Entanglement-enhanced quantum error-correcting codes

Abstract

Via explicit examples we show that the pre-existing entanglement can really enhance ͑not only behave as an assistance for͒ the efficiency of the quantum error-correcting codes ͑QECCs͒ in a single block of encoding or decoding as well as help in beating the quantum Hamming bound. A systematic approach to constructing entanglement-assisted ͑or enhanced͒ quantum error-correcting codes ͑EAQECCs͒ via graph states is also presented, and an infinite family of entanglement-enhanced codes has been constructed. Furthermore we generalize the EAQECCs to the case of not-so-perfectly protected qubit and introduce the quantity infidelity as a figure of merit and show that the EAQECCs also outperform the ordinary QECCs.

Breeding quantum error-correcting codes

“…The atoms trapped in fullerenes which tend to suffer much less noise provide another example. Especially, the logical qubits that have been preprotected by some error-correcting codes can also be looked upon as pure qubits for they have an error probability much smaller than that of normal physical qubits and can be used as the "seed" in the catalytic mode [15,19]. The EAQECCs can outperform the ordinary QECCs both catalytically and noncatalytically [19].…”

“…Especially, the logical qubits that have been preprotected by some error-correcting codes can also be looked upon as pure qubits for they have an error probability much smaller than that of normal physical qubits and can be used as the "seed" in the catalytic mode [15,19]. The EAQECCs can outperform the ordinary QECCs both catalytically and noncatalytically [19]. However the parameters of the EAQECCs need to be determined [18] by some detailed properties of the corresponding classical (linear) codes.…”

“…In fact we construct the whole family of optimal one-error-correcting breeding codes with respect to the Hamming bound [20]. Some of them can be constructed from linear classical codes and some of them from nonlinear classical additive codes, which are genuine entanglement-enhanced QECC [19].…”

“…However, just as illustrated in Ref. [19], if a qubit has a much longer decoherence time than that needed during the whole quantum information process or a much smaller error probability than that of the other qubits used in the same time, it then can be looked upon as a pure qubit. Practically, in a communication scenario [15], half qubits of the pre-existing ideal Einstein-Podolsky-Rosen (EPR) pairs can be viewed as pure qubits because they do not need to transfer through *…”

Quantum secret sharing with continuous variable graph state

“…On the other hand, graph state has been extensively studied in applications such as quantum error correction [11][12][13][14], entanglement purification [15][16][17], entanglement measurement [18][19][20], and Bell inequality [21,22]. In recent years, the implementation of QSS with graph state was introduced in [7,23] to treat three kinds of threshold QSS schemes in a unified graph state approach and to propose embedded protocols in large graph states.…”