NMR Quantum State Tomography

In standard NMR experiments the readout is made in the basis of the Zeeman interaction, i.e, the basis. Then, the basic ingredient for NMR QST would be to know how to execute the readout at different bases. However, this is completely equivalent to rotate the qubits and execute the measurements in a fixed base. Indeed  

2 NMR Quantum State Tomography in coupled spin 1/2 systems 

Another useful QST method was developed using the two-dimensional Fourier transform technique [29]. In this method the diagonal elements of the density matrix are obtained in a ID experiment where a short pulse, similar to that discussed above, is applied to retrieve the populations. The quantum coherences, including those not directly observable, are codified in a 2D spectrum S co, oyi) so that the 2D intensities depend on these coherences. This allows to extract the quantum coherences by fitting the 2D spectrum. The main advantage of such method is its scalability, as the 2D acquisition provides an efficient way of extracting the coherences even for a large number of qubits ( 5). 

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Quantum state tomography is a technique which allows the determination of all the matrix elements of the density operator of a system. Such a procedure is very important for QIP, since at the end of an algorithm or protocol, one is usually interested in knowing the quantum state of the system. In Chapter 2, density matrix was introduced in the context of NMR, and in Chapter 4 the quantum state tomography will also be discussed in the context of NMR QIP. In this chapter, these concepts are presented within the QIP formalism. 

RECONSTRUCTION OF DENSITY MATRICES IN NMR QIP QUANTUM STATE TOMOGRAPHY... 

Reconstruction of density matrices in NMR QIP Quantum State Tomography... 

Another interesting implementation of the QFT in a three-qubit system (the three of alanine) was reported by Weinstein et al. [15]. With the technique, the authors measured the periodicity of an input state, which was followed by quantum state tomography. Their result is shown in Figure 5.6. Other interesting NMR implementation of QFT can be found in Lee et al. [16] and Weinstein et al. [17]. The first applied QFT to phase estimation and quantum counting, and the second performed the quantum process tomography of QFT. 

Suppose that in a NMR experiment we produce an initial pseudopure state, and apply the quantum circuit that generates a cat state (see Chapter 3). Suppose also that we perform quantum state tomography on this state. We will find a matrix which will be similar to Equation (6.1.2), upon which one has to add the background to build the complete matrix ... 

Entanglement transfer experiment in NMR quantum information processing - This paper of 2002 by Boulant and co-workers [18] describes an experiment of entanglement transfer by NMR. The aim is to transfer an entangled state of a pair of qubits to another pair of qubits, a process which was first demonstrated using photons in 1998, by Pan and collaborators. The authors used the four nuclei of crotonic acid as qubits. The process was followed by quantum state tomography and the efficacy of the experiment was quantified by a measured called attenuated correlation. A value of 0.65 for this measure at the end of the protocol, indicated that the pseudo-entangled state was indeed transferred from one pair of qubits to the other. 

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