Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable – in contrast to the well-known undecidability for equality of PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.

The Becalm project is an open and low-cost solution for the remote monitoring of respiratory support therapies like the ones used in COVID-19 patients. Becalm combines a decision-making system based on Case-Based Reasoning with a low-cost, non-invasive mask that enables the remote monitoring, detection, and explanation of risk situations for respiratory patients. This paper first describes the mask and the sensors that allow remote monitoring. Then, it describes the intelligent decision-making system that detects anomalies and raises early warnings. This detection is based on the comparison of cases that represent patients using a set of static variables plus the dynamic vector of the patient time series from sensors. Finally, personalized visual reports are created to explain the causes of the warning, data patterns, and patient context to the healthcare professional. To evaluate the case-based early-warning system, we use a synthetic data generator that simulates patients' clinical evolution from the physiological features and factors described in healthcare literature. This generation process has been verified with a real dataset and allows the validation of the reasoning system with noisy and incomplete data, threshold values, and life/death situations. The evaluation demonstrates promising results and good accuracy (0.91) for the proposed low-cost solution to monitor respiratory patients.