The causal set and Wolfram model approaches to discrete quantum gravity both permit the formulation of a manifestly covariant notion of entanglement entropy for quantum fields. In the causal set case, this is given by a construction (due to Sorkin and Johnston) of a 2-point correlation function for a Gaussian scalar field from causal set Feynman propagators and Pauli-Jordan functions, from which an eigendecomposition, and hence an entanglement entropy, can be computed. In the Wolfram model case, it is given instead in terms of the Fubini-Study metric on branchial graphs, whose tensor product structure is inherited functorially from that of finite-dimensional Hilbert spaces. In both cases, the entanglement entropies in question are most naturally defined over an extended spacetime region (hence the manifest covariance), in contrast to the generically non-covariant definitions over single spacelike hypersurfaces common to most continuum quantum field theories. In this article, we show how an axiomatic field theory for a free, massless scalar field (obeying the appropriate bosonic commutation relations) may be rigorously constructed over multiway causal graphs: a combinatorial structure sufficiently general as to encompass both causal sets and Wolfram model evolutions as special cases. We proceed to show numerically that the entanglement entropies computed using both the Sorkin-Johnston approach and the branchial graph approach are monotonically related for a large class of Wolfram model evolution rules. We also prove a special case of this monotonic relationship using a recent geometrical entanglement monotone proposed by Cocchiarella et al.

翻译：因果集理论和沃尔夫拉姆模型对离散量子引力的研究，均可为量子场构建具有显式协变性的纠缠熵概念。在因果集理论中，纠缠熵通过基于因果集费曼传播子和泡利-约当函数构建高斯标量场的两点关联函数（由Sorkin与Johnston提出）而获得，进而通过特征分解计算纠缠熵。在沃尔夫拉姆模型中，纠缠熵则基于支流图上的Fubini-Study度量，其张量积结构由有限维希尔伯特空间函子继承而来。这两种情况下的纠缠熵自然定义于扩展时空区域（因此具有显式协变性），区别于多数连续量子场论中在单个类空超曲面上定义的非协变一般形式。本文证明，满足玻色子对易关系的自由无质量标量场的公理化场论可严格构建于多路因果图之上——这一组合结构具有充分普适性，能涵盖因果集理论和沃尔夫拉姆模型演化作为特例。我们通过数值计算表明，对于一大类沃尔夫拉姆模型演化规则，采用Sorkin-Johnston方法和支流图方法计算的纠缠熵具有单调相关关系。此外，利用Cocchiarella等人近期提出的几何纠缠单调量，我们证明了该单调关系的特例情形。