Simultaneously Minimizing Storage and Bandwidth Under Exact Repair With Quantum Entanglement

We study exact-regenerating codes for entanglement-assisted distributed storage systems. Consider an $(n,k,d,\alpha,\beta_{\mathsf{q}},B)$ distributed system that stores a file of $B$ classical symbols across $n$ nodes with each node storing $\alpha$ symbols. A data collector can recover the file by accessing any $k$ nodes. When a node fails, any $d$ surviving nodes share an entangled state, and each of them transmits a quantum system of $\beta_{\mathsf{q}}$ qudits to a newcomer. The newcomer then performs a measurement on the received quantum systems to generate its storage. Recent work [1] showed that, under functional repair where the regenerated content may differ from that of the failed node, there exists a unique optimal regenerating point that \emph{simultaneously minimizes both storage $\alpha$ and repair bandwidth $d \beta_{\mathsf{q}}$} when $d \geq 2k-2$. In this paper, we show that, under \emph{exact repair}, where the newcomer reproduces exactly the same content as the failed node, this optimal point remains achievable. Our construction builds on the classical product-matrix framework and the Calderbank-Shor-Steane (CSS)-based stabilizer formalism.