$b=u+iv$, where $x$, $y$, $u$, and $v$ are real. Then $$\langle\psi|\psi\rangle=\left(x^2+v^2+2xv\sin\alpha
+y^2+u^2-2yu\sin\alpha\right)/\cos(\alpha),
\label{e26}$$ which is explicitly positive and vanishes only if $x=y=u=v=0$.

Since $\langle u|$ denotes the $\mathcal{CPT}$ conjugate of $|u\rangle$, the completeness condition reads $$|\varepsilon_+\rangle\langle\varepsilon_+|+|\varepsilon_-\rangle\langle
\varepsilon_-|=\left(\begin{array}{cc} 1 & 0 \cr 0 &
1\end{array}\right).
\label{e27}$$ Furthermore, using the $\mathcal{CPT}$ conjugate $\langle\varepsilon_\pm|$, we have $\mathcal{C}$ as $\mathcal{C}=|\varepsilon_+\rangle\langle\varepsilon_+|-|
\varepsilon_-\rangle\langle\varepsilon_-|,$ instead of the representation in (\[e14\]), which uses the $\mathcal{PT}$ conjugate.

For the two-state system discussed here, if $\theta\to0$, then the Hamiltonian (\[e21\]) becomes Hermitian. However, in this limit $\mathcal{C}$ reduces to the parity operator $\mathcal{P}$. As a consequence, the requirement of $\mathcal{CPT}$ invariance reduces to the standard condition of Hermiticity for a symmetric matrix; namely, that $H=H^*$. This is why the hidden symmetry $\mathcal{C}$ was not noticed previously. The operator $\mathcal{C}$ emerges only when we extend a real symmetric Hamiltonian into the complex domain.

Perturbative Calculation of the $\mathcal{C}$ Operator for an $\boldsymbol{ix^3}$ Theory {#s4}
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The $\mathcal{C}$ operator can be calculated in some infinite-dimensional quantum mechanical models. For an $x^2+ix^3$ potential $\mathcal{C}$ can be obtained from the summation in (\[e14\]) using perturbative methods and for an $x^2-x^4$ potential $\mathcal{C}$ can be calculated using nonperturbative methods [@rr11]. In this paper we focus 