Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.

中文翻译：

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求最大熵相容量子态的复杂性

本文中，我们研究了出于物理观察的局限性而从一组兼容的边际中恢复密度算符的问题。鉴于兼容密度算子的集合不是奇异的，我们采用Jaynes原理并希望以最大熵来表征兼容密度算子。我们首先表明，即使对于最简单的3链情况，对于量子计算复杂度类QSZK，比较兼容密度算符的熵也是完整的。然后，我们关注量子马尔可夫链和树的特殊情况，并确定对于这些情况，在子系统数目中存在一个构造最大熵兼容密度算子的过程多项式。此外，我们将Chow-Liu算法扩展到相同的量子态子类。