Finite Precision Measurement

Only nite precision measurements are experimentally reasonable, and they cannot distinguish a dense subset from its closure. We show that the rational vectors, which are dense in S 2 , can be colored so that the contradiction with hidden variable theories provided by Kochen-Specker constructions does not obtain. Thus, in contrast to violation of the Bell inequalities, no quantum-over-classical advantage for information processing can be derived from the Kochen-Specker theorem alone. Recent theoretical and experimental work on quantum computation and quantum information theory has inspired renewed interest in fundamental results of quantum mechanics. The Horodeccy have shown, for example, that a spin-1 2 state can be teleported with greater than classical delity using any mixed two spin-1 2 state which violates some generalized Bell-CHSH inequality 1]. Quantum teleportation was rst demonstrated experimentally in quantum optics systems 2]; the parametric down-conversion techniques crucial for these experiments have also been used to verify violation of Bell's inequality directly 3]. Although the Bell-CHSH inequalities were originally derived in the context of EPR-B experiments 4] and (local) hidden variable theories 5,6], the present concern is with the diierences in information processing capability between classical and quantum systems. Analyses of EPR-B experiments from the very rst 8] have been concerned with limitations in, for example, detector eeciency 6]: The observed violations of Bell-CHSH inequalities are consequently reduced; so, too, is teleportation delity 2,9]. Logically, if not entirely chronologically prior contradictions with (noncontextual) hidden variable theories were derived by Bell 10] from a theorem of Gleason 11] and by Kochen and Specker 12]. The GHZ-Mermin three spin-1 2 state exhibits a similar incompatibility with (noncontextual) hidden variable theories 13] and reduces the communication complexity of certain problems 14]. While no quantum improvement in information processing power has yet been derived directly from the Kochen-Specker theorem, it is natural to ask for the consequences of experimental limitations on measurement in this setting. The Kochen-Specker theorem concerns the results of a (counterfactual) set of measurements on a quantum system described by a vector in a three dimensional Hilbert space. Kochen and Specker consider, for example, measurement of the squares of the three angular momentum components of a spin-1 state 12]. The corresponding operators commute and can be measured simultaneously, providing onèyes' and twòno's to the three questions, \Does the spin component along ^ a, ^ b, ^ a ^ b vanish?" for any ^ a ? ^ b …