  This paper explores the fundamental limits of unsourced random access (URA)
with a random and unknown number ${\rm{K}}_a$ of active users in MIMO
quasi-static Rayleigh fading channels. First, we derive an upper bound on the
probability of incorrectly estimating the number of active users. We prove that
it exponentially decays with the number of receive antennas and eventually
vanishes, whereas reaches a plateau as the power and blocklength increase.
Then, we derive non-asymptotic achievability and converse bounds on the minimum
energy-per-bit required by each active user to reliably transmit $J$ bits with
blocklength $n$. Numerical results verify the tightness of our bounds,
suggesting that they provide benchmarks to evaluate existing schemes. The extra
required energy-per-bit due to the uncertainty of the number of active users
decreases as $\mathbb{E}[{\rm{K}}_a]$ increases. Compared to random access with
individual codebooks, the URA paradigm achieves higher spectral and energy
efficiency. Moreover, using codewords distributed on a sphere is shown to
outperform the Gaussian random coding scheme in the non-asymptotic regime.
