Quantum Graphs as Quantum Relations
Quantum Graphs as Quantum Relations
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作为量子关系的量子图
DOI：
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发表时间：
2015
影响因子：
1.1
通讯作者：
N. Weaver
中科院分区：
文献类型：
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作者：
N. Weaver
量子纠错中出现的“非交换图”是Weaver（Quantum relations. Mem Am Math Soc 215（v-vi）：81-140，2012）。我们使用这个观点来解释Knill-Laflamme纠错条件（Knill和Laflamme在量子纠错码理论中。Phys Rev A 55：900-911，1997），以给出Stahlke的非交换图同态的内在表征（Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62：554-577，2016）和Duan、Severini和Winter的非交换二分图（Duan等人，op. Cit.通过量子信道，非交换图和量子Lovász数进行零错误通信。IEEE Trans Inf Theory 59：1164-1174，2013），并且实现与量子信道相关联的非交换混淆性图（Duan等人，op. Cit.通过量子信道，非交换图和量子Lovász数进行零错误通信。IEEE Trans Inf Theory 59：1164-1174，2013）作为对角关系的拉回。我们的框架不仅包括纯粹的经典和纯粹的量子信息理论的特殊情况下，但也出现在量子系统中的“混合”设置服从超选择规则。因此，我们能够定义非交换混淆图，给出错误校正条件，等等，这样的系统。这可能具有实用价值，因为对信息编码的超选择约束在物理上是现实的。
The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.