Editing Full Quantum Process Tomography with Linear inversion (section)

==Notation==
* <math>N</math>: Dimension of state space
* <math>\varepsilon</math>: Quantum process operator. This is the linear map which completely describes the dynamics of a quantum system, <math>\rho \xrightarrow[]{} \frac{\varepsilon(\rho)}{tr(\varepsilon(\rho))}</math>. The operator sum representation of <math>\varepsilon</math> is <math>\varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i</math>
* <math>A_i</math>: Operators acting on the system alone, yet they completely describe the state changes of the system, including any possible unitary operation.
* <math>\tilde{A_i}</math>: Fixed set of operators used to describe <math>\varepsilon</math> which form a basis for the set of operators on the state space, so that <math>A_i = \sum_m a_{im}\tilde{A_m}</math>. This is done to related <math>A_i</math> to measurable operators.
* <math>a_{im}</math>: Set of complex numbers
* <math>\rho_j</math>: A set of linearly independent basis elements for the space of N x N matrices. A convienent choice is the set of projectors <math>|m\rangle\langle n|</math>
* <math>\chi</math>: Classical error correlation matrix which is positive Hermitian by definition. <math>\varepsilon</math> is completely described by this. <math>\chi_{mn} = \sum_i a_{im}a_{in}^{*}</math>
* <math>\lambda_{jk}</math>: Parameter which can be determined from <math>\varepsilon{(\rho_j)}</math>
* <math>\beta_{jk}^{mn}</math>: Complex numbers which can be determined by standard algorithms given the <math>\tilde{A_m}</math> operators and the <math>\rho_j</math> operators. This is A <math>N^4</math> x <math>N^4</math> matrix with columns indexed <math>mn</math> and rows indexed <math>ij</math>. <math>\tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k</math>.
* <math>\kappa</math>: the generalized inverse for the matrix <math>\beta</math>, satisfying the relation <math>\beta_{jk}^{mn} = \sum_{st, xy} \beta_{jk}^{st}\kappa^{xy}_{st}\beta^{mn}_{xy}</math>
* <math>U^{\dagger}</math>: Unitary matrix which diagonalizes <math>\chi</math>