Randomly monitored quantum codes

Recent studies of monitored quantum dynamics have revealed that projective measurements, traditionally viewed as decohering operations, can instead generate and sustain long-range entanglement. Motivated by these, we ask how many physical qubits must be measured in random basis to irreversibly destroy quantum information encoded in a quantum error-correcting code. We study this problem for a broad class of stabilizer and subsystem codes, derive necessary and sufficient conditions for measurement-induced information destruction, and show that many codes, including concatenated and topological codes, achieve the maximal measurement threshold <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msubsup> <a:mi>p</a:mi> <a:mi>m</a:mi> <a:mi>th</a:mi> </a:msubsup> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:math> , meaning that the encoded information survives as long as arbitrarily small but finite fraction of physical qubits remain unmeasured. Beyond this surprising robustness, we find a structural relation underlying maximal thresholds. Namely, we prove that if the Pauli basis of the logical operator measured at full measurement does not concentrate on a single Pauli, then the measurement threshold always satisfies <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msubsup> <c:mi>p</c:mi> <c:mi>m</c:mi> <c:mi>th</c:mi> </c:msubsup> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:math> . This result uncovers a structural relation between logical measurement statistics and stability under partial measurement, revealing a general mechanism by which access to measurement outcomes enhances decodability under monitored dynamics.