Completeness of deep NN-QM representations for reflection-positive processes

Prove or disprove that every reflection-positive stochastic process admits a representation as a deep neural network quantum mechanics (deep NN-QM) in which the input processes y_t^{(i)} are symmetric Markov processes. This seeks to establish whether the class of reflection-positive processes is the closure of symmetric Markov processes under linear combinations and applications of random functions implemented by neural network architectures.

Background

The paper develops a framework in which Euclidean-time quantum mechanical systems are represented via neural networks acting on stochastic processes, with reflection positivity (RP) ensuring unitarity. It proves that neural networks preserve RP when acting on RP or Markov processes and introduces deep NN-QM by stacking such transformations.

Building on these results, the authors propose a universality-style conjecture: that any RP process can be obtained from deep NN-QM constructions starting from symmetric Markov inputs. If true, this would characterize RP processes as the closure of symmetric Markov processes under linear combinations and random neural transformations, offering an alternative definition of reflection positivity via deep neural architectures.