Production of pseudo-pure states

Conventional NMR deals with a large ensemble of spins. It means that the state of the system is in a statistical mixture, which is obviously inadequate for QIP. However, the NMR ability for manipulating spins states worked out by Cory et al. [24] and Chuang et al. [23] resulted in elegant methods for creating the so called effectively pure or pseudo-pure states. Behind the idea of the pseudo-pure states is the fact that NMR experiments are only sensitive to the traceless deviation density matrix. Thus, we might search for transformations that, applied to the thermal equilibrium density matrix, produce a deviation density matrix with the same form as a pure state density matrix. Once such state is created, all remaining unitary transformations will act only on such a deviation density matrix, which will transform as a true pure state. 

Let us start with a short discussion about the kind of transformation we are seeking for. As described in the Chapter 2, the density matrix corresponding to a pure state is a projector, which satisfies the following properties (Chapter 3) p = p and Tr(/o ) = 1. On the other hand, for a statistically mixed state, p p and Tr(p ) 1. Now, let us look at a density operator that is obtained from a mixed state operator p by a unitary transformation, p = UpU. The question is whether this operator can or cannot be a pure state operator. The trace and idempotency properties for the transformed operator become  

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The early application of volumetric data for hydrocarbons made use of the perfect gas laws. They were not sufficiently descriptive of the actual behavior to permit their widespread use at pressures in excess of several hundred pounds per square inch. The need for accurate metering aroused interest in the volumetric behavior of petroleum and its products at elevated pressures. Table II reviews references relating to the volumetric behavior of a number of components of petroleum and their mixtures. For many purposes the ratio of the actual volume to the volume of a perfect gas at the same pressure and temperature has been considered to be a single-valued function of the reduced pressure and temperature or of the pseudo-reduced (38) pressure and temperature. The proposals of Dodge (15), Lewis (12), and Brown (8) with their coworkers serve as examples of the nature of these correlations. The Beattie-Bridgeman (2) and Benedict (4) equations of state describe the volumetric behavior of many pure substances and their mixtures with an accuracy adequate (31) for most purposes. However, at pressures above 3000 pounds per square inch the accuracy of representation with existing constants leaves something to be desired. 

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