Density matrix tomography

An illustration of the use of the density matrix tomography process is shown in Figure 4.12. It is the measured truth table for a CNOT gate with control on qubits A and B, obtained from experimentally determined density matrices. 

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. 

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. 

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). 

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