Dualities and Identities for Entanglement-Assisted Quantum Codes

TL;DR

This paper explores the duality of entanglement-assisted quantum error-correcting codes, introduces optimal constructions, establishes bounds, and analyzes their performance over noisy channels.

Contribution

It introduces a duality concept for EAQEC codes, provides explicit encoding circuits, proves optimality, and establishes bounds and performance metrics for these codes.

Findings

EA repetition and accumulator codes are dual to each other.

The paper establishes Gilbert-Varshamov and Plotkin bounds for EAQEC codes.

Upper bounds on block error probability and variations of the hashing bound are derived.

Abstract

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code's information qubits with its ebits. To introduce this notion, we show how entanglement-assisted (EA) repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes.We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and we use these to examine the existence of some EAQEC codes. Finally, we provide upper bounds on the block error probability when transmitting maximal-entanglement EAQEC codes over the depolarizing channel, and we derive variations of the hashing bound for EAQEC codes, which is a lower bound on the maximum rate at which reliable communication over Pauli channels is possible with the use of pre-shared entanglement.