Density Matrices, Entanglement and the Bell Test
Density matrices describe pure states, statistical mixtures and reduced states of entangled systems. Preparations with the same reduced density matrix give identical local measurement statistics. Bell tests constrain models satisfying Bell locality and measurement independence. A reliable violation challenges those joint assumptions without providing controllable faster-than-light messages.
What a learner can do afterwards
- Writes density matrices for pure and mixed qubit states and tests purity using Tr(ρ²)
- Calculates a partial trace for an entangled pure pair and relates a local density matrix to measurement statistics
- States Bell locality and measurement independence, evaluates a CHSH expression and qualifies what a violation rules out
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Use orthonormal qubit basis |0>,|1>. For |ψ>=a|0>+b|1>, require |a|²+|b|²=1. The bra <ψ| is the conjugate-transposed row of |ψ>; * means complex conjugation. Multiply ket coefficients by bra entries: ρ=|ψ><ψ|=[[|a|²,ab*],[ba*,|b|²]]. Matrix products sum row-column products: (XY)_ij=Σ_k X_ik Y_kj; ρ² means ρρ. This pure-state projector obeys ρ²=ρ, so Tr(ρ²)=Tr(ρ)=1. Pure superpositions can have uncertain measurement outcomes.
An ensemble has ρ=Σ_j p_j|ψ_j><ψ_j|, with p_j≥0 and Σ_j p_j=1. A density matrix is Hermitian, positive semidefinite and has trace one; trace is the sum of diagonal entries. It is pure when Tr(ρ²)=1 and mixed when Tr(ρ²)<1. An incoherent preparation of |0> with probability p and |1> with probability 1-p gives diag(p,1-p), with p and 1-p on the diagonal and zeros elsewhere. Its purity is p²+(1-p)².
For two qubits, |ab>=|a>_A⊗|b>_B, with labels 0 or 1. A pure state is entangled if it cannot be written as |u>_A⊗|v>_B. Form ρ_AB=|Ψ><Ψ|. For |Ψ>=r|00>+s|11>, expand (r|00>+s|11>)(r*<00|+s*<11|) into four terms, including both cross terms. Ignoring B gives ρ_A=Tr_B(ρ_AB). Apply Tr_B(|ab><cd|)=|a><c|<d|b> to each term. Matching B basis states have inner product 1; different ones have inner product 0.
Take |Ψ>=sqrt(p)|00>+e^(iθ)sqrt(1-p)|11>, with 0<p<1. The partial trace removes cross terms because B's basis states are orthogonal. It leaves ρ_A=diag(p,1-p), independent of θ. For p=2/3, the purity is (2/3)²+(1/3)²=5/9<1. Thus this joint pure entangled state has a mixed local state. Equal positive coefficients define the Bell pair |Φ+>.
For an A-only observable M, the mean is Tr(ρ_A M). With ρ_A=diag(p,1-p) and outcomes +1 for |0> and -1 for |1>, the mean is p-(1-p). Equal local density matrices give equal statistics, even for different preparations. Being mixed alone is insufficient: diag(1/3,2/3) and diag(2/3,1/3) give different probabilities of |0>. Joint measurements can reveal correlations absent from local data.
A Bell-local model uses a shared variable λ. For settings x,y and outcomes A,B, it requires P(A,B|x,y,λ)=P(A|x,λ)P(B|y,λ). Measurement independence means λ's distribution is independent of x,y. CHSH uses two settings per side and outcomes +1 or -1. Deriving its bound also requires valid sampling. Experiments must address statistical uncertainty and relevant loopholes.
Let E_xy be the mean product of outcomes for settings x,y. Define C=E_00+E_01+E_10-E_11. Bell-local, measurement-independent models obey |C|≤2. Quantum theory permits up to 2sqrt(2). Correlations 0.7,0.7,0.7,-0.7 give C=2.8. This exceeds 2, but finite experimental data also require significance and loophole checks.
A reliable CHSH violation is incompatible with the joint assumptions behind the bound. Retaining measurement independence then excludes Bell-local models of those results. It does not select one interpretation or provide controllable faster-than-light messages. Quantum correlations retain no-signaling local statistics. Entanglement alone does not guarantee violation of a chosen Bell test.
Density matrices describe pure and mixed states. Partial traces give local states. Bell violations test specified assumptions, not signaling devices.
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