A ug 1 99 6 Comment on “ Hidden quantum nonlocality revealed by local filters

In Section 3 of his paper [1], Gisin argues that a “careless application of generalized quantum measurements can violate Bell’s inequality even for mixtures of product states.” However, the observed violation of the CHSH inequality is not in fact due to the application of generalized quantum measurements, but rather to a misapplication of the inequality itself — to conditional expectations in which the conditioning depends upon the measurements under consideration. Consider the usual setup of quantum nonlocality arguments: a system consists of two widely separated subsystems, and in each of the subsystems one of two possible experiments a, a′ and b, b′, respectively, can be performed. The system is in a quantum state ρ which is a density matrix on the Hilbert space H1 ⊗H2. We shall denote by O a the positive operator on H1 giving via tr ρ (O a ⊗ I) the probability of obtaining the value i for the measurement of a. Similarly O b denotes the positive operator on H2 for the value j of the measurement of b. A local hidden variables model for this setup consists of random variables Xa, Xa′ , Xb, and Xb′ for the experiments under consideration on some probability space (Ω, IP) such that the joint distributions of the model reproduce the quantum joint distribution P ρ IP(Xa = i, Xb = j) = P (a = i, b = j) = tr ρ (O a ⊗O b) (1) for a joint measurement of a and b, and similarly for the pairs (a, b′), (a′, b), and (a′, b′). Thus we have for the expectation value of the product a · b E(a · b) = IE(XaXb) ( = ∫