  We show that a relatively simple reasoning using von Neumann entropy
inequalities yields a robust proof of the quantum Singleton bound for quantum
error-correcting codes (QECC). For entanglement-assisted quantum
error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of
generalized quantum Singleton bound [Brun et al., IEEE Trans. Inf. Theory
60(6):3073--3089 (2014)] was believed to hold for many years until recently one
of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we
rectify this state of affairs by proving the correct generalized quantum
Singleton bound, extending the above-mentioned proof method for QECC; we also
prove information-theoretically tight bounds on the entanglement-communication
tradeoff for EAQECC. All of the bounds relate block length $n$ and code length
$k$ for given minimum distance $d$ and we show that they are robust, in the
sense that they hold with small perturbations for codes which only correct most
of the erasure errors of less than $d$ letters. In contrast to the classical
case, the bounds take on qualitatively different forms depending on whether the
minimum distance is smaller or larger than half the block length. We also
provide a propagation rule: any pure QECC yields an EAQECC with the same
distance and dimension, but of shorter block length.
