Quantum Mechanics 101 · Unit 5: Entanglement & Reality

Quantum Academy · Chapter 21: Bell's Theorem & the End of Local Realism

The experiment that let nature vote — and nature voted quantum

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Instructions. Work through these questions with the lesson closed first, then check yourself against it. For calculations, show every step and keep three significant figures. Use c = 2.998×10⁸ m/s, the CHSH combination S = E(α₀, β₀) + E(α₀, β₁) + E(α₁, β₀) − E(α₁, β₁), the quantum correlation E = cos 2(α − β) for polarisation-entangled photons, and the game relation P = 1/2 + S/8. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Local realism
2.Hidden variable
3.Locality
4.Measurement independence
5.Bell inequality
6.CHSH inequality
7.Correlation coefficient
8.Tsirelson bound
9.CHSH game
10.Loophole
11.Superdeterminism
12.No-signalling theorem

A.A two-player game in which separated players answer random questions; classical strategies win at most 75 percent, entangled ones about 85.4 percent.
B.An unverified assumption in a Bell test, such as fair sampling or the timing of settings, that a local-realist model could exploit.
C.The maximum quantum value of the CHSH combination, twice √2 ≈ 2.83, proved by Boris Tsirelson in 1980.
D.The result that one party's measurement statistics are unaffected by a distant party's choices, so entanglement cannot transmit information.
E.The hypothesis that measurement settings and hidden variables are correlated through a common past, evading Bell's theorem by denying free choice.
F.The joint assumption that measurement outcomes are fixed in advance by properties the system carries and that no influence travels faster than light.
G.A limit on correlations between distant measurements that every local hidden-variable theory must obey and quantum mechanics can violate.
H.The Clauser–Horne–Shimony–Holt form of Bell's inequality (1969): a combination S of four correlations must lie between −2 and +2.
I.A hypothetical property, beyond the quantum state, that would determine measurement outcomes in advance; conventionally written λ.
J.Matching results minus mismatched results, divided by the total, for one pair of settings; it runs from −1 to +1.
K.The assumption, also called free choice, that measurement settings are statistically independent of the hidden variables of the measured system.
L.The principle that a measurement setting or event cannot influence a distant outcome faster than light could travel between them.

Part B Questions

1.
Which of these is NOT one of the assumptions from which Bell-type inequalities are derived?Circle one
1. AMeasurement outcomes are fixed in advance by properties the particles carry
2. BNo influence travels faster than light between the two stations
3. CMeasurement settings are chosen independently of the hidden variables
4. DQuantum mechanics is a complete description of each particle
2.
For a single pair in a local hidden-variable model, the combination A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ can equal 3.Circle one
TrueFalse
3.
The maximum quantum value of the CHSH combination, twice √2 ≈ 2.83, is called the bound.Fill in the blank
4.
In the CHSH game, what is the highest win rate any classical strategy, including shared randomness, can sustain?Circle one
1. A50 percent
2. B75 percent
3. C85.4 percent
4. D100 percent
5.
If Bob rotates his polariser, the fraction of +1 results that Alice records changes immediately.Circle one
TrueFalse
6.
Which experiment first closed the locality and detection loopholes in the same test?Circle one
1. AFreedman and Clauser, Berkeley, 1972
2. BAspect, Dalibard and Roger, Orsay, 1982
3. CWeihs and colleagues, Innsbruck, 1998
4. DHanson's team, Delft, 2015
7.
In Aspect's 1982 time-varying experiment, the optical switches changed the effective polariser setting roughly every ten .Fill in the blank
8.
Cosmic Bell tests using light from distant quasars have completely closed the freedom-of-choice loophole.Circle one
TrueFalse
9.
Using the gloves-in-boxes picture, explain why perfect agreement at matched settings cannot by itself disprove local realism, and say what Bell added.Short answer
10.
State precisely what the violation of Bell inequalities rules out, and name two ways an interpretation of quantum mechanics can remain consistent with the experiments.Short answer
11.
At one pair of settings, a Bell test records 4,268 pairs with matching results and 732 with mismatched results. Calculate the correlation E.Show your working
Working
Answer:
12.
An experiment measures E(α₀, β₀) = 0.69, E(α₀, β₁) = 0.70, E(α₁, β₀) = 0.68 and E(α₁, β₁) = −0.67. Calculate S, decide whether local realism is violated, and give the equivalent CHSH-game win rate.Show your working
Working
Answer:
13.
A team playing the CHSH game reports an 80.0 percent win rate over a large number of rounds. What value of S does this imply, and is it achievable classically?Show your working
Working
Answer:
14.
Two Bell-test stations are 1.3 km apart. Both start at the same instant. What is the longest time each station may take, from choosing its setting to recording its result, if the two measurements are to be spacelike separated?Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Map the whole family of CHSH angles. Put Alice's settings at α₀ = 0 and α₁ = 2θ, and Bob's at β₀ = θ and β₁ = −θ, using E = cos 2(α − β) for polarisation-entangled photons. (a) Show that S(θ) = 3 cos 2θ − cos 6θ. (b) Writing u = cos 2θ and using cos 6θ = 4u³ − 3u, find the angle that maximises S and the maximum value. (c) Find the range of θ, between 0° and 45°, for which local realism is violated, and the CHSH-game win rate at θ = 30°.

Working

Memory aid

Mnemonic FOIL the gloves, then obey the speed limits. FOIL names the four assumptions behind Bell's inequality: Free choice of settings, One outcome per measurement, Instruction sets of pre-written answers, and Locality. Experiments say at least one letter must be torn off, and each escape route tears off a different one: superdeterminism tears off F, many-worlds tears off O, Bohmian mechanics tears off L, and, roughly speaking, Copenhagen-style and QBist readings tear off I. Then picture a highway sign for the CHSH game: SPEED LIMIT 75 for classical cars, 85 for quantum cars, and no car on this road may carry a message. To convert a win rate into S, subtract 50 and divide by 12.5: 75 gives 2, and 85.4 gives 2.83.

In one line No pre-written script obeying light-speed limits matches quantum correlations: S ≤ 2 classically, twice √2 quantum, and experiments side with quantum.