  In addition to the implementation of parity-time ($\mathcal{PT}$)-symmetric
optical systems by carefully and actively controlling the gain and loss, we
show that a $2\times 2$ $\mathcal{PT}$-symmetric Hamiltonian has a unitarily
equivalent representation without complex optical potentials in the resulting
optical coupler. Through the Naimark dilation in operator algebra, passive
$\mathcal{PT}$-symmetric couplers can thus be implemented with a refractive
index of real values and asymmetric coupling coefficients. This opens up the
possibility to implement general $\mathcal{PT}$-symmetric systems with
state-of-the-art asymmetric slab waveguides, dissimilar optical fibers, or
cavities with chiral mirrors.
