The non-unitarity of the neutrino mixing matrix is a problem related with a more fundamental question about the origin of the neutrino mass. After a brief discussion on the questions ``why do we need model-independent framework for unitarity test?'', we show interesting properties present in the oscillation probabilities of neutrinos propagating in matter with non-unitarity. That is, (1) partial unitarity and (2) universal phase correlation.
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To illuminate these points, we formulate a perturbative framework with the two expansion parameters $\epsilon \equiv \Delta m^2_{21} / \Delta m^2_{31}$ and $\alpha$ matrix elements. The complex triangular $\alpha$ matrix is introduced through the definition of $3 \times 3$ non-unitary flavor mixing matrix $N$ as
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\begin{eqnarray}
N &=&
\left( \bf{1} - \alpha \right) U =
\left\{
\bf{1} -
\left[
\begin{array}{ccc}
\alpha_{ee} & 0 & 0 \\
\alpha_{\mu e} & \alpha_{\mu \mu} & 0 \\
\alpha_{\tau e} & \alpha_{\tau \mu} & \alpha_{\tau \tau} \\
\end{array}
\right]
\right\}
U
\label{alpha-matrix-def}
\end{eqnarray}
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where $U$ is the unitary MNS mixing matrix, and hence $\alpha$ characterizes the size and the flavor dependence of unitarity violation (UV)\footnote{
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Despite that in the physics literature UV usually means ``ultraviolet'', we use UV in this manuscript as an abbreviation for ``unitarity violation'' or ``unitarity violating''.}~~caused by new physics (NP) at low- or high-scales.

~~~~The point (1) above essentially means that despite non-unitary mixing, neutrino evolution must be unitary because the three active neutrinos span a complete state space of neutral leptons. The {\em phase correlation} mentioned in the point (2) refers an intriguing property that the complex $\alpha$ parameters and $\nu$Standard Model CP phase $\delta$ always come into the oscillation probabilities in a correlated way, $e^{- i \delta } \alpha_{\mu e}$, $\alpha_{\tau e}$, and $e^{i \delta} \alpha_{\tau \mu}$, universally in all the oscillation channels. The physical meaning of this result is briefly discussed. }