## 1 Introduction  
Propagation through classical \(\mathcal{PT}\)-symmetric optical systems has been extensively studied; cf. Ref. [1] and references therein from the initial description of linear loses in directional couplers with a more complex non-Hermitian symmetry [2], the quantum-like description of classical planar waveguides [3], to all the contemporaneous work derived from the seminal introduction of optical \(\mathcal{PT}\)-symmetric structures [4]. It is well known that the propagation of electromagnetic field through a linearly active two-waveguide coupler can be described by the classical \(\mathcal{PT}\)-symmetric dimer,  
\[-i\frac{d}{dz}\left(\begin{array}{c}\mathcal{E}_{1}(z)\\ \mathcal{E}_{2}(z)\end{array}\right)=\left(\begin{array}{cc}i\gamma&g\\ g&-i\gamma\end{array}\right)\left(\begin{array}{c}\mathcal{E}_{1}(z)\\ \mathcal{E}_{2}(z)\end{array}\right), \tag{1}\]  
where the effective evanescent coupling between the two single waveguide field modes is given by the real parameter \(g\) and the effective gain and loss by the real positive parameter \(\gamma\). Such a system can be realized experimentally by balanced gain and loss in the waveguides, but it is also possible to realize it with gain-gain, loss-loss, passive-gain [5], and passive-loss [6, 7] waveguide, microcavity rings [8, 9], and electric circutits [10] setups.  
In the quantum regime [11, 12, 13, 14], the importance of adequately modeling media with linear gain or loss has been brought forward recently [15, 16]. In this regime, linear media induces quantum fluctuations, such that the ideal \(\mathcal{PT}\)-symmetric optical dimer dynamics is effectively described by quantum Langevin equations [15],  
\[-i\frac{d}{dz}\left(\begin{array}{c}\hat{a}_{1}(z)\\ \hat{a}_{2}(z)\end{array}\right)=\left(\begin{array}{cc}i\gamma&g\\ g&-i\gamma\end{array}\right)\left(\begin{array}{c}\hat{a}_{1}(z)\\ \hat{a}_{2}(z)\end{array}\right)+\left(\begin{array}{c}\hat{f}_{1}(z)\\ \hat{f}_{2}(z)\end{array}\right), \tag{2}\]in terms of the effective balanced gain and loss parameter, \(\gamma\), the effective mode coupling, \(g\), between the two modes of the optical resonators, described by the annihilation operators \(\hat{a}_{1}(z)\) and \(\hat{a}_{2}(z)\), and the delta-correlated Langevin forces introduced by the gain and loss media,  
\[\langle\hat{f}_{1}^{\dagger}(z)\hat{f}_{1}(\zeta)\rangle =2\gamma\delta(z-\zeta),\] \[\langle\hat{f}_{2}(z)\hat{f}_{2}^{\dagger}(\zeta)\rangle =2\gamma\delta(z-\zeta), \tag{3}\]  
in that order. These Gaussian fluctuations modify the well-known dynamics produced by the classical optical \(\mathcal{PT}\)-symmetric dimer and generate second order correlations in the ideal quantum optical \(\mathcal{PT}\)-symmetric dimer [15], Figure 1. Here, we want to discuss the different types of correlations that might arise from feasible experimental realizations of the quantum \(\mathcal{PT}\)-symmetric dimer beyond the balanced gain-loss setup. In the following section, we will introduce the quantum model for a generalized linear active dimer, a non-Hermitian quantum optical dimer, and provide its propagation solution. Then, we will discuss the dynamics of spontaneous photon generation as well as photon bunching and anti-bunching in the different configurations. We will study single-photon propagation with spatially separable and two-photon propagation with entangled states through the mean photon number and second order two-point correlations. Finally, we will close with a brief conclusion and discuss how mean photon propagation and second order spatial correlations might provide insight to their application in photon switching or as semi(non)-classical light sources.