dorsal/arxiv
View SchemaMixed State Entanglement and Quantum Error Correction
| Authors | Charles H. Bennett, David P. DiVincenzo, John A. Smolin, William K. Wootters |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9604024 |
| URL | https://arxiv.org/abs/quant-ph/9604024 |
| DOI | 10.1103/PhysRevA.54.3824 |
| Journal | Phys.Rev.A54:3824-3851,1996 |
Abstract
Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) 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 arbi- trary quantum state $|\xi\rangle$ can be transmitted at some rate Q through a noisy channel $\chi$ without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state $\hat{M}(\chi)$ (obtained by sharing halves of EPR pairs through a channel $\chi$) yields a QECC on $\chi$ 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 $D_1(M)$ and $D_2(M)$ 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 communica- tion, 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 com- munication. 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 com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve 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 {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.
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"abstract": "Entanglement purification protocols (EPP) and quantum error-correcting codes\n(QECC) provide two ways of protecting quantum states from interaction with the\nenvironment. In an EPP, perfectly entangled pure states are extracted, with\nsome yield D, from a mixed state M shared by two parties; with a QECC, an arbi-\ntrary quantum state $|\\xi\\rangle$ can be transmitted at some rate Q through a\nnoisy channel $\\chi$ without degradation. We prove that an EPP involving one-\nway classical communication and acting on mixed state $\\hat{M}(\\chi)$ (obtained\nby sharing halves of EPR pairs through a channel $\\chi$) yields a QECC on\n$\\chi$ with rate $Q=D$, and vice versa. We compare the amount of entanglement\nE(M) required to prepare a mixed state M by local actions with the amounts\n$D_1(M)$ and $D_2(M)$ that can be locally distilled from it by EPPs using one-\nand two-way classical communication respectively, and give an exact expression\nfor $E(M)$ when $M$ is Bell-diagonal. While EPPs require classical communica-\ntion, QECCs do not, and we prove Q is not increased by adding one-way classical\ncommunication. However, both D and Q can be increased by adding two-way com-\nmunication. We show that certain noisy quantum channels, for example a 50%\ndepolarizing channel, can be used for reliable transmission of quantum states\nif two-way communication is available, but cannot be used if only one-way com-\nmunication is available. We exhibit a family of codes based on universal hash-\ning able toachieve an asymptotic $Q$ (or $D$) of 1-S for simple noise models,\nwhere S is the error entropy. We also obtain a specific, simple 5-bit single-\nerror-correcting quantum block code. We prove that {\\em iff} a QECC results in\nhigh fidelity for the case of no error the QECC can be recast into a form where\nthe encoder is the matrix inverse of the decoder.",
"arxiv_id": "quant-ph/9604024",
"authors": [
"Charles H. Bennett",
"David P. DiVincenzo",
"John A. Smolin",
"William K. Wootters"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.54.3824",
"journal_ref": "Phys.Rev.A54:3824-3851,1996",
"title": "Mixed State Entanglement and Quantum Error Correction",
"url": "https://arxiv.org/abs/quant-ph/9604024"
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