  Non-Hermitian systems with parity-time symmetry have been found to exhibit
real spectra of eigenvalues, indicating a balance between the loss and gain.
However, such a balance is not only dependent on the magnitude of loss and
gain, but also easily broken due to external disturbance. Here, the authors
propose a transformation-optics approach to construct a unique class of
non-Hermitian systems with robustly balanced loss and gain, irrespective of the
magnitude of loss/gain and the environmental disturbance. Through
transformation-optics operators like space folding and stretching, loss and
gain can be generated and separated in the real space. While in the virtual
space, the loss and gain are still combined to each other, rendering a balance
of energy that is far more robust than other non-Hermitian systems. This
amazing feature is verified by finite-element simulations. This work reveals a
class of non-Hermitian systems in which loss and gain are balanced robustly,
thereby denoted as pseudo-Hermitian systems.
