Entanglement-Assisted Quantum Error-Correcting Codes with Imperfect Ebits

TL;DR

This paper develops methods for entanglement-assisted quantum error-correcting codes that can handle errors on receiver's ebits, compares their performance, and discusses tradeoffs with entanglement distillation.

Contribution

It introduces two schemes for EAQEC codes that correct errors on both sender and receiver qubits, extending standard stabilizer codes to more practical scenarios.

Findings

EAQEC codes can be adapted to correct errors on receiver's ebits.

Optimal EAQEC codes are identified and shown to be equivalent to standard stabilizer codes.

Channel fidelity can be efficiently estimated using Monte Carlo methods.

Abstract

The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error-free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer codes can be applied. Several EAQEC codes of this type are found to be optimal. In a second scheme, the receiver uses a standard stabilizer code to protect the ebits, which we call a "combination code." The performances of different quantum codes are compared in terms of the channel fidelity over the depolarizing channel. We give a formula for the channel fidelity over the depolarizing channel (or any Pauli error channel), and show that it can be efficiently approximated by a Monte Carlo calculation. Finally, we discuss the tradeoff between performing extra entanglement distillation and applying an EAQEC code with imperfect ebits.