Parameterization of a density operator \( \rho\) by a complex matrix \( T\) such that \( \rho = T T^\dagger \) and with \( T\) satisfying \( \mathrm{tr}(TT^\dagger)=1 \).

The matrix \( T\) is obviously not unique but has a unitary freedom: \( T' = TU\) is also possible parameterization for any unitary \( U\). You can choose a gauge to fix this unitary freedom. Two are common:

Force \( T\) to be positive semidefinite. Then \( T = \rho^{1/2} \).
Force \( T\) to be a lower triangular matrix. You can obtain \( T\) by performing a Cholesky (or LLT or LDLT) decomposition.

Throughout the project, if we refer to a ‘T parameterization,’ we do not imply any particular gauge. For example, the class Tomographer::DenseDM::TSpace::LLHMHWalker does not fix the gauge and performs the random walk on all valid T matrices.

See, for example, Tomographer::DenseDM::fidelityT().