  The classical Strichartz estimates for the free Schr\"odinger propagator have
recently been substantially generalised to estimates of the form \[
\bigg\|\sum_j\lambda_j|e^{it\Delta}f_j|^2\bigg\|_{L^p_tL^q_x}\lesssim\|\lambda\|_{\ell^\alpha}
\] for orthonormal systems $(f_j)_j$ of initial data in $L^2$, firstly in work
of Frank--Lewin--Lieb--Seiringer and later by Frank--Sabin. The primary
objective is identifying the largest possible $\alpha$ as a function of $p$ and
$q$, and in contrast to the classical case, for such estimates the critical
case turns out to be $(p,q) = (\frac{d+1}{d},\frac{d+1}{d-1})$. We consider the
case of orthonormal systems $(f_j)_j$ in the homogeneous Sobolev spaces
$\dot{H}^s$ for $s \in (0,\frac{d}{2})$ and we establish the sharp value of
$\alpha$ as a function of $p$, $q$ and $s$, except possibly an endpoint in
certain cases, at which we establish some weak-type estimates. Furthermore, at
the critical case $(p,q) = (\frac{d+1}{d-2s},\frac{d(d+1)}{(d-1)(d-2s)})$ for
general $s$, we show the veracity of the desired estimates when $\alpha = p$ if
we consider frequency localised estimates, and the failure of the
(non-localised) estimates when $\alpha = p$; this exhibits the difficulty of
upgrading from frequency localised estimates in this context, again in contrast
to the classical setting.
