Signal-to-interference-plus-noise ratio is an important performance measure, which is defined as
$$\mathrm{SINR} = \frac{P}{I+N}$$
where $P$ is the power of signal of interest, I is the interference power, and N is the noise power.
With a linear MIMO receiver, e.g., ZF and MMSE, the estimated symbol can be written as
$$\mathbf{y}=\mathbf{D}\mathbf{s}+\mathbf{Z}\mathbf{s}+\mathbf{W}\mathbf{n}$$
where $\mathbf{D}$ is a diagonal matrix, $\mathbf{Z}$ is a matrix which contains interferences between data streams and $\mathbf{W}\mathbf{n}$ is the filtered noise. Thus, the signal power, the interference power and the noise power can be obtained by $$P=\mathrm{E}[\Vert \mathbf{D}\mathbf{s} \Vert^2 ]$$ $$I=\mathrm{E}[\Vert \mathbf{Z}\mathbf{s} \Vert^2 ]$$ $$N=\mathrm{E}[\Vert \mathbf{W}\mathbf{n} \Vert^2 ]$$
The post-processing SINR can be expressed as $$\mathrm{SINR} = \frac{E(\Vert \mathrm{D}\mathbf{s} \Vert^2) }{E_s \cdot \mathrm{trace}(\mathbf{Z}\mathbf{Z}^{H})+\sigma_{n}^2 \cdot \mathrm{trace} (\mathbf{W}\mathbf{W}^{H})}$$
For a ZF detector, due to $\mathbf{Z} = 0$ and $\mathbf{D}=\mathbf{I}$, the formula can be expressed as $$\mathrm{SINR_{ZF}} = \frac{1}{\sigma_{n}^2 \cdot \mathrm{trace}(\mathbf{W}_{ZF}\mathbf{W}^{H}_{ZF})} = \frac{1}{\mathrm{MSE}_{ZF}}$$ For a MMSE detector, assuming that $E_s = 1$, the expression can be represented as $$\mathrm{SINR_{MMSE}} = \frac{\mathrm{trace}(\mathbf{D}\mathbf{D}^{H})}{\mathrm{trace}(\mathbf{Z}\mathbf{Z}^{H})+\sigma_{n}^2 \cdot \mathrm{trace} (\mathbf{W}_{MMSE}\mathbf{W}^{H}_{MMSE})}$$ For SIC methods, similar to ZF/MMSE, but the effect of error propagation has to be additionally addressed.