Editing Full Quantum state tomography with Maximum Likelihood Estimation (section)

==Outline==
Quantum state tomography attempts to characterize an unknown state ρ by measuring its components, usually in the Pauli basis or a different selected basis of measurement operators. Multiple identical copies of the quantum state are required as different measurements need to be performed on each copy. To fully reconstruct the density matrix for a mixed state, this method is used. This procedure can be used to completely characterize an unknown apparatus.

When [[matrix inversion]] technique is used, some of the results produced can violate important basic properties such as positivity leading to a density matrix which is not valid. To avoid this problem, the maximum likelihood estimation of density matrices is employed. In practice, analytically calculating this maximally likely state is prohibitively difficult, and a numerical search is necessary. Three elements are required: a manifestly legal parametrization of a density matrix, a likelihood function which can be maximized, and a technique for numerically finding this maximum over a search of the density matrix’s parameters.

This method consists of the following steps:

* An unknown state is prepared multiple times so as to create many copies.
* The experimenter picks a basis of measurement operators.
* The measurement corresponding to each measurement operator in the selected basis is taken multiple times. This is used to approximate the probability of measuring the state in the value corresponding to the measurement operator.
* Generate a formula for an explicitly physical and valid density matrix, i.e., a matrix that has the three important properties of normalization, Hermiticity, and positivity.
* Introduce a likelihood function which quantifies how good the final density matrix is in relation to the experimental probability data. This likelihood function is a function of the certain real parameters according to the described formula and of the experimental probability data. This, in general, will depend on the specific measurement apparatus used and the physical implementation of the qubit (as these will determine the statistical distributions of counts, and therefore their relative weightings). To provide a likelihood function here, we assume Gaussian counting statistics and that each of our measurements is taken for the same amount of time.
* Using standard numerical optimization techniques, find the optimum set of variables for which the likelihood function has its maximum value. The best estimate for the density matrix is then determined using this function.

This method is generally preferred over the quantum state tomography using linear matrix inversion.