dorsal/arxiv
View SchemaThe Minimum Distance Problem for Two-Way Entanglement Purification
| Authors | Andris Ambainis, Daniel Gottesman |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0310097 |
| URL | https://arxiv.org/abs/quant-ph/0310097 |
| DOI | 10.1109/TIT.2005.862089 |
| Journal | IEEE Trans. Info. Theory vol. 52, issue 2, 748-753 (2006) |
Abstract
Entanglement purification takes a number of noisy EPR pairs and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can achieve the rate k/n = 1 - (t/n) \log_2 3 - h(t/n) - O(1/n) (asymptotically reaching the quantum Hamming bound), where the EPP produces at least k good pairs out of n total pairs with up to t arbitrary errors, and h(x) = -x \log_2 x - (1-x) \log_2 (1-x) is the usual binary entropy. In contrast, the best known lower bound on the rate of QECCs is the quantum Gilbert-Varshamov bound k/n \geq 1 - (2t/n) \log_2 3 - h(2t/n). Indeed, in some regimes, the known upper bound on the asymptotic rate of good QECCs is strictly below our lower bound on the achievable rate of 2-EPPs.
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"abstract": "Entanglement purification takes a number of noisy EPR pairs and processes\nthem to produce a smaller number of more reliable pairs. If this is done with\nonly a forward classical side channel, the procedure is equivalent to using a\nquantum error-correcting code (QECC). We instead investigate entanglement\npurification protocols with two-way classical side channels (2-EPPs) for finite\nblock sizes. In particular, we consider the analog of the minimum distance\nproblem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound\nand the quantum Singleton bound. We also show that 2-EPPs can achieve the rate\nk/n = 1 - (t/n) \\log_2 3 - h(t/n) - O(1/n) (asymptotically reaching the quantum\nHamming bound), where the EPP produces at least k good pairs out of n total\npairs with up to t arbitrary errors, and h(x) = -x \\log_2 x - (1-x) \\log_2\n(1-x) is the usual binary entropy. In contrast, the best known lower bound on\nthe rate of QECCs is the quantum Gilbert-Varshamov bound k/n \\geq 1 - (2t/n)\n\\log_2 3 - h(2t/n). Indeed, in some regimes, the known upper bound on the\nasymptotic rate of good QECCs is strictly below our lower bound on the\nachievable rate of 2-EPPs.",
"arxiv_id": "quant-ph/0310097",
"authors": [
"Andris Ambainis",
"Daniel Gottesman"
],
"categories": [
"quant-ph"
],
"doi": "10.1109/TIT.2005.862089",
"journal_ref": "IEEE Trans. Info. Theory vol. 52, issue 2, 748-753 (2006)",
"title": "The Minimum Distance Problem for Two-Way Entanglement Purification",
"url": "https://arxiv.org/abs/quant-ph/0310097"
},
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