Density Matrices, Entanglement and the Bell Test · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Pure states, mixed states and Bell correlations

Science · Quantum & Modern Physics · ages 23-24
Name ______________________   Date ____________

  1. For |Ψ>=sqrt(1/10)|00>+sqrt(9/10)|11>, trace out qubit B. What is A's density matrix in the {|0>,|1>} basis?

    • ρ_A=diag(9/10,1/10)
    • ρ_A=diag(1/2,1/2)
    • ρ_A=diag(1/10,9/10)

  2. A preparation randomly supplies |0> with probability 1/5 and |1> with probability 4/5, without a coherent relative phase. Which density matrix describes the ensemble?

    • ρ=[[4/5,0],[0,1/5]]
    • ρ=[[1/5,0],[0,4/5]]
    • ρ=[[1/5,2/5],[2/5,4/5]]

  3. Any two mixed density matrices give the same statistics for every local measurement.

    Circle one:   True   False

  4. Qubit A has ρ_A=diag(1/4,3/4). A basis measurement assigns +1 to |0> and -1 to |1>. What is the mean outcome?

    • -1/2, the probability-weighted mean
    • +1/2, the probability-weighted mean
    • 0, the probability-weighted mean

  5. For |ψ>=(3|0>-4|1>)/5, which pair gives ρ=|ψ><ψ| and its purity Tr(ρ²)?

    • ρ=(1/25)[[9,-12],[-12,16]]; purity 1
    • ρ=(1/25)[[16,-12],[-12,9]]; purity 1
    • ρ=(1/25)[[9,0],[0,16]]; purity 337/625

  6. Which assumption pair underlies the CHSH bound |C|≤2 for a shared-variable model, with valid sampling?

    • Bell locality with settings dependent on λ
    • Bell locality with perfect detector precision
    • Bell locality with settings independent of λ

  7. A reliable CHSH violation is observed. A friend says it selects one interpretation and provides controllable faster-than-light messaging. Which response is supported?

    • It selects a unique interpretation and permits controllable faster-than-light messages
    • It selects no unique interpretation and supplies no controllable faster-than-light messages
    • It selects a unique interpretation but forbids controllable faster-than-light messages

  8. A CHSH test gives E_00=E_01=E_10=0.6 and E_11=-0.6. The excess over the bound is statistically significant and relevant loopholes are addressed. What follows?

    • C=2.4; the tested model satisfies the bound
    • C=2.4; the tested model violates the bound
    • C=1.2; the tested model satisfies the bound
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Pure states, mixed states and Bell correlations W1-mt_qwUJ2qV0l3-s1

  1. ρ_A=diag(1/10,9/10) · Tracing B removes the cross terms. The squared coefficient magnitudes give diagonal entries 1/10 and 9/10, in that order.
  2. ρ=[[1/5,0],[0,4/5]] · The incoherent preparation gives (1/5)|0><0|+(4/5)|1><1|=diag(1/5,4/5). It has no coherent off-diagonal entries.
  3. False · Local indistinguishability requires equal density matrices. Two different mixed matrices can give different measurement probabilities.
  4. -1/2, the probability-weighted mean · The probabilities are 1/4 and 3/4. The mean is (1/4)(+1)+(3/4)(-1)=-1/2.
  5. ρ=(1/25)[[9,-12],[-12,16]]; purity 1 · The outer product gives diagonal entries 9/25 and 16/25 and off-diagonal entries -12/25. This normalized pure-state projector has Tr(ρ²)=1.
  6. Bell locality with settings independent of λ · The model uses Bell-local factorization and a distribution of λ independent of the measurement settings.
  7. It selects no unique interpretation and supplies no controllable faster-than-light messages · The violation rejects the joint assumptions behind the bound. It neither selects one interpretation nor provides controllable faster-than-light messaging.
  8. C=2.4; the tested model violates the bound · C=0.6+0.6+0.6-(-0.6)=2.4>2. With the stated checks, the results exclude the joint model assumptions, not every no-signaling theory.
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