We propose a deterministic method to find all holographic entropy inequalities and prove the completeness of our method. We use a triality between holographic entropy inequalities, contraction maps and partial cubes. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method. We also demonstrate interesting by-products, most notably, a procedure to generate candidate quantum entropy inequalities.

翻译：我们提出了一种确定性方法来寻找所有全息熵不等式，并证明了该方法的完备性。我们利用了全息熵不等式、收缩映射和部分立方体之间的三重对应关系。具体而言，全息熵不等式的有效性可由收缩映射的存在性导出，而我们证明了这等价于寻找收缩图的等距嵌入。因此，基于收缩映射证明方法的完备性，寻找所有全息熵不等式的问题等价于寻找所有收缩映射的问题，我们将此转化为寻找所有像图部分立方体的问题。我们给出了该问题的算法解，并刻画了方法的复杂度。我们还展示了若干有趣的副产品，其中最显著的是生成候选量子熵不等式的流程。