Quantum state tomography

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. 

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The interest in the FD quantum-optical states has been stimulated by the progress in quantum-optical state preparation and measurement techniques [36], in particular, by the development of the discrete quantum-state tomography [37-42]. There are several other reasons for studying states in FD spaces ... 

F.A. Bonk, E.R. de Azevedo, R.S. Sarthour, J.D. Bulnes, J.C.C. Freitas, A.P. Guimard, I.S. Ohveira, T.J. Bonagamba, Quantum logical operations for spin 3/2 quadrupolar nuclei monitored by quantum state tomography, J. Magn. Reson. 175 (2005) 226. 

Therefore, performing measurements of observables which are the products of the Pauli matrices, it is possible to determine all the elements of the density matrix operator p, with an arbitrary precision. This process is referred to as Quantum State Tomography, and is a procedure for measuring the quantum state of a system. 

The Wigner function is a distribution for the position q) and momentum (p) of a system. From the knowledge of the Wigner function of a system, its density matrix can be determined in a kind of quantum state tomography. 

U. Leonhardt, Quantum-state tomography and discrete Wigner function, Phys. Rev. Lett. 74 (1995) 4101. 

U. Leonhardt, Discrete Wigner function and quantum-state tomography, Phys. Rev. A S3 (1996) 2998. 

Figure 4.8 shows experimental results for the deviation density matrix obtained after applying each operation for a 2-qubit system Uq, U, and U2 as well as the average state (see also Problems P4.3 and P4.4). The deviation density matrices were obtained using the quantum state tomography process, which will be described in the next section. As it can be seen, the final averaged deviation density matrix is very similar to that of the pure state 100). 

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

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

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

R. Das, T.S. Mahesh, A. Kumar, Efficient quantum-state tomography for quantum-information processing using a two-dimensional Fourier-transform technique, Phys. Rev. A 67 (6) (2003) Art. No. 62304-1. 

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. 

Besides directly observing the correct phase interference in the detected signals (Figure 6.5), the authors perform quantum state tomography, from which a fidelity of 0.99 was obtained for if ". 

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