Bounds for Pure Disjoint $(r,\delta )$-Quantum Locally Recoverable Codes

We study pure disjoint $(r,\delta )$-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill–Laflamme conditions for recovery from up to $\delta -1$ erasures within a recovery block, and introduce blockwise Shor–Laflamme and unitary weight enumerators that capture how error weight is distributed across recovery sets. We establish several properties of these enumerators and use them to derive a Singleton-like bound that strengthens the known bound for disjoint $(r,\delta )$-qLRCs under a purity assumption, as well as a linear-programming upper bound on the code dimension. These results provide a non-stabilizer, weight-enumerator-based approach to the study of pure disjoint $(r,\delta )$-qLRCs.