  Recently, open systems with balanced, spatially separated loss and gain have
been realized and studied using non-Hermitian Hamiltonians that are invariant
under the combined parity and time-reversal ($\mathcal{PT}$) operations. Here,
we model and investigate the effects of a local, two-state, quantum degree of
freedom, called a pseudospin, on a one-dimensional tight-binding lattice with
position-dependent tunneling amplitudes and a single pair of non-Hermitian,
$\mathcal{PT}$-symmetric impurities. We show that if the resulting Hamiltonian
is invariant under exchange of two pseudospin labels, the system can be
decomposed into two uncoupled systems with tunable threshold for $\mathcal{PT}$
symmetry breaking. We discuss implications of our results to systems with
specific tunneling profiles, and open or periodic boundary conditions.
