# Mutual Compatibility/Incompatibility of Quasi-Hermitian Quantum Observables
## 4 Problem of Compatibility/Incompatibility
### Criteria  
In a way not noticed in [7], the test and verification of coincidence (14) are facilitated by the solution of Equation (11). This enables us to rewrite Equation (14) as relation  
\[\Omega_{1}^{\dagger}\,\mathfrak{c}_{1}^{2}\,\Omega_{1}=\Omega_{2}^{\dagger}\, \mathfrak{c}_{2}^{2}\,\Omega_{2} \tag{15}\]in which one can only vary the \(2N\) diagonal matrix elements of \(\epsilon_{1}^{2}\) and \(\epsilon_{2}^{2}\).  
One of the consequences of the set of constraints (15) is that we can immediately eliminate half of the variable parameters since we can write, say,  
\[\epsilon_{2}^{2}=M^{\dagger}\,\epsilon_{1}^{2}\,M\,,\quad M=\Omega_{1}\, \Omega_{2}^{-1}\,. \tag{16}\]  
Both sides of this relation are Hermitian matrices that have two parts. The diagonal parts are elementary and provide simply an explicit definition of the diagonal matrix \(\epsilon_{2}^{2}\) in terms of the known matrix \(M\) and the variable diagonal matrix \(\epsilon_{1}^{2}\).  
The \(N-\)plet of the real and positive variables in \(\epsilon_{1}^{2}\) is then still constrained by the rest of relation (16), which can be read, at the larger \(N\), as an over-determined set of \(N(N-1)/2\) constraints. They may be written, say, as a multiplet of linear equations  
\[\left(M^{\dagger}\,\epsilon_{1}^{2}\,M\right)_{m,n}=0\,,\quad\forall m<n\,. \tag{17}\]  
In practice, the implementation of such a criterion would be only straightforward in the two-dimensional case since at \(N=2\) we would just have a single constraint (17) imposed upon two real parameters. In general, in contrast, the two arbitrarily chosen independent matrices \(A_{1}\) and \(A_{2}\) will not be mutually compatible at \(N\geq 4\) (and more so when \(N=\infty\)) since \(N(N-1)/2>N\). In these cases, one will have to rely on the purely numerical techniques.  
For some purposes, a less pragmatic alternative approach to the problem could be based on our following final result.  
Lemma 2: _The two \(N\) by \(N\) matrix candidates \(A_{1}\) and \(A_{2}\) for observables can be declared compatible (i.e., simultaneously quasi-Hermitian) if and only if the \(2N-\)parametric matrix_  
\[\mathcal{U}(\epsilon_{1},\epsilon_{2})=\epsilon_{1}\,M\,\epsilon_{2}^{-1} \tag{18}\]  
_is unitary._  
Proof: The unitarity is equivalent to requirement (16).  
One of the consequences of the latter observation would be constructive. At a given Hilbert-space dimension \(N\), indeed, one could choose an arbitrary unitary matrix \(\mathcal{U}\) and, having recalled Equation (18), one would be able to define the matrix \(M\). After its pre-multiplication by some \(2N\) arbitrary constants, a more or less arbitrary factorization would yield, in the light of Equation (16), the doublet of matrices \(\Omega_{1}\) and \(\Omega_{2}\). In terms of any one of them, one could define the \"shared\" metric \(\Theta\). An arbitrary choice in the two diagonal matrices, \(\alpha_{1}\) and \(\alpha_{2}\), would then finally lead to the construction of the compatible pair of operators, \(A_{1}\) and \(A_{2}\), via the respective relations (8).  
In the opposite direction of the constructive analysis, the process of implementation of the unitarity criterion (18) could be based on the most general unitary-matrix ansatz \(\mathcal{U}\) for the left-hand side of Equation (18). Thus, in the simplest example with \(N=2\), such an ansatz would be four-parametric,  
\[\mathcal{U}=\exp\!\left[\begin{array}{cc}i\alpha&i\gamma+\delta\\ i\gamma-\delta&i\beta\end{array}\right]\,. \tag{19}\]  
This means that, at any given two-by-two dynamical-input-information matrix \(M^{(2)}\), one could search for the four unknown parameters \(\kappa_{j}^{[n]}\) with \(j=1,2\) and \(n=1,2\) via the solution of the set of constraints (18), forming a quadruplet of coupled nonlinear algebraic equations.