Quantum Mechanics 101 · Unit 5: Entanglement & Reality

Quantum Academy · Chapter 20: Quantum Entanglement

Two particles, one story — Einstein's 'spooky action at a distance'

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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 cos²θ for the agreement of photon pairs in |Φ+⟩, E = −cos θ for spin singlets, hc = 1239.84 eV·nm, h = 6.626×10⁻³⁴ J·s and c = 2.998×10⁸ m/s. 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.Entanglement
2.Product state
3.Bell states
4.Spin singlet
5.Correlation coefficient
6.EPR argument
7.Element of reality
8.Local hidden-variable theory
9.No-signalling
10.Spontaneous parametric down-conversion
11.Ebit
12.Monogamy of entanglement

A.A joint state that factorises into a definite state for each part; also called a separable state.
B.The four maximally entangled two-qubit states, Φ+, Φ−, Ψ+ and Ψ−, such as |Φ+⟩ = (1/√2)(|00⟩ + |11⟩).
C.The two-particle spin state of total spin zero, (1/√2)(|↑↓⟩ − |↓↑⟩), which gives opposite results along any common axis.
D.One unit of entanglement: a single maximally entangled pair of qubits, such as a Bell state.
E.A theory in which pre-set properties carried by each particle determine its results, with no dependence on distant detector settings.
F.The average product of two ±1 measurement results, equal to P(same) minus P(different); it runs from −1 to +1.
G.The rule that a qubit maximally entangled with one partner cannot be entangled with any other system.
H.In the EPR paper, a physical quantity whose value can be predicted with certainty without in any way disturbing the system.
I.The 1935 argument of Einstein, Podolsky and Rosen that, if locality holds, quantum mechanics is an incomplete description of reality.
J.A property of a joint quantum state of two or more systems that cannot be written as a product of states of the individual systems.
K.Splitting of one pump photon into a signal and an idler photon inside a nonlinear crystal, conserving energy and momentum.
L.The fact that one party's local measurement statistics cannot be changed by anything done to a distant entangled partner.

Part B Questions

1.
Which of these two-qubit states is entangled?Circle one
1. A(1/√2)(|00⟩ + |01⟩)
2. B(1/√2)(|01⟩ + |10⟩)
3. C(1/2)(|00⟩ + |01⟩ + |10⟩ + |11⟩)
4. D|1⟩|0⟩
2.
With matched analysers on |Φ+⟩ photon pairs, Alice can predict her own next result because she knows the pairs always agree.Circle one
TrueFalse
3.
Einstein, Podolsky and Rosen argued in 1935 that quantum mechanics is , because quantities that can be predicted with certainty are not all assigned definite values.Fill in the blank
4.
In John Bell's essay about Bertlmann's socks, what explains the correlation between the colours of the two socks?Circle one
1. AA signal that travels from one sock to the other when one is seen
2. BA superposition of colours that collapses when observed
3. CProperties fixed in advance, when Bertlmann got dressed
4. DThe observer's expectation about the colours
5.
By looking only at his own list of results, Bob can tell whether Alice has measured her photons yet.Circle one
TrueFalse
6.
In spontaneous parametric down-conversion, one pump photon splits into a signal photon and an photon.Fill in the blank
7.
Why do photons emerging where the two type-II down-conversion cones intersect carry polarisation entanglement?Circle one
1. ABecause the crystal stamps both photons with a shared hidden polarisation as they are born
2. BBecause the pump's polarisation is copied onto both daughters, so they always match
3. CBecause the cones overlap, so the photons collide and swap polarisations as they leave
4. DBecause nothing records which cone each photon came from, so the two alternatives add as amplitudes
8.
Quantum teleportation requires an ordinary classical channel in addition to a shared entangled pair.Circle one
TrueFalse
9.
A source emits either |HH⟩ or |VV⟩ at random. Explain how Alice and Bob can distinguish it experimentally from a source of |Φ+⟩ pairs.Short answer
10.
State the EPR criterion of reality and explain where the assumption of locality enters their argument.Short answer
11.
Alice and Bob measure 2,000 |Φ+⟩ photon pairs with analysers 22.5 degrees apart. How many pairs should agree, and what is the correlation E? What does the straight-line local hidden-variable model, E = 1 − 4θ/π, predict instead?Show your working
Working
Answer:
12.
Two spin-singlet particles are measured with Stern–Gerlach axes 45 degrees apart. Find the probability of opposite results, the probability of matching results, and the correlation E.Show your working
Working
Answer:
13.
A 532 nm green laser pumps a down-conversion crystal, and the signal photon is detected at 810 nm. At what wavelength is the idler, and how do the photon energies balance?Show your working
Working
Answer:
14.
Bob keeps his analyser at 0 degrees. Alice switches hers between 25 degrees and 70 degrees, hoping to send a signal. For each of Alice's settings, find the probability that Bob's photon passes, splitting it into the cases where Alice's photon passes and where it is blocked.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Build the local hidden-variable model of Part 4 and catch it out. Each photon pair carries a hidden polarisation angle λ, chosen uniformly between 0 and 180 degrees. An analyser at angle α outputs 'pass' if α lies within 45 degrees of λ (counting angles modulo 180 degrees), and 'blocked' otherwise. (a) Show that each side alone passes half the time, whatever its setting. (b) Show that, for analysers θ apart with θ between 0 and 90 degrees, the probability of disagreement is θ ÷ 90°, so E = 1 − θ ÷ 45° (in radians, E = 1 − 4θ/π). (c) Compare with quantum mechanics, E = cos 2θ: at which angles do they agree, at what angle is the gap largest, and how large is it? (d) Using the standard error √((1 − E²) ÷ N) for the model's E, roughly how many pairs at 22.5 degrees would make the gap five standard errors wide?

Working

Memory aid

Mnemonic Think TWINS, and picture two identical twins raised on opposite sides of the world. T, Together: the pair has one joint state that will not factor into two separate ones. W, Whatever the angle: correlation depends only on the angle between the settings, cos²θ agreement for photons, E = −cos θ for spin singlets. I, Individually random: each twin alone answers like a fair coin. N, No signal: nothing either twin does changes the other's own statistics; the pattern appears only when notes are compared. S, Source: entangled twins are born together, from a down-converting crystal, a calcium cascade or a spin-zero parent. Run any entanglement claim through TWINS and the hype falls out.

In one line Entangled pairs share one state: matched settings give perfect correlation, each result alone is random, correlation follows the angle, and no message passes.