Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over $\mathbb {F}_{p^k}$

Abstract

We provide a procedure to construct entanglement-assisted Calderbank–Shor–Steane (CSS) codes over qudits from the parity check matrices of two classical codes over $\mathbb {F}_q$ , where $q=p^k$ , $p$ is prime, and $k$ is a positive integer. The construction procedure involves the proposed Euclidean Gram–Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over $\mathbb {F}_q$ . We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.