  Measurement-induced phase transitions (MIPTs) are known to be described by
non-unitary conformal field theories (CFTs) whose precise nature remains
unknown. Most physical quantities of interest, such as the entanglement
features of quantum trajectories, are described by boundary observables in this
CFT. We introduce a transfer matrix approach to study the boundary spectrum of
this field theory, and consider a variety of boundary conditions. We apply this
approach numerically to monitored Haar and Clifford circuits, and to the
measurement-only Ising model where the boundary scaling dimensions can be
derived analytically. Our transfer matrix approach provides a systematic
numerical tool to study the spectrum of MIPTs.
