The problem of characterizing quantum channels arises in a number of contexts such as quantum process tomography and quantum error correction. However, direct approaches to parameterizing and optimizing the Choi matrix representation of quantum channels face a curse of dimensionality: the number of parameters scales exponentially in the number of qubits. Recently, Torlai et al. [2020] proposed using locally purified density operators (LPDOs), a tensor network representation of Choi matrices, to overcome the unfavourable scaling in parameters. While the LPDO structure allows it to satisfy a complete positivity' (CP) constraint required of physically valid quantum channels, it makes no guarantees about a similarly requiredtrace preservation' (TP) constraint. In practice, the TP constraint is violated, and the learned quantum channel may even be trace-increasing, which is non-physical. In this work, we present the problem of optimizing over TP LPDOs, discuss two approaches to characterizing the TP constraints on LPDOs, and outline the next steps for developing an optimization scheme.