Let $A\_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r\_i, i=1, \ldots, N$ be independent Rademacher random variables. 
The following inequality gives a bound on the expectation of the p-th Schatten norm:
$$
E\left\|\sum\_{i=1}^N r\_i A\_i\right\|\_{S\_{p}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum\_{i=1}^NA\_iA\_i^\*\right)^{1/2}\right\|\_{S\_{p}}^{p}, \, \left\|\left(\sum\_{i=1}^NA\_i^\*A\_i\right)^{1/2}\right\|\_{S\_{p}}^{p}\right\}.
$$
Question: whether some similar bounds known for p-th moment of the fixed q-th Schatten norm, i.e. something like
$$
E\left\|\sum\_{i=1}^N r\_i A\_i\right\|\_{S\_{q}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum\_{i=1}^NA\_iA\_i^\*\right)^{1/2}\right\|\_{S\_{q}}^{p}, \, \left\|\left(\sum\_{i=1}^NA\_i^\*A\_i\right)^{1/2}\right\|\_{S\_{q}}^{p}\right\}.
$$



Matrix inequalities for the moment of the fixed Shatten norm
If you do not care about the precise constants, the answer is yes, and this follows directely from the Kahane inequality (which says that, in any normed space and $1 \leq p,q<\infty$, the $L\_p$-norm of $\sum r\_i x\_i$ is dominated by its $L\_q$-norm, up to constant depending on $p,q$ only). The conclusion is that for every $1 \leq p, q <\infty$, there is a constant $C\_{p,q}$ such that 
$$
\left(E \|\sum\_{i=1}^N r\_i A\_i\|\_{S\_{q}}^{p}\right)^{\frac 1 p}\leq C\_{p,q} \max\left\{\left\|\left(\sum\_{i=1}^NA\_iA\_i^\*\right)^{1/2}\right\|\_{S\_{q}}, \, \left\|\left(\sum\_{i=1}^NA\_i^\*A\_i\right)^{1/2}\right\|\_{S\_{q}}\right\}.
$$


Observe that it is necessary to take the $1/p$-th power if you want to have $C\_{p,p} \leq C \sqrt{p}$.


If you care about the constants, what you ask for is way too strong. It is clearly necessary to have $C\_{p,q} \to \infty$ as $q \to \infty$ (as otherwise by considering diagonal matrices it would give Khintchine inequalities in $\ell\_\infty$).


