physical fact in the history of mankind.” Relativity’s spacetime structure likely needs revision (e.g., a preferred foliation) to accommodate this fact. Einstein would have been upset, but “distress is not an argument.”
