Synthesis and reconstruction of quantum states (6)

In the sixth lesson , we studied the estimation and reconstruction of states of a harmonic oscillator evolving in a space subtended by an infinite number of states. We began with a reminder of the Wigner function W, which gives a description of the oscillator in its phase space, as close as possible to the classical description. This description is perfectly equivalent to that of the density operator 𝛒, since the Wigner function W and 𝛒 can be deduced from each other by simple mathematical transformations. We recalled the essential properties of W and described the Wigner functions of coherent states and a few non-classical states of the field (Fock states and Schrödinger cat states). In particular, we emphasized the signature of "non-classicality" constituted by the existence of negative-valued non-Gaussian structures of the Wigner function.

We then described two standard methods for directly reconstructing the Wigner function of a field mode state when a large number of copies are available. The so-called quantum tomography method relies on the measurement of field quadrature distributions along multiple directions in phase space. Having made these quadrature measurements, the Wigner function is calculated using an inversion procedure analogous to that used to reconstruct medical tomography images from absorption measurements of X-rays passing through the body at different angles. Quadrature measurement is performed using a homodyning method that mixes the field to be measured with a reference field (local oscillator) of variable phase. This method is well suited to the measurement of free-propagating optical fields or optical fibers. Another method involves translating the field into phase space (which, like homodyning, involves mixing the field to be measured with a coherent reference) and then measuring the parity of the photon number in the resulting field. This method directly gives the value of the Wigner function at various points corresponding to the amplitudes of the translational fields used. It requires no inversion procedure and is well suited to the study of microwave fields trapped in a cavity.

Direct reconstruction methods are based on an exact mathematical formulation that "rigidly" constrains the values of the Wigner function, or what amounts to the same thing, the elements of the density matrix of the field under study, when a large number of measurements have been made. They involve identifying the frequencies of the measurements obtained (given by experimental histograms) with the exact theoretical values expected (given by their expression as a function of the density matrix or Wigner function). When using quantum tomography, this identification involves solving a large number of coupled equations. If the measurements are marred by systematic noise, or if the samples are not large enough, leading to significant statistical fluctuations, the equations to be solved may lead to a grossly false or even unphysical state (density matrix with negative eigenvalues). A fundamental cause of error lies in the fact that measurements are carried out with non-ideal equipment. Its imperfections are, however, quantifiable and can be measured by preliminary calibration operations. Introducing this information into the reconstruction procedure enables us to reconstruct the prepared state, which is different from the "raw" measured state. A natural way of proceeding is to use a statistical estimation method to determine the state, based on the principles discussed in Lesson 2. We have shown how the principle of maximum likelihood can be adapted to the reconstruction of the system state. We describe the measurements of the various observables by giving the frequencies with which each eigenvalue was obtained, and infer from these results the density operator (and therefore the Wigner function) that best reproduces the observed statistic. This is done by constructing a likelihood function associated with the measurement results, which depends on the a priori unknown state of the system, and maximizing this function with respect to the state. This procedure automatically matches measured frequencies with theoretical probabilities as closely as possible. In practice, the search for the density operator can be carried out using an iterative method well suited to computer calculation. The result is a density operator that is necessarily physical, positive and normalized by construction. It is also well-suited to taking systematic measurement errors into account.