Abstract
We explore the legal purity parameters for the joint measurements. Instead of direct unsharpening the measurements, we perform the quantum cloning before the sharp measurements. The necessary fuzziness in the unsharp measurements is equivalently introduced in the imperfect cloning process. Based on the information causality and the consequent noisy nonlocal computation, one can derive the information-theoretic quadratic inequalities that must be satisfied by any physical theory. On the other hand, to guarantee the classicality, the linear Bell-type inequalities deduced by these quadratic ones must be obeyed. As for the joint measurability, the purity parameters must be chosen to obey both types of inequalities. Finally, the quadratic inequalities for purity parameters in the joint measurability region are derived.
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