Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S03 / 3 / c / Solution

When $p_0,p_1>0$, the observed relative frequencies converge, for large sample size, to the conditional probabilitiesEquality says that the second binary outcome has the same distribution whichever first outcome occurred; this is independence of these two particular measurement results. Cross-multiplication givesFor finite samples the two empirical fractions fluctuate; they are estimates, not exact deterministic equalities. If $p_0=0$ or $p_1=0$, the missing conditional fraction is undefined, but one outcome is constant, so the joint measurement results are independent and the displayed modulus identity still holds.

This criterion only addresses the computational-basis statistics. For examplehas all four joint probabilities equal to $1/4$, yet is maximally entangled: its first-qubit density matrix is $I/2$ and $\alpha \delta =-1/4\ne 1/4=\beta \gamma$. Moreover classical correlations of mixed separable states also exist, for example $\frac{1}{2}|00⟩⟨00|+\frac{1}{2}|11⟩⟨11|$. Correlations in a single basis alone neither certify nor exclude quantum entanglement.