Questions: Density Matrices and the Density Operator

Question 1

Beam A contains particles each in the superposition (|↑⟩ + |↓⟩)/√2. Beam B contains a 50/50 mixture — half the particles in |↑⟩ and half in |↓⟩, with no quantum superposition. How do their density matrices differ?

• A. They are identical, since both beams have equal probability of measuring spin-up or spin-down
• B. Beam A has off-diagonal coherences in ρ; Beam B has ρ proportional to the identity matrix with no coherences
• C. Beam B has larger off-diagonal entries because the classical uncertainty is greater
• D. Both have Tr(ρ²) = 1, indicating they are both pure states

Answer: (B)

Both beams give 50% spin-up and 50% spin-down on measurement, so the diagonal entries of ρ are the same. The difference is in the off-diagonal entries (coherences): Beam A's superposition state produces nonzero coherences that encode quantum interference effects; Beam B's classical mixture has a density matrix proportional to the identity — no coherences. Only Beam A satisfies Tr(ρ²) = 1 (pure state); Beam B has Tr(ρ²) = 1/2 (mixed state). Interference experiments distinguish them.

Question 2

A density matrix ρ for a quantum system satisfies Tr(ρ²) = 0.7. What does this tell you about the system?

• A. The system is in a pure quantum state
• B. The system is in a mixed state — a classical statistical ensemble of quantum states
• C. The system is in a superposition of exactly two states with unequal weights
• D. The density matrix is unphysical and violates the normalization condition

Answer: (B)

A pure state satisfies ρ² = ρ and therefore Tr(ρ²) = 1. Any value Tr(ρ²) < 1 indicates a mixed state — a classical probability distribution over quantum states. The value 0.7 tells us some but not complete information about the system's quantum state. (Tr(ρ) = 1 still holds for both pure and mixed states, so the matrix is physical.) The closer Tr(ρ²) is to 1/d (where d is the Hilbert space dimension), the more mixed the state.

Question 3

A 50/50 quantum superposition (|↑⟩ + |↓⟩)/√2 and a 50/50 classical mixture of |↑⟩ and |↓⟩ are physically equivalent — they predict identical measurement outcomes for most possible experiments.

• T. True
• F. False

Answer: False

This is the central misconception about density matrices. Both states give 50% probability of measuring spin-up or spin-down in the z-basis. But the superposition has off-diagonal coherences in ρ that produce quantum interference, which shows up in measurements along other axes. For example, measuring the superposition in the x-basis gives a definite outcome; measuring the mixture in the x-basis gives 50/50. The density matrix captures this difference: the superposition has Tr(ρ²) = 1; the mixture has Tr(ρ²) = 1/2.

Question 4

The expectation value formula ⟨Â⟩ = Tr(ρÂ) applies to both pure states and mixed states, making the density operator a unified framework for computing observables.

• T. True
• F. False

Answer: True

This universality is the central advantage of the density operator formalism. For a pure state ρ = |ψ⟩⟨ψ|, Tr(ρÂ) = ⟨ψ|Â|ψ⟩, recovering the standard expectation value. For a mixed state ρ = Σᵢ pᵢ|ψᵢ⟩⟨ψᵢ|, Tr(ρÂ) = Σᵢ pᵢ⟨ψᵢ|Â|ψᵢ⟩, correctly weighting each state's expectation value by its classical probability. The single formula handles both cases without needing to track whether the state is pure or mixed.

Question 5

What is the fundamental physical difference between a quantum superposition and a classical mixture, and how does the density matrix capture this distinction?

Model answer: A quantum superposition places a system in a coherent combination of states — the system genuinely has no definite value for the superposed property, and quantum interference effects are possible. A classical mixture represents ignorance — the system is in one definite state, we just don't know which. The density matrix captures this: a pure state (superposition) has nonzero off-diagonal entries (coherences) and satisfies Tr(ρ²) = 1; a mixed state has ρ with Tr(ρ²) < 1 and the off-diagonal coherences are absent or reduced.

This distinction is not merely formal — it is experimentally testable. Interference experiments (like double-slit or spin rotations) distinguish superpositions from mixtures: coherences contribute to interference patterns that mixtures cannot produce. The density matrix formalism makes both cases tractable within one framework, which is why it is indispensable for open quantum systems and quantum information theory.