A spin-1/2 particle is prepared in |+x⟩ = (|↑⟩ + |↓⟩)/√2. A second particle is in a statistical mixture: spin up 50% of the time and spin down 50%, with no tracking of which. Both give identical measurement statistics in the z-basis. What physically distinguishes these two states?
AThey are physically identical — different preparations of the same quantum state
BThe pure state has off-diagonal coherences in its density matrix; the mixed state does not
CThe mixed state has Tr(ρ²) = 1; the pure state has Tr(ρ²) < 1
DThe pure state has equal diagonal elements; the mixed state has unequal diagonal elements
Both states show identical 50/50 z-measurement statistics — the diagonal elements are the same. The distinction lives in the off-diagonal coherences: the pure state has them (reflecting the definite phase relationship between |↑⟩ and |↓⟩), the mixed state does not. This difference becomes visible when measuring in a different basis: the pure state gives a definite outcome in the x-basis, while the mixture gives 50/50 in every basis. Option C reverses the Tr(ρ²) relationship — pure states have Tr(ρ²) = 1, mixed states have Tr(ρ²) < 1.
Question 2 Multiple Choice
A quantum computer qubit begins in a pure superposition state with Tr(ρ²) = 1.0. After interacting with its environment, Tr(ρ²) drops to 0.52. What has happened?
AThe qubit was measured, collapsing it to a definite computational basis state
BThe qubit entangled with environmental degrees of freedom; tracing out the environment leaves a mixed reduced state
CThe qubit is now more stable because decoherence has suppressed quantum noise
DTr(ρ²) decreasing means the qubit has gained quantum information from the environment
Decoherence occurs when a quantum system entangles with its environment. Even though the joint system+environment state may remain pure, tracing out the environment from the joint density matrix eliminates the off-diagonal coherences, leaving a mixed reduced state for the qubit. Tr(ρ²) < 1 confirms the state is mixed — quantum coherence has been lost. Option A describes projective measurement, which would yield a new pure state with Tr(ρ²) = 1 after collapse. Decoherence is a gradual degradation, not a discrete event.
Question 3 True / False
A quantum state that shows 50/50 probabilities for spin-up and spin-down should be a mixed state.
TTrue
FFalse
Answer: False
The pure state |+x⟩ = (|↑⟩ + |↓⟩)/√2 gives exactly 50/50 probabilities in the z-basis while being a pure state with Tr(ρ²) = 1 and nonzero off-diagonal coherences. Equal measurement probabilities in one basis say nothing about purity — purity is determined by whether coherences exist, not by the diagonal probabilities. The distinction requires probing a different measurement basis, where the pure state gives a definite outcome and the mixture gives 50/50.
Question 4 True / False
For a pure state, the density matrix ρ satisfies ρ² = ρ.
TTrue
FFalse
Answer: True
A pure state has ρ = |ψ⟩⟨ψ|, so ρ² = |ψ⟩⟨ψ|ψ⟩⟨ψ| = |ψ⟩⟨ψ| = ρ since ⟨ψ|ψ⟩ = 1. This means ρ is a projector onto the pure state, and Tr(ρ²) = Tr(ρ) = 1. For a mixed state, ρ² ≠ ρ and Tr(ρ²) < 1. The identity ρ² = ρ is the precise algebraic signature that distinguishes pure states from mixtures.
Question 5 Short Answer
Explain why a pure quantum superposition and a classical statistical mixture can give identical measurement statistics in one basis but differ in another. What does this reveal about quantum coherence?
Think about your answer, then reveal below.
Model answer: A pure superposition encodes definite phase relationships between basis states through off-diagonal coherences in the density matrix. A classical mixture has the same diagonal entries (same probabilities for a given basis measurement) but zero off-diagonal terms — it represents classical uncertainty about which pure state the system is in. Measuring in the original basis only probes the diagonal and cannot distinguish them. Measuring in a rotated basis probes the off-diagonal coherences and reveals the difference: the pure state gives a definite outcome in the rotated basis, the mixture gives 50/50. Coherences are the operational signature of quantum superposition — they enable interference. Their presence or absence, not the diagonal probabilities, is what separates quantum from classical uncertainty.
This is the conceptual core of the density matrix formalism. Tr(ρ²) = 1 iff the state is pure (coherences intact); Tr(ρ²) < 1 means some quantum information has been lost to the environment. Decoherence is precisely the process of losing these off-diagonal terms through entanglement with the environment.