Metrisable assessment of the course of stream-systemic processes in vector form in industry 4.0

The goal of this paper is to present an innovative conception how to use metrisable vector structure of a manufacturing process, based on quantitative relations between the activity of input streams, features of the product, and effect of losses; all of which are excellent practical solution for Industry 4.0, and in turn intelligent factories. This solution can be a usefull way in the process of building sustainable organization. A vector representation of manufacturing processes was formulated, one which is based in system engineering. Three manufacturing system state vectors were proposed. These are: input stream vector $${\upphi }$$ ϕ , product features vector $$\overrightarrow {{\text{ P}}}$$ P → which is also referred to as quality vector, and losses vector $$\overrightarrow {{\text{ S}}}$$ S → . Scalar, vector, and mixed products of these vectors may form constitutive equations of manufacturing processes. The relations between the vectors $$\upphi$$ ϕ , $$\overrightarrow {{\text{ P}}}$$ P → ,$$\overrightarrow {{\text{ S}}}$$ S → provide a possibility for a metrisable, complex analysis and assessment of a contemporary manufacturing process. The paper shows practical methods for defining the size of the vector values within the process. The demonstrated vector description of stream-systemic processes can also be applied to non-material manufacturing.

Graph Approach to Quantum Teleportation Dynamics

Quantum teleportation plays a key role in modern quantum technologies. Thus, it is of much interest to generate alternative approaches or representations that are aimed at allowing us a better understanding of the physics involved in the process from different perspectives. With this purpose, here an approach based on graph theory is introduced and discussed in the context of some applications. Its main goal is to provide a fully symbolic framework for quantum teleportation from a dynamical viewpoint, which makes explicit at each stage of the process how entanglement and information swap among the qubits involved in it. In order to construct this dynamical perspective, it has been necessary to define some auxiliary elements, namely virtual nodes and edges, as well as an additional notation for nodes describing potential states (against nodes accounting for actual states). With these elements, not only the flow of the process can be followed step by step, but they also allow us to establish a direct correspondence between this graph-based approach and the usual state vector description. To show the suitability and versatility of this graph-based approach, several particular teleportation examples are examined in detail, which include bipartite, tripartite, and tetrapartite maximally entangled states as quantum channels. From the analysis of these cases, a general protocol is devised to describe the sharing of quantum information in presence of maximally entangled multi-qubit system.

History operators in quantum mechanics

It is convenient to describe a quantum system at all times by means of a “history operator” [Formula: see text], encoding measurements and unitary time evolution between measurements. These operators naturally arise when computing the probability of measurement sequences, and generalize the “sum over position histories” of the Feynman path-integral. As we argue in this work, this description has some computational advantages over the usual state vector description, and may help to clarify some issues regarding nonlocality of quantum correlations and collapse. A measurement on a system described by [Formula: see text] modifies the history operator, [Formula: see text], where [Formula: see text] is the projector corresponding to the measurement. We refer to this modification as “history operator collapse”. Thus, [Formula: see text] keeps track of the succession of measurements on a system, and contains all histories compatible with the results of these measurements. The collapse modifies the history content of [Formula: see text], and therefore modifies also the past (relative to the measurement), but never in a way to violate causality. Probabilities of outcomes are obtained as [Formula: see text]. A similar formula yields probabilities for intermediate measurements, and reproduces the result of the two-vector formalism in the case of the given initial and final states. We apply the history operator formalism to a few examples: entangler circuit, Mach–Zehnder interferometer, teleportation circuit and three-box experiment. Not surprisingly, the propagation of coordinate eigenstates [Formula: see text] is described by a history operator [Formula: see text] containing the Feynman path-integral.