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arXiv:0912.2150 · Quantum Information Processing

Nonlocal quantum information in bipartite quantum error correction

Mark M. Wilde, David A. Fattal

Published Mar 31, 2010• 16 citations
arXivHTMLPDF
DOI
10.1007/s11128-010-0175-0

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We show how to convert an arbitrary stabilizer code into a bipartite quantum code. A bipartite quantum code is one that involves two senders and one receiver. The two senders exploit both nonlocal and local quantum resources to encode quantum information with local encoding circuits. They transmit their encoded quantum data to a single receiver who then decodes the transmitted quantum information. The nonlocal resources in a bipartite code are ebits and nonlocal information qubits, and the local resources are ancillas and local information qubits. The technique of bipartite quantum error correction is useful in both the quantum communication scenario described above and in fault-tolerant quantum computation. It has application in fault-tolerant quantum computation because we can prepare nonlocal resources offline and exploit local encoding circuits. In particular, we derive an encoding circuit for a bipartite version of the Steane code that is local and additionally requires only nearest-neighbor interactions. We have simulated this encoding in the CNOT extended rectangle with a publicly available fault-tolerant simulation software. The result is that there is an improvement in the "pseudothreshold" with respect to the baseline Steane code, under the assumption that quantum memory errors occur less frequently than quantum gate errors. © 2010 Springer Science+Business Media, LLC.
One of the most promising physical properties for implementation of quantum technology is light polarization. However, since light polarization is a fragile property, the use of quantum error correction (QEC) is a crucial issue in order to make quantum information over optical network feasible. In this direction, this paper discusses optical setups for quantum error correction in quantum communication setups based on light polarization. In particular, we show how to use the QEC setup to provide error-free entanglement distribution and its use in an error-free probabilistic teleportation setup. Then, we discuss the performance of the error-correction setup in a very noisy channel. Finally, we show that the same QEC setup can also be used to correct bipartite of qubit states.
Entanglement distillation from noisy bipartite states and reliable quantum communication over noisy channels are fundamental tasks in quantum information theory. Yet existing coding schemes typically require prior knowledge of the underlying state or channel, or assume independent or weakly correlated quantum information sources. Here, using Schur-Weyl duality, we establish a universal quantum error-correction principle that overcomes these limitations. We show that the irreducible permutation spaces arising from the local Schur-Weyl decomposition of multiple copies of an unknown bipartite mixed state form maximally entangled quantum code spaces whose error structure is governed entirely by representation theory. Correcting these symmetry-resolved errors yields protocols for entanglement distillation and quantum communication that achieve coherent-information rates with exponentially vanishing error and optimal second-order corrections, even in the case of almost i.i.d. resources. Our approach also provides sharp finite-copy guarantees that are worst-case optimal at every finite block length, including in the sample-efficient regime, where the number of copies scales only polynomially with the number of qubits. Together, these results establish Schur-Weyl duality as a general mechanism for universal, robust, and optimal quantum coding, unifying entanglement distillation, quantum error correction, and quantum communication within a common representation-theoretic framework.
Abstract Encoding and decoding quantum information in a multipartite quantum system are indispensable for quantum error correction and also play crucial roles in multiparty tasks in distributed quantum information processing such as quantum secret sharing. To quantitatively characterize nonlocal properties of multipartite quantum transformations for encoding and decoding, we analyze the entanglement costs of encoding and decoding quantum information in a multipartite quantum system distributed among spatially separated parties connected by a network. This analysis generalizes previous studies of entanglement costs for preparing bipartite and multipartite quantum states and implementing bipartite quantum transformations by entanglement‐assisted local operations and classical communication (LOCC). We identify conditions for the parties being able to encode or decode quantum information in the distributed quantum system deterministically and exactly, when inter‐party quantum communication is restricted to a tree‐topology network. In our analysis, we reduce the multiparty tasks of implementing the encoding and decoding to sequential applications of one‐shot zero‐error quantum state splitting and merging for two parties. While encoding and decoding are inverse tasks of each other, our results suggest that a quantitative difference in entanglement cost between encoding and decoding arises due to the difference between quantum state merging and splitting.