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. In our Tomographer project, we usually use the positive semidefinite gauge in practice. But you should double-check before blindly assuming this!

See, for example, Tomographer::fidelity_T().