Entanglement-assisted Quantum Reed-Muller Tensor Product Codes

Priya J. Nadkarni1, Praveen Jayakumar1,2, Arpit Behera1,3, and Shayan Srinivasa Garani1

1Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012
2Department of Chemistry, University of Toronto, Toronto, Canada, M5S 3H6
3Department of Complex Systems, Weizmann Institute of Science, Rehovot, Israel, 7610001

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We present the construction of standard entanglement-assisted (EA) qubit Reed-Muller (RM) codes and their tensor product variants from classical RM codes. We show that the EA RM codes obtained using the CSS construction have zero coding rate and negative catalytic rate. We further show that EA codes constructed from these same classical RM codes using the tensor product code (TPC) construction have positive coding rate and provide a subclass of EA RM TPCs that have positive catalytic rate, thus establishing the coding analog of superadditivity for this family of codes, useful towards quantum communications. We also generalize this analysis to obtain conditions for EA TPCs from classical codes to have positive catalytic rate when their corresponding EA CSS codes have zero rate.

The phenomenon of superadditivity of quantum channel capacity involving two zero-capacity quantum channels that achieve a positive quantum capacity does not have a classical bearing. A coding analog of this phenomenon was first demonstrated for quantum CSS codes. Two classical codes individually yielding zero-rate quantum codes were combined to obtain a positive rate quantum code using the quantum tensor product construction. The present study further demonstrates that the coding analog of superadditivity is applicable for constructing quantum codes from well-studied Reed-Muller codes that individually yield zero-rate quantum codes. Conditions for observing this phenomena with polar codes and other classical codes are provided. Our findings enable the construction of many algebraic quantum codes with good properties, useful towards quantum communications.

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