Inequality on spectral norm of sum of tensor products I am looking for a lower bound on
$$\left\|\sum\nolimits_i M_i \otimes M_i\right\|_{\infty}$$
when the $M_i$'s are positive semi-definite matrices and $\|.\|_{\infty}$ denote the spectral norm. I suspect that the inequality $\left\|\sum\nolimits_i M_i \otimes M_i\right\|_{\infty} \geq \left\|\sum\nolimits_i M_i^2\right\|_{\infty}$ that holds when the matrices commute is not true in general. I'm interested in any counter-example to that, and alternative inequalities.
 A: Since you have
$$
0\leq M_k\otimes M_k\leq\sum_jM_j\otimes M_j, 
$$
and $\|M_k\|^2=\|M_k\otimes M_k\|$, you have the inequality
$$
\|M_k\|^2\leq\Big\|\sum_jM_j\otimes M_j\Big\|,
$$
which gives
$$
\max_k\|M_k\|^2\leq\Big\|\sum_jM_j\otimes M_j\Big\|.
$$
This bound is sharp. For instance suppose that $P_1,\ldots,P_r$ are pairwise orthogonal projections. Then the $P_j\otimes P_j$ are also pairwise orthogonal projections. Define $M_k=\frac1k\,P_k$. Then
$$
\Big\|\sum_jM_j\otimes M_j\Big\|=\Big\|\sum_j\frac1{j^2}\,P_j\otimes P_j\Big\|=1=\|M_1\otimes M_1\|=\|M_1\|^2. 
$$
And it doesn't look like one can improve this. In particular, the bound in the question does not hold in general. Consider
$$
M_1=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\qquad M_2=\begin{bmatrix} 1&1\\1&1\end{bmatrix}. 
$$
We have
$$
\|M_1\|^2=1,\qquad\|M_2^2\|=4.
$$
Also,
$$
\|M_1^2+M_2^2\|=\left\|\begin{bmatrix}3&2\\2&2 \end{bmatrix} \right\|=\frac{5+\sqrt{17}}2,
$$
while
$$
\|M_1\otimes M_1+M_2\otimes M_2\|=\left\|\begin{bmatrix} 2&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{bmatrix}\right\|=\frac{5+\sqrt{13}}2
$$
