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arXiv:0804.1404 · Physical Review A

Optimal entanglement formulas for entanglement-assisted quantum coding

Mark M. Wilde, Todd A. Brun

Published Jun 19, 2008• 189 citations
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Abstract

We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corollaries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.

DOI
10.1103/physreva.77.064302

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Entanglement-assisted quantum error-correcting codes are not only theoretically interesting, but also of great importance in practical physically applications. In this work, two class of entanglement-assisted quantum codes are constructed. The first class of entanglement-assisted quantum codes is minimal ebits entanglement-assisted quantum codes, which require only one copy of maximally entangled state no matter how large the code length is. The second class of entanglement-assisted quantum codes is maximal entanglement entanglement-assisted quantum codes, which can achieve the entanglement-assisted hashing bound asymptotically.
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.