ONE-PARTY QUANTUM-ERROR-CORRECTING CODES FOR UNBALANCED ERRORS: PRINCIPLES AND APPLICATION TO QUANTUM DENSE CODING AND QUANTUM SECURE DIRECT COMMUNICATION
In this article, we present unbalanced-quantum-error-correcting codes (one-party QECCs) — a novel idea for correcting unbalanced quantum errors. In some quantum communication tasks using entangled pairs, the error distributions between two parts of the pairs are unbalanced, and one party holds the whole entangled pairs at the final stage, and he or she is able to perform joint measurements on the pairs. In this situation the proposed one-party QECCs can improve error correction by allowing a higher-tolerated error rate. We have established the general correspondence between linear classical codes and the one-party QECCs, and we have given the general definition for these types of quantum-error-correcting codes. It has been shown that the one-party QECCs can correct errors as long as the error threshold is not larger than 0.5. They work even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples of the one-party QECCs. We provide the applications of the one-party QECCs in quantum dense coding, so that it can function in noisy channels. As a result, a large number of quantum secure direct communication protocols based on dense coding are also able to be protected by this new type of one-party QECCs.
In this article, we present unbalanced-quantum-error-correcting codes (one-party QECCs) — a novel idea for correcting unbalanced quantum errors. In some quantum communication tasks using entangled pairs, the error distributions between two parts of the pairs are unbalanced, and one party holds the whole entangled pairs at the final stage, and he or she is able to perform joint measurements on the pairs. In this situation the proposed one-party QECCs can improve error correction by allowing a higher-tolerated error rate. We have established the general correspondence between linear classical codes and the one-party QECCs, and we have given the general definition for these types of quantum-error-correcting codes. It has been shown that the one-party QECCs can correct errors as long as the error threshold is not larger than 0.5. They work even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples of the one-party QECCs. We provide the applications of the one-party QECCs in quantum dense coding, so that it can function in noisy channels. As a result, a large number of quantum secure direct communication protocols based on dense coding are also able to be protected by this new type of one-party QECCs.
In this article, we present the unbalanced quantum error correcting codes(one-party-QECC), a novel idea for correcting unbalanced quantum errors. In some quantum communication tasks using entangled pairs, the error distributions between two parts of the pairs are unbalanced, and. one party holds the whole entangled pairs at the final stage, and he or she is able to perform joint measurements on the pairs. In this situation the proposed one-party-QECC can improve error correction by allowing a higher tolerated error rate. We have established the general correspondence between linear classical codes and the one-party-QECC, and we have given the general definition for this type quantum error correcting codes.It has been shown that the one-party-QECC can correct errors as long as the error threshold is not larger than 0.5. The one-party-QECC works even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples of the one-party-QECC. We provide the applications of one-party-QECC in quantum dense coding so that it can function in noisy channels. As a result, a large number of quantum secure direct communication protocols based on dense coding is also able to be protected by this new type of one-party-QECC.
Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts D1(M) and D2(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.
Introduction Quantum error-correcting codes (QECCs) have turned out to have many applications; but their primary purpose, of course, is to protect quantum information from noise. This need manifests itself in two rather different situations. The first is in quantum computing. The qubits in a quantum computer are subject to noise due both to imprecision in the operations (or gates) and to interactions with the external environment. They undergo errors when they are acted upon, and when they are just being stored (though hopefully not at the same rate). All the qubits in the computer must be kept error-free long enough for the computation to be completed. This is the principle of fault tolerance. A rather different situation occurs in quantum communication . Here, the sender and receiver (Alice and Bob) are assumed to be physically separated, and the qubits travel from the sender to the receiver over a (presumably noisy) channel. This channel is assumed to be the dominant source of errors. While the qubits may undergo processing before and after transmission, errors during this processing are considered negligible compared to the channel errors. This picture of transmission through a channel is similar in spirit to the paradigm underlying much of classical information theory.
In quantum communication systems, quantum error-correcting codes (QECCs) are known to exhibit improved performance with the use of error-free entanglement bits (ebits). In practical situations, ebits inevitably suffer from errors, and as a result, the error-correcting capability of the code is diminished. Previous studies have proposed two different schemes as a solution. One study uses only one QECC to correct errors on the receiver side (i.e., Bob) and sender side (i.e., Alice). The other uses different QECCs on each side. In this paper, we present a method to correct errors on both sides by using single nonadditive entanglement-assisted codeword stabilized quantum error-correcting code (EACWS QECC). We use the property that the number of effective error patterns decreases as much as the number of ebits. This property results in a greater number of logical codewords using the same number of physical qubits.
Pre-shared non-local entanglement dramatically simplifies and improves the performance of quantum error correction via entanglement-assisted quantum error-correcting codes (EAQECCs). However, even considering the noise in quantum communication only, the non-local sharing of a perfectly entangled pair is technically impossible unless additional resources are consumed, such as entanglement distillation, which actually compromises the efficiency of the codes. Here we propose an error-correcting protocol assisted by two-way noisy communication that is more easily realisable: all quantum communication is subjected to general noise and all entanglement is created locally without additional resources consumed. In our protocol the pre-shared noisy entangled pairs are purified simultaneously by the decoding process. For demonstration, we first present an easier implementation of the well-known EAQECC [[4, 1, 3; 1]]. Then, we construct for the first time a 1-error-correcting code of 4 physical qudits capable of encoding 1 qubit of information, which is impossible in standard quantum error correction, demonstrating that our protocol can also improve the encoding rate in some sense. A systematic construction of two-way-noisy-communication-assisted codes is also provided.
Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein-Podolsky-Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.