This work introduces the Tensor Diagrammatic Calculus (TDC), a graphical-mathematical language for representing multi-scale tensorial structures through networks of contractions, transformations, and emergence operators. Building on Penrose’s graphical tensor notation, TDC defines four primitive entities: Glyphons (elementary tensorial units), Morphons (transformation operators), Symbions (emergent relational structures), and Indices (connection channels). The framework provides a formal correspondence between algebraic tensor expressions and diagrammatic networks, and introduces two operators—emergence (E) and immersion (I)—to represent transitions between microscopic and macroscopic levels of description. We show that every TDC diagram admits an algebraic translation and that contraction rules preserve index compatibility. The formalism is compatible with existing tensor network notation and category-theoretic approaches to physical processes. Workedexamples from quantum mechanics, statistical mechanics, and classical field theory are provided. TDC is proposed as a representational tool for symbolic physics, not as a replacement for existing formalisms.
This work introduces the Tensor Diagrammatic Calculus (TDC), a graphical-mathematical language for representing multi-scale tensorial structures through networks of contractions, transformations, and emergence operators. Building on Penrose’s graphical tensor notation, TDC defines four primitive entities: Glyphons (elementary tensorial units), Morphons (transformation operators), Symbions (emergent relational structures), and Indices (connection channels). The framework provides a formal correspondence between algebraic tensor expressions and diagrammatic networks, and introduces two operators—emergence (E) and immersion (I)—to represent transitions between microscopic and macroscopic levels of description. We show that every TDC diagram admits an algebraic translation and that contraction rules preserve index compatibility. The formalism is compatible with existing tensor network notation and category-theoretic approaches to physical processes. Workedexamples from quantum mechanics, statistical mechanics, and classical field theory are provided. TDC is proposed as a representational tool for symbolic physics, not as a replacement for existing formalisms.
The development of Penrose tensor diagrams has transformed the representation of mathematical structures in contemporary physics, providing a graphical language that unites algebraic abstraction with geometric intuition. Over the past decades, these diagrams have become essential tools in quantum physics, field theory, and spacetime geometry, allowing compact descriptions of tensor contractions and correlations. Authors such as Biamonte, Bergholm, Kissinger, and Coecke have extended this formalism to computational and categorical contexts, establishing a framework of graphical calculus that facilitates the visualization and manipulation of complex physical systems. Within this context, Martín Pomares Calero introduces the concept of Symbolic Physics, an epistemological and ontological approach that interprets Penrose diagrams as symbolic structures of the universe. From this perspective, the symbol is not a mere representational sign but an active component of physical and cognitive processes, capable of encoding the fundamental laws of reality. This approach proposes a synthesis between tensor calculus, scientific semiotics, and visual cognition, oriented toward a physics of meaning. This paper examines the main research lines connecting tensorial thought and symbolic interpretation, emphasizing how the integration between mathematical and symbolic frameworks opens new paths for the theoretical understanding of the universe.
High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional generalizations of matrices. While tensors provide a natural representation for multi-modal structure, their direct manipulation quickly becomes challenging as the order grows: the number of parameters increases exponentially, and algebraic expressions involving many indices become difficult to interpret and implement. Tensor networks (TNs) provide an effective framework for addressing these challenges. Originally introduced by Penrose and developed extensively in quantum physics, the graphical language of tensor networks encodes contractions as edges in a graph, reducing notational overhead and revealing structural properties obscured by index notation. Despite the central role of high-dimensional tensors in modern machine learning and numerical analysis, tensor network diagrams remain underutilized outside quantum computing, partly due to the lack of a self-contained mathematical reference accessible to a broad technical audience. This manuscript provides a self-contained guide to tensor networks and their use in tensor algebra. We present the main operations on tensors, contractions, products, and reshaping through, graphical notation, and show how classical tensor decompositions and related computations are naturally expressed in this framework. We also illustrate how tensor networks simplify the derivation of gradients and the manipulation of high-dimensional probability distributions. Throughout, we show that the diagrammatic approach yields genuinely shorter and more transparent proofs of classical identities, rank bounds, and gradient formulas that would otherwise require laborious index manipulation.
From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalyzed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation, but cannot generally be computed with directly. In this paper we introduce the ZXH-calculus, a graphical language based on the ZX-calculus, that we use to represent and reason about many-body states entirely graphically. As a demonstration, we express the one-dimensional (1D) AKLT-state, a symmetry protected topological state, in the ZXH-calculus by developing a representation of spins higher than 1/2 within the calculus. By exploiting the simplifying power of the ZXH-calculus rules we show how this representation straightforwardly recovers the AKLT matrix-product state representation, the existence of topologically protected edge states, and the nonvanishing of a string-order parameter. Extending beyond these known properties, our diagrammatic approach also allows us to analytically derive that the Berry phase of any finite-length 1D AKLT chain is . In addition, we provide an alternative proof that the two-dimensional (2D) AKLT-state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum-computing resource. Lastly, we build 2D higher-order topological phases diagrammatically, which we use to illustrate a symmetry-breaking phase transition. Our results show that the ZXH-calculus is a powerful language for representing and computing with physical states entirely graphically, paving the way to develop more efficient many-body algorithms and giving a novel diagrammatic perspective on quantum phase transitions.
We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.
We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to: permutational quantum computing, quantum machine learning, condensed matter physics, and quantum gravity. Respectively, we analyse permutational computing transition amplitudes, evaluate barren plateaus for SU(2) symmetric ansätze, study properties of AKLT states, and derive the minimum quantised volume in loop quantum gravity.
Our starting point is the mixed-dimensional ZX calculus, a purely diagrammatic language that has been proven to be complete for finite-dimensional Hilbert spaces. That is, any equation that can be derived in the Hilbert space formalism, can also be derived in the mixed-dimensional ZX calculus. We embed the Spin-ZX calculus inside the mixed-dimensional ZX calculus, rendering it a quantum information flavoured diagrammatic language for the quantum theory of angular momentum, i.e. SU(2) representation theory. We diagrammatically derive the fundamental spin coupling objects - such as Clebsch-Gordan coefficients, symmetrising mappings between qubits and spin spaces, and spin Hamiltonians - under this embedding.
Our results establish the Spin-ZX calculus as a powerful tool for representing and computing with SU(2) systems graphically, offering new insights into foundational relationships and paving the way for new diagrammatic algorithms for theoretical physics.
The content of this paper can be roughly organized into a three-level hierarchy of generality. At the first, most general level, we introduce a new language which allows us to express various categorical structures in a systematic and explicit manner in terms of so-called 2-schemes. Although 2-schemes can formalize categorical structures such as symmetric monoidal categories, they are not limited to this, and can be used to define structures with no categorical analogue. Most categorical structures come with an effective graphical calculus such as string diagrams for symmetric monoidal categories, and the same is true more generally for interesting 2-schemes. In this work, we focus on one particular non-categorical 2-scheme, whose instances we refer to as tensor types. At the second level of the hierarchy, we work out different flavors of this 2-scheme in detail. The effective graphical calculus of tensor types is that of tensor networks or Penrose diagrams, that is, string diagrams without a flow of time. As such, tensor types are similar to compact closed categories, though there are various small but potentially important differences. Also, the two definitions use completely different mechanisms despite both being examples of 2-schemes. At the third level of the hierarchy, we provide a long list of different families of concrete tensor types, in a way which makes them accessible to concrete computations, motivated by their potential use in physics. Different tensor types describe different types of physical models, such as classical or quantum physics, deterministic or statistical physics, many-body or single-body physics, or matter with or without symmetries or fermions.