T Parameterization

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().