  Quantum measurement has conventionally been regarded as the final step in
quantum information processing, which is essential for reading out the
processed information but collapses the quantum state into a classical state.
However, recent studies have shown that quantum measurement itself can induce
novel quantum phenomena. One seminal example is a monitored random circuit,
which can generate long-range entanglement faster than a random unitary
circuit. Inspired by these results, in this paper, we address the following
question: When quantum information is encoded in a quantum error-correcting
code, how many physical qubits should be randomly measured to destroy the
encoded information? We investigate this question for various quantum
error-correcting codes and derive the necessary and sufficient conditions for
destroying the information through measurements. In particular, we demonstrate
that for a large class of quantum error-correcitng codes, it is impossible to
destroy the encoded information through random single-qubit Pauli measurements
when a tiny portion of physical qubits is still unmeasured. Our results not
only reveal the extraordinary robustness of quantum codes under measurement
decoherence, but also suggest potential applications in quantum information
processing tasks.
