  The Prime state of $n$ qubits, $|\mathbb{P}_n\rangle$, is defined as the
uniform superposition of all the computational-basis states corresponding to
prime numbers smaller than $2^n$. This state encodes, quantum mechanically,
arithmetic properties of the primes. We first show that the Quantum Fourier
Transform of the Prime state provides a direct access to Chebyshev-like biases
in the distribution of prime numbers. We next study the entanglement entropy of
$|\mathbb{P}_n\rangle$ up to $n=30$ qubits, and find a relation between its
scaling and the Shannon entropy of the density of square-free integers. This
relation also holds when the Prime state is constructed using a qudit basis,
showing that this property is intrinsic to the distribution of primes. The same
feature is found when considering states built from the superposition of primes
in arithmetic progressions. Finally, we explore the properties of other
number-theoretical quantum states, such as those defined from odd composite
numbers, square-free integers and starry primes. For this study, we have
developed an open-source library that diagonalizes matrices using floats of
arbitrary precision.
