Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^k}$</tex-math></inline-formula>

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 Fq, where q = pk, 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 Fq. 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.

Keywords

• Entanglement-assisted (EA) codes
• entanglement-assisted Calderbank–Shor–Steane (CSS) code
• extension procedure
• quantum error correction
• tensor product codes (TPCs)