  Finite-dimensional quantum theory serves as the theoretical foundation for
quantum information and computation. Mathematically, it is formalized in the
category FHilb, comprising all finite-dimensional Hilbert spaces and linear
maps between them. However, there has not been a graphical language for FHilb
which is both universal and complete and thus incorporates a set of rules rich
enough to derive any equality of the underlying formalism solely by rewriting.
In this paper, we introduce the qufinite ZXW calculus - a graphical language
for reasoning about finite-dimensional quantum theory. We set up a unique
normal form to represent an arbitrary tensor and prove the completeness of this
calculus by demonstrating that any qufinite ZXW diagram can be rewritten into
its normal form. This result implies the equivalence of the qufinite ZXW
calculus and the category FHilb, leading to a purely diagrammatic framework for
finite-dimensional quantum theory with the same reasoning power. In addition,
we identify several domains where the application of the qufinite ZXW calculus
holds promise. These domains include spin networks, interacting
mixed-dimensional systems in quantum chemistry, quantum programming, high-level
description of quantum algorithms, and mixed-dimensional quantum computing. Our
work paves the way for a comprehensive diagrammatic description of quantum
physics, opening the doors of this area to the wider public.
