Continuity of the Maximum-Entropy Inference
Continuity of the Maximum-Entropy Inference
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DOI：
10.1007/s00220-014-2090-1
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发表时间：
2014-06
影响因子：
2.4
通讯作者：
W. Stephan
中科院分区：
文献类型：
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作者：
W. Stephan
本文研究了利用连续排序函数的最大化，从一组固定的可观测值的期望值推断有限能级量子系统状态的反问题。我们已经证明了最大熵推理可以是期望值凸集到状态凸集的不连续映射，因为图像中包含了支持度降低的状态，而这个映射限制为完全支持状态的吉本氏族的光滑参数化。本文证明了对任意排序函数的推理是连续的，直至边界点。这是从状态到期望值的受限线性映射的开放性的连续性条件中得出的。开放条件还表明，具有不连续推理的排序函数是典型的。进一步证明了该推理在任何多面体的限制下是连续的，这意味着不连续属于非交换可观测的量子域，吉本族的测地闭包等于最大熵态的集合。我们讨论了最大熵态集的八种描述，并给出了精度证明和偏差分析。
We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non-commutative observables and that a geodesic closure of a Gibbsian family equals the set of maximum-entropy states. We discuss eight descriptions of the set of maximum-entropy states with proofs of accuracy and an analysis of deviations.