Weak measurements of non local variables

Methods for measuring the weak value of non local variables are investigated. We analyze local (indirect) measurement methods for obtaining the weak values. We also describe some new (direct) methods (Non local weak measurements) for measuring the weak values of some non local variables. Non local variables are variables of a composite system with parts in two or more remote locations. An example of a non local variable is the sum of the spin components of a spin singlet state with one spin 1/2 particle on earth and the other on the moon. Non local variables cannot always be 1 ar X iv :0 81 1. 17 06 v1 [ qu an tph ] 1 1 N ov 2 00 8 measured in a direct way because such measurements can sometimes contradict relativistic causality. Those non local variables that can be measured require the use of a specific, entangled measuring device for the (non local) measurement . Weak measurements are best described as standard measurements with a weakened coupling. Unlike the standard (strong) measurement whose result is one of the eigenvalues, the result of weak measurements is the weak value , a complex number. Weak measurements have an inherently large uncertainty, and must therefore be made on a large ensemble of identical systems. The weak values under investigation here are those of a non local sum and a non local product of two observables. For the sum of two observables we describe a method for measuring the weak value in a direct way. We also describe an indirect method for obtaining the weak value using the relation between the sum of the local weak values and the weak value of the non local sum. This indirect method requires two local measuring devices, one at each location. The indirect (local) method is compared to the direct (non local) method and is shown to be inefficient. For the product of two observables, there is no direct method for measuring the non local weak values. In some special cases the methods we found for the measurement of a sum can be used to measure the product. Using these methods to measure the product requires some prior knowledge of the system to be measured. We give a critical analysis of a recent method for measuring the product (called “joint weak values”). This method for obtaining the weak value of the product using local weak measurements is shown to be even more ineffi2 cient than the local method described for the sum. We compare this method with a (non physical) direct method and show that it requires much larger resources.