Quantum Mechanics 101 · Unit 3: The Mathematical Machinery

Quantum Academy · Chapter 15: Superposition & Measurement

Both, neither, and the moment the dice are thrown

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Instructions. Work through every question with the lesson closed, then check yourself. In the X basis, |+⟩ = (1/√2)(|0⟩ + |1⟩) and |−⟩ = (1/√2)(|0⟩ − |1⟩). For a qubit cos(θ/2)|0⟩ + sin(θ/2)|1⟩ with relative phase φ, the probability of 0 in Z is cos²(θ/2) and the probability of plus in X is one half of (1 + sin θ · cos φ). Unpolarized light loses half its intensity at the first polarizer, and polarized light passes a polarizer at angle θ to its polarization with probability cos²θ. Show every step in the calculation questions. 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.Superposition
2.Probability amplitude
3.Normalization
4.Basis
5.Relative phase
6.Mixture
7.Projective measurement
8.Collapse
9.Incompatible observables
10.Quantum Zeno effect
11.Measurement problem
12.Decoherence

A.The projection postulate: the update of the state to the eigenstate matching a measurement result; whether it is a physical process is interpretation-dependent.
B.A complete set of perfectly distinguishable states, such as |0⟩ and |1⟩ or |+⟩ and |−⟩, in terms of which any state can be written.
C.A probabilistic blend of definite states that reflects ignorance, like a coin under a cup; unlike a superposition, it produces no interference.
D.The loss of observable interference when a system becomes entangled with its environment, which records information about it.
E.A state written as a weighted sum of other states, such as α|0⟩ + β|1⟩; a single definite state whose outcomes are uncertain only for measurements in a basis that does not contain it.
F.The conflict between linear, unitary evolution, which predicts superpositions of pointer readings, and the single definite outcomes every experiment records.
G.Observables whose operators do not commute, such as Z and X, so that no state has definite values of both and measuring one randomizes the other.
H.A complex coefficient in a superposition; the squared magnitude of an amplitude gives the probability of the matching outcome.
I.The suppression of a quantum transition by frequent measurements, which repeatedly return the system to its initial state.
J.An ideal measurement whose results are the eigenvalues of an observable, with probabilities given by the Born rule, leaving the system in the matching eigenstate.
K.The phase difference between the amplitudes in a superposition; invisible to some measurements but decisive for interference.
L.The requirement that the squared magnitudes of all amplitudes add to one, so that the outcome probabilities sum to 1.

Part B Questions

1.
Which of these is a properly normalized qubit state?Circle one
1. A0.5|0⟩ + 0.5|1⟩
2. B0.8|0⟩ + 0.8|1⟩
3. C0.6|0⟩ − 0.8i|1⟩
4. D|0⟩ + |1⟩
2.
Box P holds qubits in the state |+⟩; box Q holds qubits secretly set to |0⟩ or |1⟩ with even odds. Which single type of measurement, repeated on many qubits from each box, tells you which box is which?Circle one
1. AZ measurements, because the superposition gives 0 more often
2. BX measurements, because only the superposition gives plus every time
3. CTwo Z measurements in a row on each qubit, to see whether the answer changes
4. DNone, because the two boxes are physically identical
3.
A qubit is measured in Z and gives 0. It is then measured in X and gives minus. What is the probability that a third measurement, in Z, gives 0?Circle one
1. A0
2. B0.25
3. C0.5
4. D1
4.
Two qubit states with the same values of |α|² and |β|² give the same statistics for every possible measurement.Circle one
TrueFalse
5.
An ideal projective Z measurement repeated immediately on the same qubit can give 0 the first time and 1 the second.Circle one
TrueFalse
6.
Decoherence explains why interference between macroscopically different states becomes unobservable, but by itself it does not select which single outcome occurs.Circle one
TrueFalse
7.
A probabilistic blend of definite states, whose probabilities reflect only our ignorance, is called a .Fill in the blank
8.
The rule that leaves a system in the eigenstate matching the measured result is called the projection postulate, or .Fill in the blank
9.
State the measurement problem in your own words. Your answer should mention linearity.Short answer
10.
A friend says the quantum Zeno effect proves that a conscious observer controls reality. Use the NIST experiment to answer them.Short answer
11.
A qubit is prepared in |ψ⟩ = (1/√3)|0⟩ + √(2/3)|1⟩. Show that it is normalized, find the probability of each Z result, and predict the number of 1 results in 600 runs, with the binomial standard deviation √(N·p·(1 − p)).Show your working
Working
Answer:
12.
A qubit is prepared with θ = 90° and φ = 60°. Find the probability of 0 in a Z measurement and of plus in an X measurement. What would a fifty-fifty mixture of |0⟩ and |1⟩ give in X?Show your working
Working
Answer:
13.
A drive would rotate a qubit fully from |0⟩ to |1⟩ in a time T, and you make N equally spaced Z measurements during it. (a) Find the probability the qubit is found in |0⟩ at every check for N = 20. (b) Use the approximation 1 − π²/(4N) to estimate how many checks keep the survival probability at 99 percent or more, then check your estimate with the exact formula.Show your working
Working
Answer:
14.
An 8.0 mW unpolarized beam passes four polarizers in a row, with axes at 0°, 30°, 60° and 90°. What power emerges? Compare with two crossed polarizers at 0° and 90° alone.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The measurement problem meets decoherence. A qubit starts in α|0⟩ + β|1⟩ with real amplitudes, and a detector records it, so that the combined state becomes α|0⟩|A₀⟩ + β|1⟩|A₁⟩, where the pointer states are normalized and have real overlap c = ⟨A₀|A₁⟩. (a) Rewrite the combined state using |0⟩ = (1/√2)(|+⟩ + |−⟩) and |1⟩ = (1/√2)(|+⟩ − |−⟩), and show that the probability of finding the qubit in |+⟩ is one half of (1 + 2αβc). (b) Evaluate it for α = β = 1/√2 when the detector records nothing (c = 1), records partially (c = 0.6), and records perfectly (c = 0). (c) Evaluate it for α = 0.6, β = 0.8 and c = 0.5. (d) Explain what this has to do with why a dust grain in air never shows interference.

Working

Memory aid

Mnemonic Walk through the doors of the quantum CASINOS. C: Coefficients squared give the odds, and they must sum to 1. A: Amplitudes Add before squaring, so superpositions interfere and coins under cups do not. S: Snap, the state jumps to the eigenstate matching the result. I: Immediate repeats give Identical answers. N: New basis, New dice, because an incompatible measurement re-randomizes. O: Often-checked qubits freeze, the Zeno effect. S: Smooth versus Snap, the measurement problem, with the environment's records hiding the other branches. To use it, run the seven letters in order on any measurement question: compute the odds, look for interference, snap the state, repeat, switch basis, watch the clock, then ask which rule you just used.

In one line A superposition is one definite state that can interfere; measurement picks one answer at random and snaps the state to match.